modeling and analysis of cooperative and large-scale

Modeling and Analysis of Cooperative andLarge-scale Molecular Communication Systems

Yuting Fang

April 2020

A Thesis Submitted for the Degree of

Doctor of Philosophy

of The Australian National University

Research School of Electrical,

Energy and Materials Engineering

The Australian National University

c© Copyright by Yuting Fang 2020

Declaration

The contents of this thesis are the results of original research and have not been submitted for

a higher degree to any other university or institution.

Much of the work in this thesis has been published or has been submitted for publication as

journal articles or conference proceedings.

The research work presented in this thesis has been performed jointly with Dr. Nan Yang (The

Australian National University), Dr. Adam Noel (University of Warwick), Prof. Weisi Guo

(University of Warwick), Prof. Andrew Eckford (York University), Prof. Rodney A. Kennedy

(The Australian National University), and Dr. Matteo Icardi (University of Nottingham). The

substantial majority of this work was my own.

Yuting Fang

Research School of Electrical,

Energy and Materials Engineering,

The Australian National University,

Canberra, ACT 2601,

AUSTRALIA

iii

Acknowledgments

I would like to first express my heartfelt gratitude to my primary supervisor Dr. Nan Yang and

external co-supervisor Dr. Adam Noel (University of Warwick), for their constant guidance,

support, and encouragement throughout my PhD study. Their rigor and patience impact on my

PhD study and future career. I feel more than fortunate to be their student.

They are gratefully acknowledged below:

• During my PhD study, Nan provides caring supervision, numerous discussions, patient

guidance for my research and paper writing. Besides that, Nan always trains me to be an

independent researcher by providing me chances to funding proposal writing and student

supervision. Nan also uses his own networking to expand my research connections,

which makes me establish close collaborations with internationally leading researchers

in molecular communications.

• Adam’s patience never stops to impress me. Since we are geographically apart, we

usually correspond with each other via emails. Although writing response in the text is

very time-consuming, Adam always replies to my email timely with a very detailed and

thorough response. These responses guide me to successfully finish all research work

and paper writing during my PhD. I also really appreciate the time we spent together

when I visited the University of Warwick.

I would like to sincerely appreciate Prof. Rodney A. Kennedy, a highly-esteemed re-

searcher, for his insightful technical judgments on our coauthored papers and his time spent

on evaluating my annual progress. I am also very grateful to Dr. Sean Zhou and Dr. Salman

Durrani for their useful suggestions on my PhD study, teaching, and future career during our

casual chat.

I would like to sincerely thank Prof. Weisi Guo at the University of Warwick, Prof. An-

drew Eckford at York University, and Prof. Dimitrios Makrakis at the University of Ottawa for

their kind invitation to visiting their research group and numerous discussion during our col-

laborations. Special thanks to Prof. Weisi Guo and Prof. Andrew Eckford for their generous

financial support for my overseas trips to the UK and Canada, respectively. I also would like to

sincerely thank Dr. Matteo Icardi at the University of Nottingham for his expert comments on

fluid mechanics. Collaborating with Prof. Weisi Guo, Prof. Andrew Eckford, Prof. Dimitrios

Makrakis, and Dr. Matteo Icardi has stimulated many fancy ideas and broadened my research

horizon.

v

I would like to thank my colleagues at ANU for creating a warm and relaxing study envi-

ronment. Thank you to my colleagues both past and present for their help for my study and

life, especially, Prof. Parastoo Sadeghi, Prof. Thushara D. Abhayapala, Dr. Jihui Zhang, Dr.

Zhuo Sun, Dr. Noman Akbar, Dr. Xiaofang Sun, Dr. Jianwei Hu, Dr. Zohair Abu-Shaban,

Dr. Alice Bates, Dr. Ni Ding, Dr. Hanchi Chen, Dr. Shihao Yan, Dr. Biao He, Dr. Wen

Zhang, Dr. Wanchun Liu, Dr. Mingchao Yu, Dr. Nicole Sawyer, Dr. Abbas Koohian, Dr.

Usama Elahi, Chunhui Li, Simin Xu, Yiran Wang, Hang Yuan, Xinyu Huang, Yucheng Liu,

Haoran Jiang, Fei Ma, Khurram Shahzad, Sheeraz Alvi, Sahar Idrees, Huiyan Sun, and Yanyan

Wang. I appreciate my officemate, Xiaohui Zhou for making a friendly and encouraging office

atmosphere. I also would like to thank my friends at the University of Warwick, especially,

Mahmoud Abbaszadeh, Dr. Baohua Shao, and Hua Yan for their help during my visit in the

UK.

Special thanks to the Australian National University for providing me the PhD scholarship,

2017 ANU Dean’s Travel Grant, 2018 Vice-Chancellor HDR Travel Grant to support my study

in ANU, conference attendance, overseas academic visits. I am also grateful to the Chinese

government for providing stipends during my PhD.

I would like to express the deepest gratitude to my parents Tianyong Fang and Li Tan for

their unconditional love, support, and encouragement since I was born. Special thanks to my

husband Xiaobo Wu for his continuous support for my PhD study and his frequent travel to

Canberra and accompany me. This thesis is dedicated to my family.

Abstract

Molecular communications (MC) is the use of molecules as carriers of information between

devices. In MC, there are several main research challenges: 1) Low reliability of diffusion-

based MC systems, 2) optimal MC system design, 3) understanding cooperation among the

microscopic population with noisy signaling, and 4) realistic molecular information propaga-

tion environments. To deal with these challenges in MC, this thesis focuses on the four main

issues: 1) How to improve reliability of diffusion-based MC systems, 2) how to design prac-

tical suboptimal cooperative MC systems, 3) what the impact of noisy molecular signaling

on the bacterial cooperation behavior is, and 4) how the communication performance changes

when molecules transport in a realistic environment.

First, we study cooperative detection among multiple distributed receivers (RXs) in a

diffusion-based MC system. Unlike most existing studies that consider one-phase noisy trans-

mission or one-symbol transmission for simplicity, we consider multiple-symbol transmission

and two-phase noisy transmission from a transmitter (TX) to a fusion center (FC) via multiple

RXs. The FC uses hard fusion rules to arrive at a final decision. We derive the system error

probability. We formulate the suboptimal convex optimization problems to determine the op-

timal decision thresholds. We show that the system error performance is greatly improved by

combining the detection information of distributed RXs.

Second, we propose symbol-by-symbol maximum likelihood (ML) detection for a coop-

erative diffusion-based MC system. Different from the first work, the FC uses the likelihood

of its observations from all RXs to make a decision on the transmitted symbol in each inter-

val. We propose three ML detection variants according to different RX behaviors and different

knowledge at the FC. We derive the system error probabilities for two ML detector variants.

We also optimize the molecule allocation among RXs for one variant. We show that simpler

and non-ML cooperative variants studied in the first work have error performance comparable

to ML detectors.

Third, we present an analytically tractable model for predicting the statistics of the number

of cooperative microorganisms. Unlike prior studies that considered abstract signal propaga-

tion channels among microorganisms, we use diffusion-reaction equations to accurately char-

acterize signal received at each microorganism due to independent diffusion and degradation of

molecules. Microorganisms are randomly distributed in a two-dimensional (2D) environment

where each one continuously releases molecules at random times. We derive the 2D channel

response due to one bacterium or randomly-distributed bacteria. We then derive the expected

probability of cooperation at the bacterium. We finally derive the moment generating function

vii

viii

and different statistics for the number of cooperators. Our model can be used to predict the

impact of noisy signaling, e.g., diffusion coefficient and reaction rate, on the statistics of the

number of responsive cooperators in QS.

Last, we investigate the communication through realistic porous channels for the first time

via statistical breakthrough curves. Assuming that the number of arrived molecules can be

approximated as a Gaussian random variable and using fully resolved computational fluid dy-

namics results for the breakthrough curves, we present the numerical results for the throughput,

mutual information, error probability, and information diversity gain. We reveal the unique

characteristics of the porous medium channel.

This thesis serves an unprecedented way to enable 1) high-accuracy disease detection and

health monitoring and 2) bacterial infection prevention and new environmental remediation. It

also provides useful insights for designing the optimal MC systems through porous media and

the optimal cooperative MC systems.

List of Publications

The work in this thesis has been published or has been submitted for publication as journal

articles or conference papers. These papers are:

Journal Articles

J1. Y. Fang, A. Noel, N. Yang, A. W. Eckford, and R. A. Kennedy, “Convex optimization

of distributed cooperative detection for multi-receiver molecular communication,” IEEE

Trans. Mol., Bio. Multi-Scale Commun., vol. 3, no. 3, pp. 166–182, Sep. 2017.

J2. Y. Fang, A. Noel, N. Yang, A. W. Eckford, and R. A. Kennedy, “Symbol-by-symbol

maximum likelihood detection for cooperative molecular communication,” IEEE Trans.

Commun., vol. 67, no. 7, pp. 4885–4899, Jul. 2019.

J3. Y. Fang, W. Guo, M. Icardi, A. Noel, and N. Yang, “Molecular information delivery in

porous media,” IEEE Trans. Mol., Bio. Multi-Scale Commun., vol. 4, no. 4, pp. 257–

262, Dec. 2018.

J4. Y. Fang, A. Noel, A. Eckford, N. Yang, and Jing Guo, “Characterization of cooperators in

quorum sensing with 2D molecular signal analysis,” submitted to IEEE Trans. Commun.

Conference Papers

C1. Y. Fang, A. Noel, A. Eckford, and N. Yang, “Expected density of cooperative bacteria in

a 2D quorum sensing based molecular communication system,” in Proc. IEEE GLOBE-

COM 2019, Waikoloa, HI, Dec. 2019, pp. 1–6.

C2. Y. Fang, A. Noel, N. Yang, A. Eckford, and R. A. Kennedy, “Maximum likelihood de-

tection for collaborative molecular communication,” in Proc. IEEE ICC 2018, Kansas

City, MO, May 2018, pp. 1–7.

C3. Y. Fang, A. Noel, Y. Wang, and N. Yang, “Simplified cooperative detection for multi-

receiver molecular communication,” in Proc. IEEE ITW 2017, Kaohsiung, Taiwan

(ROC), Nov. 2017, pp. 1–5.

C4. Y. Fang, A. Noel, N. Yang, A. Eckford, and R. A. Kennedy, “Distributed cooperative

detection for multi-receiver molecular communication,” in Proc. IEEE GLOBECOM

2016, Washington, DC, Dec. 2016, pp. 1–7.

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The papers J1-J4 and C1 are used in the thesis. The other conference papers are related to

the thesis but not explicitly used in this thesis. The following publications are also the results

from my PhD study but not related to this thesis:

Journal Articles

J5. A. Noel, Y. Fang, N. Yang, D. Makrakis, and A. W. Eckford, “Using game theory for real-

time behavioral dynamics in microscopic populations with noisy signaling,” submitted

to Journal of the Royal Society Interface.

J6. X. Huang, Y. Fang, N. Yang, and A. Noel, “Channel characterization for 1D molecular

communication with two absorbing receivers,” IEEE Commun. Lett., accepted to appear.

Conference Papers

C5. A. Noel, Y. Fang, N. Yang, D. Makrakis, and A. Eckford, “Effect of local signaling

reliability on cooperation in bacteria,” in Proc. IEEE ITW 2017, Kaohsiung, Taiwan

(ROC), Nov. 2017, pp. 1–5.

Acronyms

MC molecular communications

QS quorum sensing

TX transmitter

RX receiver

FC fusion center

RV random variable

PDF probability density function

PMF probability mass function

CDF cumulative distribution function

CCDF complementary CDF

ML maximum likelihood

FS free space

PM porous medium

PSD positive semidefinite

PPP Poisson point process

PGFL probability generating functional

ISI inter-symbol interference

1D one-dimensional

2D two-dimensional

3D three-dimensional

DF decode-and-forward

AF amplify-and-forward

xi

MD-ML DF with multi-molecule-type and ML detection at the FC

SD-ML DF with single-molecule-type and ML detection at the FC

SA-ML AF with single-molecule-type and ML detection at the FC

MGF moment generating function

CGF cumulant generating function

MDP missed detection probability

FAP false alarm probability

i.i.d. independent and identically distributed

UCA uniform concentration assumption

Pe Péclet number

Notations

Pr(·) probability

Pr(·|·) conditional probability

∈ is an element of

dxe smallest integer greater than or equal to x

bxc greatest integer smaller than or equal to x

bxe nearest integer to x

| · | Euclidean norm

erf(·) error function

Γ(a,x) incomplete Gamma function Γ(a,x) =∫

∞

x ta−1 exp(−t)dt

Γ(a) Gamma function Γ(a) = Γ(a,x)|x=0

min· minimum value of a totally ordered set

R set of real numbers

Rn n dimensional Euclidean space

∇ nabla operator

0 PSD

Φ point process

⊂ is a subset of

E· expectation

Ex· expectation with respect to x

Var· variance

exp(·) exponential function

log (·) natural logarithm

xiii

|A| cardinality of a set A

Kn(·) modified nth order Bessel function of the second kind

MK(t) MGF of a RV K

KK(t) CGF of a RV K

, defined as

∏ product

∑ summation

Contents

Declaration iii

Acknowledgments v

Abstract vii

List of Publications ix

Acronyms xi

Notations xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Introduction of Molecular Communications . . . . . . . . . . . . . . . 1

1.1.2 Research Challenges for MC . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2.1 Low Reliability . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2.2 Performance Limit . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2.3 Predict Node Behavior . . . . . . . . . . . . . . . . . . . . . 5

1.1.2.4 Realistic Propagation Environments . . . . . . . . . . . . . . 6

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Distribution of Received Molecular Signals . . . . . . . . . . . . . . . 6

1.2.2 Convex Optimization for MC Systems . . . . . . . . . . . . . . . . . . 8

1.2.3 Randomly-Distributed MC Systems . . . . . . . . . . . . . . . . . . . 9

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Cooperative MC Systems . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 ML detection for MC . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Bacterial Behavioral Analysis . . . . . . . . . . . . . . . . . . . . . . 12

1.3.4 Realistic Propagation Environments . . . . . . . . . . . . . . . . . . . 13

1.3.5 Limitation of Existing Studies . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 15

xv

xvi Contents

2 Convex Optimization of Cooperative MC Systems 212.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Error Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Perfect Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1.1 TX−RXk Link . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1.2 Global Error Probability . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Noisy Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2.1 TX−RXk−FC Link . . . . . . . . . . . . . . . . . . . . . 27

2.2.2.2 Global Error Probability . . . . . . . . . . . . . . . . . . . . 29

2.3 Error Performance Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 30


2.3.2 Noisy Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.2.1 Optimal ξRX . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.2.2 Joint Optimal ξRX and ξFC . . . . . . . . . . . . . . . . . . . 36

2.3.3 Average Error Performance Optimization . . . . . . . . . . . . . . . . 39

2.4 Numerical Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . 40


2.4.2 Noisy Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Symbol-by-Symbol ML Detection for Cooperative MC 493.1 System Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 ML Detection Design and Derivation . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.1 MD-ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 SD-ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 SA-ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.4 Comparison of Complexity . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Error Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 SD-ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2 SA-ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Error Performance Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 60


3.5.1 Symmetric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.2 Asymmetric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Contents xvii

4 Characterization of Cooperators in QS with 2D Molecular Signal Analysis 714.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 2D Channel Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.1 One Point TX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.2 Randomly Distributed TXs . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Cooperating Probability at a Fixed-Located Bacterium . . . . . . . . . . . . . 80

4.3.1 Exact Cooperating Probability . . . . . . . . . . . . . . . . . . . . . . 80

4.3.2 Approximate Cooperating Probability . . . . . . . . . . . . . . . . . . 82

4.4 Characterization of Number of Cooperative Bacteria . . . . . . . . . . . . . . 83

4.4.1 Moment and Cumulant Generating Functions . . . . . . . . . . . . . . 84

4.4.2 Moments and Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.3 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.4 Pairs of Two Nearest Nodes Both Cooperating . . . . . . . . . . . . . 89


4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Molecular Information Delivery in Porous Media 995.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.1 Channel Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Conclusions 1096.1 Thesis Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.1 Theoretical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.2 Validation of Theoretical Work . . . . . . . . . . . . . . . . . . . . . . 113

A Appendix A 115A.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.3 Proofs of Theorem 2.3 and Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . 116

B Appendix B 119B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

B.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.3 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xviii Contents

B.4 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

B.5 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

B.6 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

B.7 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C Appendix C 127C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128





C.7 Proof of Remark 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

C.8 Proof of Remark 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

D Appendix D 133D.1 Derivation of Performance metrics . . . . . . . . . . . . . . . . . . . . . . . . 133

D.2 Proof of Corollary 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

D.3 Proof of Corollary 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography 139

List of Figures

1.1 Illustration of MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 An example of bacteria coordinating their behavior via QS where grey and

blue circles denote noncooperative and cooperative bacteria, respectively. The

core of QS: 1) Each bacterium includes a synthase that emits the signaling

molecules and receptors that can bind with the molecules. 2) The molecules

diffuse into and out of the bacteria. 3) If the number of molecules that are

bound exceeds a threshold, a receptor is activated to regulate target genes for a

cooperative response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 An example of a cooperative MC system with K = 5, where the transmission

from the TX to the RXs is represented by solid arrows and the decision report-

ing from the RXs to the FC is represented by dashed arrows. . . . . . . . . . . 22

2.2 Average global error probability QFC of different fusion rules versus the deci-

sion threshold at RXs ξRX with K=3 in the perfect reporting scenario. . . . . . 42

2.3 Optimal average global error probability Q∗FC of different fusion rules versus

the number of cooperative RXs K in the perfect reporting scenario. . . . . . . . 43

2.4 Average global error probability QFC of different fusion rules versus the deci-

sion threshold at RXs ξRX with K = 3 in the noisy reporting scenario. . . . . . . 45

2.5 Expected average global error probability QFC versus the decision threshold

at RXs ξRX and the decision threshold at the FC ξFC with K = 3 in the noisy

reporting scenario for (a) OR rule, (b) AND rule, and (c) majority rule. In

(a)–(c), ‘’ is the optimal QFC achieved by ξ ∗RX and ξ ∗FC, obtained by exhaustive

search, and ‘’ is the approximated QFC achieved by ξ RX and ξ FC. . . . . . . . 46

2.6 Optimal average global error probability QFC of different fusion rules versus

the radius of the FC rFC with K = 3 in the noisy reporting scenario. . . . . . . . 47

xix

xx LIST OF FIGURES

3.1 An example of a cooperative MC system with 2 RXs. The transmission from

the TX to the RXs is represented by black dashed arrows. “D” and “A” denotes

the RXs making decisions and amplifying observations, respectively, and Ak

denotes the type of released molecule. The transmission from the RXs to the

FC in MD-ML, SD-ML, and SA-ML are represented by red, blue, and green

arrows, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 An example of the timing schedule for the system with MRX = 5 and MFC = 5. . 51

3.3 Optimal average error probability Q∗FC versus the number of samples by FC

MFC for (a) SD-ML and SA-ML, (b) MD-ML and the majority rule, (c) SD-

ML and SD-Constant, and (d) SA-ML and SA-Constant. The analytical error

performance of the majority rule and SD-Constant is presented in [1] and [2],

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Optimal average error probability Q∗FC of different variants versus the number

of RXs K. The analytical error performance of SD-Constant and the majority

rule is presented in [2] and [1], respectively. . . . . . . . . . . . . . . . . . . . 67

3.5 Optimal average error probability Q∗FC of different variants versus the distance

dTX3 between the TX and RX3. RX1 and RX2 are fixed at (2µm,0,0.6µm) and

(2µm,0,−0.6µm), respectively. The locations of RX3 are (1) (2µm,0.6µm,0),

(2) (1.6µm,0.48µm,0), (3) (1.2µm,0.36µm,0), (4) (0.8µm,0.24µm,0), (5)

(0.4µm,0.12µm,0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6 Error probability QFC[ j] of SD-ML versus the number of molecules released by

RX1, S1, for different locations of RX1: (a) (2µm,0.6µm,0µm), (b) (1.5µm,0.45µm,0µm),

(c) (1µm,0.3µm,0µm), (d) (0.5µm,0.15µm,0µm). The location of RX2 is

fixed at (2µm,−0.6µm,0µm). . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.7 Error probability QFC[ j] of SD-ML versus the number of molecules released

by RX1, S1, the number of molecules released by RX2, S2, and the number

of molecules released by RX3, S3. The X-axis, Y-axis, and Z-axis coordi-

nates of ‘’ are the solutions to problem (3.26). RX1, RX2, and RX3 are at

(1.915µm,0.58µm,0), (1.827µm,0.579µm,0), and (1.265µm,0.328µm,0),

respectively. The x-axis and y-axis coordinates of the locations of the RXs are

randomly generated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1 A population of bacteria randomly distributed on a circle according to a 2D

spatial point process, where each bacterium acts as a point TX and as a circular

passive RX. The molecules diffuse into and out of the bacteria. . . . . . . . . . 72

4.2 An example of release times due to continuous emission of molecules at a

bacterium according to a random process. . . . . . . . . . . . . . . . . . . . . 73

LIST OF FIGURES xxi

4.3 The time-varying expected number of molecules observed N†agg (~xi, t|λ ) versus

time t. R1 = 20 µm, λ = 7.9×10−2/µm2, and~xi = (10 µm,10 µm). For other

simulation details, please see Sec. 3.5. . . . . . . . . . . . . . . . . . . . . . . 73

4.4 The expected number of molecules observed at the RX N(~b, t)

versus time

due to the emission of one TX located at (0,0). In Fig. 4.4(a), we consider one

impulse emission with 105 molecules and molecular degradation is considered.

We consider three cases of the RX in Fig. 4.4(a): Case a) the circular RX lo-

cated at (0,5 µm), Case b) the square RX located at (0,5 µm), and Case c) the

circular RX located at (0,0). In Fig. 4.4(b), we consider continuous emission

and the circular RX is considered. We also consider three cases of the RX in

Fig. 4.4(b): Case d) the RX located at (0,5 µm) with molecular degradation,

Case e) the RX located at (0,5 µm) without molecular degradation, and Case

f) the RX located at (0,0) with molecular degradation. . . . . . . . . . . . . . 91

4.5 The expected number of molecules observed at the RX, E

Nagg

(~b|λ

), in

Fig. 4.5(a) and the corresponding cooperating probability at the RX, Pr(N†

agg(~xi|λ ) ≥ η),

in Fig. 4.5(b) due to continuous emission at randomly-distributed TXs. For dif-

ferent environmental radii R1 = 50 µm, R1 = 100 µm, and R1 = 150 µm, the

RX’s location is (R12 , R1

2 ). For R1 = 50 µm, we also consider the RX located at

the center of environment, i.e., (0,0). . . . . . . . . . . . . . . . . . . . . . . 92

4.6 The expected number of cooperators over spatial PPP EK versus threshold

η for different population radii R1. . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 The variance VarK and different orders of moments E(K)n of number of

cooperators versus threshold η for different population radii R1. . . . . . . . . 96

4.8 The PDF of number of cooperators for different population radii R1 and differ-

ent thresholds η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.9 The probability when the number of cooperators is larger than 10%, 20%, 30%

of all bacteria versus threshold η for different population radii R1. . . . . . . . 98

4.10 The number of Pairs of any node and its nth nearest node both cooperating

P1(n) versus the population radius R1 for different thresholds η . . . . . . . . . 98

xxii LIST OF FIGURES

5.1 (a): A 2D sketch of the considered system model, where L is the distance be-

tween the TX and RX. (b): A 3D sample of a PM [3]. (c): Illustration of molec-

ular transport through a PM with heterogeneous advection [4], where the red

lines represent streamlines of the laminar flow; the shading of the background

denotes the flow velocity which decreases from light to dark; the horizontal

arrow denotes transport of molecules over the length of a pore in streamwise

direction; and the vertical arrow indicates transport of molecules across stream-

lines into low velocity zones in the wake of the solid grains. In (b) and (c), the

grains are represented in grey and black, respectively. . . . . . . . . . . . . . . 100

5.2 The CDF and PDF f (t) of the arrival time of the molecule versus time t in the

PM channel for different Pe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3 The PDF f (t) of the arrival time of the molecule versus time in the PM and FS

channels for different Pe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 The average mutual information I and the average error probability Q of the

MC system via the PM versus the threshold ξ for different Pe: Pe= 3,30,300,1000.

N = 100 and T = 400s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 The throughput C of the MC system via the PM and FS channels versus the

symbol slot T with N = 105 for different Pe: (a) Pe = 3, (b) Pe = 30, (c)

Pe = 300, and (d) Pe = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6 The optimal average error probability Q∗ of the MC system versus the number

of molecules N released for bit “1” for different symbol slots: T = 300s, T =

350s, and T = 400s with Pe = 3. . . . . . . . . . . . . . . . . . . . . . . . . 107

Chapter 1

Introduction

1.1 Motivation

1.1.1 Introduction of Molecular Communications

Over the past decades, information transmission has been an essential problem for organized

society. The conventional electromagnetic signals are not appropriate in many biological and

chemical engineering environments. This is because 1) electromagnetic signals quickly de-

cay in such environments and 2) the unwanted effects caused by the electromagnetic signals

may pose a health risk. In fact, molecules are used by many cells, tissues, organisms, and

plants to signal each other and share information in nature, e.g., neuromuscular junction. In

the neuromuscular junction, i.e., the contact between neurons and muscle fiber, nerve cells

release Vesicle of Acetylcholine molecules and these molecules diffuse through the junction

and bind with receptors of muscle cells. Using this communication method, neurons are able

to transmit a signal to the muscle fiber, for causing muscle contraction. Another example is

quorum sensing (QS), which is a very common method for microorganisms to communicate

with each other. Bacteria use QS to coordinate their behaviors when cell densities change.

In QS, bacteria assess the population density by releasing and recapturing molecules in their

environment. Inspired by nature, molecular communications (MC) has been proposed. In the

following, we focus on three issues of MC, i.e., what is MC, why study MC, and how to study

MC, to elaborate on the definition, the research motivation, and the unique features of MC.

• What is MC? – MC refers to communication systems where the information transmission

between a transmitter (TX) and a receiver (RX) is realized through the exchange of

chemical molecules, as shown in Fig. 1.1. The process of MC includes the following

steps: 1) Information emission: A TX generates molecules and releases molecules. 2)

Information transmission: The released molecules propagate in a fluid medium until they

bind with the receptors on the surface of a RX. 3) Information decode: The RX finally

recovers the information encoded in these molecules.

• Why study MC? – 1) Theoretical reasons: MC is an interdisciplinary research topic

1

2 Introduction

Receptor

Transmitter

Synthase

Signaling

Molecule

Receiver

Figure 1.1: Illustration of MC

which lies across wireless communications, signal processing, mass diffusion, and bi-

ology. We note that MC bridges the gap between biological signaling and conventional

communications by using conventional communication theory and techniques to study

and design systems that use chemical molecules to transmit information. Therefore,

studying MC helps us to understand biological systems from the aspects of signal pro-

cessing and information theory, interface with biological systems, and gain the inspira-

tion for the design of synthetic biological networks. 2) Practical reasons: MC has two

unique potential benefits, i.e., bio-compatibility and low energy consumption [5], since

MC is an existing communication method used by many organisms and no external en-

ergy is required by free diffusion of molecules. Due to these unique benefits, MC has

been acknowledged as one of the most promising communication methods in nanoscale.

The resulting network, i.e., nanonetwork, will advance a diverse number of potential

applications in i) medical and healthcare area, such as targeted drug delivery, health

monitoring, disease detection, and nanomedicine, and ii) environmental area, such as

biosensor and actuator networks, environmental monitoring, and pollution control [6].

• How to study MC? – We first note that there are fundamental differences between con-

ventional communications and MC. For example, the noise of conventional communica-

tions mainly comes from fading and thermal noise, but noise in MC mainly comes from

the randomness of molecular movement and the chemical reactions occurred during the

movement. Also, unlike conventional communications, the propagation media of MC is

usually a fluid boundary environment. Based on [7], we summarize the unique features

of MC in Table 1.1. Based on these features, MC distinguishes conventional communi-

cation analysis from the following aspects: 1) Diffusion trajectory of each molecule is

uncertain, 2) Some environmental phenomena, e.g., chemical reactions and fluid flow,

§1.1 Motivation 3

Table 1.1: Unique Features of MC

Property Conventional Communications MC

Media Air Fluid

Signal Type Electrical or Optical Chemical

Propagation Speed Speed of light (3×108 ms) On order of µm/s

Propagation Range m-km nm-µm

Energy Consumed High Low

Directionality No boundary Boundary

Noise Fading, Thermal noise Diffusive, Chemical

may affect molecular propagation, 3) Transceiver devices are made by modified cells

which have limited computational resources, and 4) A particle-based simulation method

[8] is used to verify theoretical results.

1.1.2 Research Challenges for MC

To fully understand MC, the following challenges need to be tackled: 1) Low reliability of

diffusion-based MC systems, 2) practical suboptimal MC system design and parameter op-

timization, 3) behavior predict of the microscopic population with noisy signaling, and 4)

realistic molecular information propagation environments. This thesis tackles these research

challenges in Chapters 2–5 and the thesis outline is shown in Fig. 1.2.

1.1.2.1 Low Reliability

The simplest molecular propagation mechanism is free diffusion where the information-carrying

molecules propagate from the TX to the RX via Brownian motion. In free diffusion, molecules

transport by colliding with other molecules. Therefore, no external energy is required for free

diffusion and it is used by many processes in cells, e.g., QS. One of the primary challenges

posed by diffusion-based MC is that its reliability rapidly decreases when the TX-RX distance

increases.

A common approach to enhancing communication reliability is using multiple RXs sharing

common information to help transmission. In biological environments, some cells or organ-

isms indeed share common information to achieve a specific task [9], e.g., calcium (Ca2+)

signaling [10]. In one process regulated by Ca2+ signaling, named excitation-contraction cou-

pling, the cells in skeletal muscle share Ca2+ ions to induce the contraction of myofibrils [11].

4 Introduction

Maximum Likelihood Detection, Optimization of molecule allocation

Performance Limit

Predict Node Behavior

Realistic Propagation Environment

Cooperative MC, Optimization of ThresholdsLow Reliability

Characterization of Cooperators, 2D Molecular Signal Analysis

Performance Analysis of MCin Porous Media

Research Challenge My Work Chapter

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Figure 1.2: Thesis Outline

In wireless communications, the cooperation among distributed detectors has been identified

as an effective means of improving reliability. For example, cooperative spectrum sensing is

achieved by allowing multiple secondary users to share sensing data to improve the detection

quality of a primary user [12]. Generally, in a distributed detection system the data of the indi-

vidual detectors are shared at a fusion center (FC). This data may be hard (binary) decisions,

soft (multi-level) decisions, or quantized observations. The FC then appropriately combines

the received data to yield a global inference [13] using a fusion rule, such as the AND rule

and OR rule for hard decisions. In fact, logic operations and corresponding computations re-

quired at the FC, e.g., AND, OR, and addition operations, can be implemented at the molecular

level [14, 15]. Therefore, MC is a suitable domain to apply distributed detection to improve

transmission reliability and such cooperative detection in MC needs to be studied.

1.1.2.2 Performance Limit

Practical suboptimal MC system design: After we explore the reliability benefits brought

by cooperative detection in MC, we are interested in the best error performance that can be

achieved by a practical cooperative MC system. In wireless communications, the maximum

likelihood (ML) detector is commonly used to optimize detection performance [16, Ch, 5]. In

the MC domain, the ML sequence detector has been considered for optimality in several stud-

ies, e.g., [17, 18]. However, the high complexity of sequence detection is a significant barrier

to implementation in the MC domain, even when applying simplified algorithms. Therefore,

a lower computational complexity, yet suboptimal ML detector, e.g., symbol-by-symbol ML

§1.1 Motivation 5

Target genes

SynthaseReceptor

Low Cell Density High Cell Density

Signaling

Molecule

Noncooperative

Bacterium

Cooperative

Bacterium

Figure 1.3: An example of bacteria coordinating their behavior via QS where grey and blue circles de-note noncooperative and cooperative bacteria, respectively. The core of QS: 1) Each bacterium includesa synthase that emits the signaling molecules and receptors that can bind with the molecules. 2) Themolecules diffuse into and out of the bacteria. 3) If the number of molecules that are bound exceeds athreshold, a receptor is activated to regulate target genes for a cooperative response.

detector, needs to be investigated.

Parameter optimization: To maximize the benefits brought by cooperative detection in MC,

it is of significance to investigate the optimal value of system parameters, e.g., the detection

thresholds at the RXs. Also, the resources of molecules are usually constrained in the realistic

biology environment. Hence, it is also of interest to study the optimal number of molecules

used for transmission in MC. It is worthwhile to note that optimizing the environmental and

system parameters, especially for complex MC network, is a challenging yet important prob-

lem.

1.1.2.3 Predict Node Behavior

MC is present in nature and used by many biological entities and systems, such as cells and

microorganisms. As mentioned in Section 1.1.1, QS is a ubiquitous approach for microbial

communities to respond to a variety of situations in which monitoring the local population

density is beneficial. In QS, bacteria assess the number of other bacteria they can interact

with by releasing and detecting a molecular signal in their environment, as shown in Fig.

1.3. This is because a higher density of bacteria leads to more molecules that can be detected

before they diffuse away. If the number of molecules detected exceeds a threshold, bacteria

express target genes for a cooperative response. QS enables coordination within large groups

of cells, potentially increasing the efficiency of processes that require a large population of

cells working together.

Microscopic populations utilize QS to complete many collaborative activities, such as vir-

ulence, bioluminescence, biofilms, and resistance of antibiotics. These tasks play a crucial role

in bacterial infections, environmental remediation, and wastewater treatment [19]. We note

6 Introduction

that the QS process is highly dependent on signaling molecules. Therefore, the accurate char-

acterization of releasing, diffusion, degradation, and reception of such molecules across the

environment in which bacteria live is very important to understand, control, and predict bacte-

rial behaviors in QS, which eventually can help us to prevent undesirable bacterial infections

and lead to new environmental remediation methods [20].

1.1.2.4 Realistic Propagation Environments

Significant research has been conducted to investigate molecular signal propagation in both

free space (FS) and simple bounded environments, e.g., [21, 22]. These papers have been

suitable for establishing tractable limits on communication performance by assuming that

molecules propagate in environments without obstacles. However, in many biological (e.g.,

tissue membrane [23]) and chemical engineering (e.g., catalyst bed [24]) environments, the

channel consists of porous medium (PM) materials. The PM is a solid with pores (i.e., voids)

distributed more or less uniformly throughout the bulk of the body [25]. Many natural and

man-made substances, e.g., rocks, soils, and ceramics, can also be classified as PM materials

[26].

Porous channels are fundamentally different from FS channels due to the intricate network

of pores. The molecules undergo complex trajectories and experience heterogeneous advection

as they propagate through pores of different sizes and lengths, causing so-called mechanical

dispersion [25, 27], which is an augmented effective diffusion caused by velocity fluctuations.

More importantly, particles may become trapped in immobile or re-circulation zones in the

vicinity or the wake of solid grains [4, 28], therefore taking some time to exit, and causing

non-trivial anomalous transport phenomena, such as long tails in the arrival time distributions.

Therefore, it is of fundamental importance to investigate what impact these PM flow and trans-

port properties have on the MC performance.

1.2 Background

In this section, we provide the background information of the techniques used in this thesis

which largely makes the thesis self-contained.

1.2.1 Distribution of Received Molecular Signals

In this subsection, we provide the distribution used for approximating the number of molecules

observed at a RX, which are fundamentals for evaluating the performance metrics (e.g., error

probability and throughput) of MC systems.

Binomial distribution: We denote X as the number of molecules observed by the RX at

some time. By considering that each molecule behaves independently and has the probability

§1.2 Background 7

of being observed by a RX at some time, it is accurate to assume X is Binomial distributed;

see [29, Ch. 5]. We then evaluate the probability mass function (PMF) of X as

Pr(X = k) =(

nk

)pk(1− p)n−k, (1.1)

where n is the total number of molecules, p is the probability of a given molecule being ob-

served, k ∈ 0,1, . . . ,n. The cumulative distribution function (CDF) of X is evaluated as [29,

Eq. (5.1.4)]

Pr(X ≤ i) =i

∑k=0

(nk

)pk(1− p)n−k. (1.2)

Binomial→Poisson: We note that the calculation of (1.1) and (1.2) requires high compu-

tational complexities. Therefore, we approximate the Binomial random variable (RV) X as a

new Poisson RV X with mean np. This approximation is valid when n is sufficiently large and

p is sufficiently small with np < 10; see [29, Ch. 5]. Considering X is a Poisson RV with mean

np, we write its PMF as [29, Eq. (5.2.1)]

Pr(X = k) =(np)k exp(−np)

k!, (1.3)

and its CDF is

Pr(X ≤ i) =i

∑k=0

(np)k exp(−np)k!

. (1.4)

Since discrete distributions of the number of observed molecules x make taking derivative

with respect to x cumbersome, we need to use continuous functions to approximate the PMF

and CDF of x. In the following, we present the continuous approximations for the CDF of x.

The PMF of x can be obtained by Pr(X ≤ k)−Pr(X ≤ k−1).

Gamma→(1.4): Using [30, Eqs. (26.4.19),(26.4.21)], we approximate (1.4) as

Pr(X ≤ i) =Γ(bi+ 1c,np)

Γ(bi+ 1c), (1.5)

where Γ(a,x) and Γ(a) denote incomplete Gamma function Γ(a,x) =∫

∞

x ta−1 exp(−t)dt and

Gamma function Γ(a) = Γ(a,x)|x=0, respectively.

Binomial→Gaussian: Based on the central limit theorem [29, Ch. 6], we can approxi-

mate a Binomial RV as a Gaussian RV with mean np and variance np(1− p). By adding the

8 Introduction

continuity correction, we approximate the CDF of Binomial RV X , i.e., (1.2), as

Pr(X ≤ i) =12

[1+ erf

(i+ 0.5−np√

2np(1− p)

)]. (1.6)

Poisson→Gaussian: Similarly, based on the central limit theorem [29, Ch. 6], we can ap-

proximate a Poisson RV as a Gaussian RV with mean np. By adding the continuity correction,

we approximate the CDF of Poisson RV X as

Pr(X ≤ i) =12

[1+ erf

(i+ 0.5−np√

2np

)]. (1.7)

1.2.2 Convex Optimization for MC Systems

In this subsection, we review the general form of optimization problems, the standard form of

convex optimization problems, and the conditions to ensure convexity of functions based on

[31, Ch. 3 and Ch. 4]. These techniques can be used to find the optimal value of parameters

that maximizes the communication performance of MC systems.

General optimization problem: We use the following notations

minx

f0(x)

s.t. fi(x) ≤ 0, i = 1, . . . ,m

hi(x) = 0, i = 1, . . . , p,

(1.8)

to describe the problem of finding an x that minimizes f0(x) among all x that satisfy the condi-

tions fi(x)≤ 0, i = 1, . . . ,m, and hi(x) = 0, i = 1, . . . , p. We refer to x ∈Rn as the optimization

variable, the function f0 : Rn → R as the objective function, the inequalities fi(x) ≤ 0 as

inequality constraints, and the equations hi(x) = 0 as the equality constraints. A point x is

feasible if it satisfies the constraints hi(x) = 0, i = 1, . . . , p and fi(x)≤ 0, i = 1, . . . ,m. We

refer to the set including all feasible points as the feasible set.

Convex optimization problem: We note that a convex optimization problem is one of the

forms given by [31, Eq. (4.15)]

minx

f0(x)

s.t. fi(x) ≤ 0, i = 1, . . . ,m

aTi x = bi, i = 1, . . . , p,

(1.9)

where f0, . . . , fm are convex functions. Different from the general standard form problem (1.8),

the convex problem (1.9) has three extra requirements: i) the objective function must be con-

vex; ii) the inequality constraint functions fi(x) must be convex; and iii) the equality constraint

§1.2 Background 9

functions hi(x) = aTi x− bi must be affine. According to these requirements, the key require-

ment for formulating a convex optimization problem is ensuring that f0, . . . , fm are convex

functions.

Convexity of functions: We finally discuss the conditions for ensuring the convexity of

functions. Both first-order conditions and second-order conditions can ensure the convexity,

here we only discuss the second-order conditions since they are easier and more common to

be used. Assuming that f is twice differentiable, then f is convex if and only if its Hessian is

positive semidefinite (PSD) [31, Ch. 3.1.4], i.e.,

∇2 f (x) 0. (1.10)

For the function f (x) on R, (1.10) reduces to the simple conditions f′′(x) ≥ 0. It means

that the derivative is nondecreasing. For the function f (x) on R2, (1.10) reduces to that all of its

principal minors of its Hessian are nonnegative [32]. Thus, the joint convexity of f (x,y) with

respect to x and y can be proven by finding ∂ 2 f (x,y)∂ (x)2 ≥ 0, ∂ 2 f (x,y)

∂x2 ≥ 0, and(

∂ 2 f (x,y)∂x2

)(∂ 2 f (x,y)

∂y2

)−(

∂ 2 f (x,y)∂xy

)2≥ 0.

1.2.3 Randomly-Distributed MC Systems

In this subsection, we discuss models and tools used to account for the MC systems where

nodes (e.g., TXs, RXs, and bacteria) are randomly distributed. It is motivated by the fact that

nodes may move in realistic biological environments and their locations may be not fixed.

Point process: Different from the model with deterministic topology, point process models

deal with topological randomness. A point process model is abstracted to be a collection of

nodes residing in a certain place [33]. The locations of these nodes are not deterministic but

subject to uncertainty [34]. In the point process model, the locations of nodes are changing

from one realization to another realization. The occurrences of realizations follow certain

probabilities [35, 36].

Poisson point process: There are several types of point processes and the most popular

process is the Poisson point process (PPP) [37] due to its easy-to-use properties and well-

known theorems. The PPP is formally defined in Definition 1.1[38].

Definition 1.1 (PPP). A point process Φ = xi; i = 1,2,3, ... ⊂Rd is a PPP if and only if the

number of points inside any compact set B ⊂ Rd is a Poisson RV, and the number of points

in disjoint sets is independent. When the node density is constant, the PPP is known as the

homogeneous PPP.

Stochastic geometry: In the point process model, the performance metrics of systems are

changing from one realization to another realization, since the locations of nodes are changing

in different realizations. As a result, the average result or distribution of such performance

10 Introduction

metrics can better reflect their characterizations [33]. We note that stochastic geometry is a

powerful mathematical tool to deal with random spatial topologies, by providing numerous

methods to compute the spatial expectation and distribution of such quantities [39]. Here,

the spatial expectation means that the average is taken over a large number of realizations of

nodes. In the following, we present theorems to evaluate the expectation of a random sum and

a random product over PPP, based on [38].

Theorem 1.1 (Campbell Theorem–Random Sum). We assume that Φ is a point process on Rn

and f : Rn→R is a measurable function. We then have E∑x∈Φ f (x) given by [38, Ch. 4.2]

E∑x∈Φ

f (x)=∫

Rnf (x)Λ(dx), (1.11)

where Λ(dx) denotes the intensity measure of the point process Φ and x ∈Rn.

Theorem 1.2 (Probability Generating Functional–Random Product). We assume that f : Rn→R is a measurable function and Φ∈Rn is a PPP. Based on the probability generating functional

(PGFL), we evaluate E∏x∈Φ f (x) as [38, Ch. 4.2]:

E∏x∈Φ

f (x)= exp−∫

Rn(1− f (x))Λ(dx)

. (1.12)

1.3 Literature Review

In this section, we review the relevant work in the literature on cooperative MC systems, ML

detection for MC, bacterial behavioral analysis, and realistic propagation environments.

1.3.1 Cooperative MC Systems

The majority of the existing MC studies have focused on the modeling of the communication

between one TX and one RX [40]. Recent studies, [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,

52], have considered a cooperative MC system which consists of multiple RXs.

One-phase cooperative MC: Among these studies, [41] first proposed a molecular broad-

cast channel where a single TX communicates with multiple noncooperative RXs and derived

the corresponding capacity expression. Following [41], [42] evaluated various approaches to

maximizing the probability of information molecules and the rate of information reaching mul-

tiple RXs for a molecular broadcast system. In [43], communication between two populations

of bacteria through a diffusion channel was studied where each population acts as a TX or a

RX. Focusing on the communication between a group of TXs and a group of RXs, [44] opti-

mized the transmission rates that maximize the throughput and efficiency. In [45], simulations

were performed to demonstrate the feasibility of a bio-nanosensor network where bacteria-

based bio-nanomachines perform target detection and tracking. [46] investigated the impact of

§1.3 Literature Review 11

the interference on the MC between a TX and a RX pair which are connected through a mi-

crofluidic channel containing the fluid flow. A new stochastic model named reaction-diffusion

master equation with exogenous input was proposed in [47] to characterize an MC system with

multiple TXs and RXs. [48] designed a multiple-input multiple-output MC system and inves-

tigated the inter-symbol and inter-link interference therein, but did not consider cooperation

between the links. [49] designed an MC network where source, intermediate, and destination

bio-nanomachines exchange molecular packets.

Two-phase cooperative MC: Compared to [41, 42, 43, 44, 45, 46, 47, 48, 49], [50, 51, 52]

considered more complex two-phase cooperative MC systems. In particular, [50] suggested a

two-tier nano abnormality detection scheme by assuming that the sensor nano-machines have

independent Poisson observations and analyzed its detection performance. [51] reinvestigated

the similar detection scheme in [50], but assumed that the noise among sensor nano-machines

are correlated. [52] explored the in vivo distributed detection of an undesired biological agent’s

biomarkers by a group of biological sized nanomachines in an aqueous medium under drift.

While [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] stand on their own merits, [41, 42,

43, 44, 45, 46, 47, 48, 49] only considered one-phase cooperative MC systems and [50, 51, 52]

only considered the single transmission of one symbol. This indicates that the role of the

cooperation among multiple RXs in determining the TX’s intended symbol sequence in a two-

phase MC system has not been established in the literature.

1.3.2 ML detection for MC

In conventional communications, the ML detector is commonly used to achieve the optimal

detection performance; see [16, Ch, 5]. In the MC domain, the ML sequence detector has been

considered for optimality in several studies, e.g., [17, 18, 52, 53, 54, 55, 56, 57, 58].

ML detection at one RX: [17, 18, 53, 54] considered the ML sequence detector at a single

RX. In particular, [17] proposed a sequence detection method based on an ML criterion to re-

cover the transmitted information distorted by inter-symbol interference (ISI) and noise. [18]

derived the optimal sequence detector in an ML sense between a single TX and a RX, and

presented a modified Viterbi algorithm to reduce the computational complexity of optimal de-

tection. A near ML sequence detector using the reduced-State Viterbi algorithm was proposed

in [53]. In [54], simple RX models based on ML estimation are studied for the additive inverse

Gaussian noise channel with molecular diversity. However, the high complexity of sequence

detection is a significant barrier to implement it in the MC domain, even when applying simpli-

fied algorithms; see [18, 53]. The (suboptimal) symbol-by-symbol ML detector requires less

computational complexity than the ML sequence detector. Motivated by this, [55, 56] consid-

ered symbol-by-symbol ML detection at a single RX for MC. [55] analyzed a strength-based

optimum signal detection model with a symbol-by-symbol ML detector. [56] investigated an

12 Introduction

M-ary modulation scheme for end-to-end communication between one TX and one RX with

symbol-by-symbol ML detection.

Cooperative ML detection: The optimal ML detection is rarely applied to the cooperative

MC communication system in most of the literature except for recent studies in [52, 57, 58].

[57] considered cooperative ML detection where sensors detect an event and send molecules

to an FC over an anomalous diffusion channel. However, [57] ignored the detection process of

event and simply assumed a constant detection probability between the observed event and a

RX, and a constant ISI, independent of time intervals. [58] considered cooperative abnormality

detection via a diffusive MC network consisting of sensors and an FC, by assuming a constant

probability for sensed value at each sensor and the constant expected number of molecules

received at the FC in each time slot. [52] considered an optimal fusion rule, i.e., log-likelihood

ratio test, at the FC, but considered the single transmission.

Therefore, [17, 18, 53, 54, 55, 56] only considered one-RX ML detection, [57, 58] con-

sidered one-phase communication, i.e., the RXs communicate with an FC but do not detect

information from a TX, and [52, 58] considered one-symbol transmission, i.e., the FC makes

a single decision about the presence of an abnormality. This indicates that the symbol-by-

symbol ML detection has not been applied to a cooperative MC system with multiple RXs and

multiple communication phases.

1.3.3 Bacterial Behavioral Analysis

As mentioned in Sections 1.1.1 and 1.1.2.3, bacteria use molecular signals to coordinate their

cooperative behaviors. In both areas of MC and biology, there are growing research efforts

from the aspects of experiments and theoretical modeling to study the coordination of bacteria

via QS, e.g., [20, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68].

Simulations and experiments: Among them, [20, 59, 60] investigated the cooperative be-

havior of bacteria using simulation or biological experiments. [20] relied on simulation meth-

ods to model the evolution of QS regulated cooperation. [59] discussed confusion over the

terms kin selection, mutualism, mutualism, cooperation, altruism, and punishment using a

simple game theory model. [60] presented the first empirical evidence that high population

density can favor cooperation in a spatially distributed environment.

Theoretical modeling: [61, 62, 63, 64, 65, 66, 67, 68] mathematically modeled bacterial

behavior coordination. [61] proposed a simple game to predict cooperation in a bacterial pop-

ulation under population uncertainty. [62] introduced a game-theoretic model to show how

individual links in a bacterial network could form. [63] studied the effects of cooperation

and uncertainty on communication efficiency within a nanoscale network. [64] considered an

optimization-based framework to study QS as a networked decision system. [65, 66] consid-

ered a queueing model to analyze the dynamics of bacterial behaviors. [67] studied the effects

§1.3 Literature Review 13

of bacterial social interactions on information delivery in bacterial nanonetworks. [68] de-

veloped a modeling framework for the spatio-temporal dynamics of the associated metabolic

circuit for cells in a colony.

We note that [61, 62, 63, 64, 65, 67] relied on abstract or simplifying models to represent

the molecular diffusion channel, i.e., did not consider the motion of each signaling molecules

based on Fick’s law, in order to focus on understanding how behavior evolves over time. A

molecular diffusion channel between two clusters of bacteria was considered in [43], but [43]

did not consider the bacteria behavioral response. Recently, [69] identified opportunities to

combine MC and game theory to analyze behavioral dynamics in microscopic populations with

noisy signaling. To the best of our knowledge, analyzing the responsive cooperative behavior

at bacteria, taking into account the chemical reaction and diffusion of each molecule based on

reaction-diffusion equations, is not available in the literature.

1.3.4 Realistic Propagation Environments

Most MC studies consider the FS channel, i.e., infinite boundary conditions without drift.

However, for many biological and medicare applications, e.g., drug delivery inside the human

body, the realistic propagation channel may be bounded, flow-assisted, and consisting of a

network of nonuniformly distributed solid grains. Only limited studies, e.g., [70, 71, 72, 73,

74, 75], considered bounded propagation environments, and a few studies, e.g., [76, 77, 78, 79,

80, 81, 82], considered flow-assisted propagation environments.

Bounded environments: [70] showed that molecular information can transmit more reliably

in complex and confined structural environments, compared to conventional electromagnetic-

based systems. [71] considered information carrying molecules are used to transfer informa-

tion on a microfluidic chip in a confined space and measured the achievable information rates

of such MC systems. [72] considered MC in confined environments, but assumed that the

number of molecules within the environment fixed. [73] performed simulations on the motion

of particles in confined spaces since the required analytical solution in confined spaces are gen-

erally not available in closed form. [74] proposed a generalized model for the ligand-receptor

protein interaction in three-dimensional (3D) spherically bounded and diffusive biological mi-

croenvironments. [75] evaluated the mean and covariance of the output signal for a reversible

reaction RX in a voxelated 3D cubic bounded medium is obtained using reaction-diffusion

master equation.

Flow-assisted environments: [76] proposed a symbol interval optimization algorithm in

MC with drift. [77] derived closed-form expressions for the molecular reception rate of the

active and passive RXs. [78] proposed an ML estimator for the clock offset between two

nanomachines in an MC system with drift. [80] derived the probability density function (PDF)

of the first hitting time for a fully-absorbing point RX in a one-dimensional (1D) flow-induced

14 Introduction

diffusive MC system where both the TX and RX can move. [81] analyzed channel impulse

response for a drift-diffusion fluid system with the periodic flow. [82] derived analytical ex-

pressions for the end-to-end symbol error probability and the capacity for an M-ary modulation

scheme in a flow-aided diffusive environment with molecular degradation.

Although [70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82] stand on their own merits,

molecular information delivery over an intricate network of pores with heterogeneous advec-

tion in a bounded environment has not been considered in the literature.

1.3.5 Limitation of Existing Studies

Although the existing studies stand on their own merits, there are still various remaining open

problems in the area of MC. In the thesis, we aim to fill in several important blanks in the

literature by raising and tackling the following key questions:

1. How do we improve the reliability of diffusion-based MC systems? How do we quantify

the reliability improvement brought by combining observation at multiple RXs? What

is the optimal decision threshold at the nano-devices which achieves the best error per-

formance?

2. How do we design practical suboptimal detection methods for a cooperative MC sys-

tem since the complexity at nano-device is limited in biological environments? What

is the best error performance can be achieved by such a practical cooperative detection

MC system? What is the optimal molecule allocation among distributed RXs to maxi-

mize the benefits brought by such a practical cooperative detection MC system? Is the

error performance achieved by lower-complexity detection method comparable to that

achieved by suboptimal detection methods?

3. How many molecules each bacterium can receive if a population of randomly distributed

bacteria continuously emits molecules at random times? What is the impact of noisy

population estimation on the bacterial cooperation behavior due to noisy molecular sig-

naling? What is the mean and distribution of the number of cooperative bacteria under

the noisy molecular signaling among a population of randomly distributed bacteria?

4. What is the unique characteristic of porous channels compared to the FS channel for

molecular information delivery? What are error performance, throughput, and capacity

for molecular information delivery over porous media? Is there any diversity gain for

PM channel since molecules may be trapped in the vicinity of solid grains and have

different propagation paths?

§1.4 Thesis Outline and Contributions 15

1.4 Thesis Outline and Contributions

This thesis mainly focuses on modeling, analysis, and optimization of cooperative and large-

scale MC systems for future applications of high-accuracy health monitoring and undesirable

bacterial infection prevention. Under this focus, we consider cooperative detection in MC,

bacterial cooperation modeling with noisy signaling, and molecular information delivery over

porous media. The specific contributions of each chapter are detailed below:

Chapter 2 – Convex Optimization of Cooperative MC Systems

In Chapter 2, we for the first time quantify and maximize the benefits of multi-RX cooper-

ation in a cooperative diffusion-based MC system. Our goal is to establish a fundamental

understanding of the reliability improvement brought by combining the detection results of

distributed RXs at an FC. In our considered system, for each symbol transmitted from the

TX, the RXs first independently make local hard decisions on the transmitted symbol and then

report their decisions to the FC. After receiving the local hard decisions, the FC fuses all de-

cisions to make a global decision on the transmitted symbol using an N-out-of-K fusion rule.

Here, we consider two different reporting scenarios from the RXs to the FC, namely, perfect

reporting and noisy reporting. In this work, we assume that the FC does not feedback its global

decision to RXs.

To maximize the benefits of multi-RX cooperation in the system, we determine the jointly

optimal decision thresholds at the RXs and FC such that the expected global error probability

is minimized. We note that it is mathematically intractable to derive analytical expressions

for such optimal thresholds. Therefore, we resort to convex optimization as an efficient and

effective method to solve the joint optimization problem. Since the expected global error

probability is not necessarily convex with respect to thresholds at the RXs and FC, we conduct

a new convex analysis of the error performance for the system having a symmetric topology.

Based on this analysis, we formulate convex optimization problems and find the solution via

an efficient convex optimization algorithm.

The primary contributions of this chapter are summarized as follows:

1. We derive closed-form expressions for the expected global error probabilities of the

cooperative MC system in the perfect and noisy reporting scenarios. We clarify that a

symbol-by-symbol detection with a constant decision threshold at all RXs and the FC is

considered in this derivation.

2. We derive new approximated expressions for the expected global error probability of the

cooperative MC system in both reporting scenarios. We also derive additional convex

constraints under which the approximated expressions are jointly convex with respect to

the decision thresholds at the RXs and the FC.

16 Introduction

3. Based on the derived convex approximations and constraints, we formulate suboptimal

convex optimization problems for a given transmitted symbol sequence. For the sake

of practicality, we then extend the formulated convex optimization problems such that a

single optimal threshold is determined to minimize the average error performance over

all realizations of transmitted symbol sequences.

The results in this chapter have been presented in [1], which is listed again for ease of

reference:

[1] Y. Fang, A. Noel, N. Yang, A. W. Eckford, and R. A. Kennedy, “Convex optimization of

distributed cooperative detection for multi-RX molecular communication,” IEEE Trans. Mol.,

Bio. Multi-Scale Commun., vol. 3, no. 3, pp. 166–182, Sep. 2017.

Chapter 3 – Symbol-by-Symbol ML Detection for Cooperative MC

In Chapter 3, we present symbol-by-symbol ML detection for a cooperative diffusion-based

MC system, based on [1, 2, 83], which consists of one TX, K RXs, and an FC. The significance

of this chapter is that our results provide lower bounds on the error performance that can be

achieved by the detectors considered in [1, 2, 83]. To the best of the authors’ knowledge,

combined with our previous work in [84], this work is the first to apply symbol-by-symbol ML

detection to a cooperative MC system with multiple communication phases.

In this chapter, we present three symbol-by-symbol ML detectors: 1) decode-and-forward

(DF) with multi-molecule-type and ML detection at the FC (MD-ML), 2) DF with single-

molecule-type and ML detection at the FC (SD-ML), and 3) Amplify-and-forward (AF) with

single-molecule-type and ML detection at the FC (SA-ML). ML detection in the current sym-

bol interval requires knowing the previously-transmitted symbols by the TX (and by all RXs

for DF). For convenience, we refer to the FC-estimated previous symbols as local history and

the perfect knowledge of the previous symbols as genie-aided history. Our major contributions

are summarized as follows:

1. We present novel symbol-by-symbol ML detection designs for the cooperative MC sys-

tem with all detector variants, i.e., SD-ML, MD-ML, and SA-ML. For practicality, we

consider the FC chooses the current symbol using its local history and design the meth-

ods for the FC to obtain the local history. We also derive the likelihood of observations

for all detectors.

2. We derive analytical expressions for the system error probability for SD-ML and SA-ML

using the assumption of genie-aided history. This assumption leads to tractable error

performance analysis. The error probabilities for SD-ML with K = 1 and SA-ML are

given in closed forms. The error performance of MD-ML is mathematically intractable.


3. We determine the optimal molecule allocation among RXs to minimize the system error

probability of SD-ML, which provides a lower bound on the system error probability

that can be achieved in practice. To this end, we formulate and solve a joint optimization

problem in terms of molecule allocation and a constant threshold. In this problem, the

objective function is the closed-form approximation of error probability of SD-ML since

there is no closed-form expression for the error probability of SD-ML. We also analyti-

cally prove that the equal distribution of molecules among two symmetric RXs achieves

the local minimal error probability of SD-ML.

4. We validate the accuracy of our analytical expressions of error probability via a particle-

based simulation method where we track the motions of molecules over time due to

diffusion. Using simulation and numerical results, we also demonstrate the FC’s effec-

tiveness in estimating the previously-transmitted symbols and confirm the effectiveness

of our optimization method.


reference:

[85] Y. Fang, A. Noel, N. Yang, A. W. Eckford, and R. A. Kennedy, “Symbol-by-symbol max-

imum likelihood detection for cooperative molecular communication,” IEEE Trans. Commun.,

vol. 67, no. 7, pp. 4885–4899, Jul. 2019.

Chapter 4 – Characterization of Cooperators in QS with 2D Molecular Sig-nal Analysis

In Chapter 4, by leveraging the knowledge of QS, mass diffusion, stochastic geometry, and

probability processes, we develop an analytically tractable model for predicting the statistics

of the number of responsive cooperative bacteria (i.e., cooperators) in random locations by

accounting for the random walk and the random degradation of QS molecules. Since our

model accounts for the random motion of molecules based on reaction-diffusion equations,

our model can be used to predict with high accuracy the effect of diffusion and chemical

reaction of molecules on the concentration of molecules observed by bacteria and statistics of

the number of responsive cooperators.

We consider a 2D environment. In this environment, bacteria are randomly spatially dis-

tributed where each one continuously releases molecules at random times. Developing the

analytical model in this paper is theoretically challenging since we need to address the random

received signal at bacteria in random locations due to randomness in the motion and degra-

dation of molecules, randomness in the locations of many TXs, and randomness in times of

emitting molecules. Despite these challenges, we make the following contributions:

1. We analytically derive the channel response (i.e., the expected number of molecules

18 Introduction

observed) at a RX due to continuous emission or an impulse emission of molecules

at one point TX. Built on this, we then derive the channel response at a RX due to

continuous emission of molecules at randomly-distributed point TXs on a circle in a

two-dimensional (2D) environment.

2. Using the results in 1), we first derive the exact expression for the expected probability

of cooperation at the bacterium in a fixed location due to the emission of molecules from

randomly-distributed bacteria by using the Laplace transform of the random aggregate

received molecules. We then derive the approximate expression of such a probability,

which is easier to compute than the exact expression, yet has good accuracy when the

population density is high based on our numerical results.

3. Based on the results in 2), we derive the approximate expressions for the moment gen-

erating function (MGF) and cumulant generating function (CGF) of the number of co-

operators. Using the MGF and the CGF, we derive the approximate expressions for the

nth moment and cumulant of the number of cooperators. We study the convergence of

the number of cooperators to a Gaussian distribution via the higher order statistics. We

compare the Poisson and Gaussian distributions with the derived statistics to approxi-

mate the PMF and CDF of the number of cooperators. We show that the Poisson distri-

bution provides the overall best approximation, especially when the population density

is low based on our numerical results. In addition, we derive the expected number of

pairs of two nearest nodes both cooperating, which can be used to study the neighboring

cooperative bacteria in a QS system.

The results in this chapter have been presented in [86] and [87], which is listed again for

ease of reference:

[86] Y. Fang, A. Noel, A. Eckford, N. Yang, and Jing Guo, “Characterization of cooperators

in quorum sensing with 2D molecular signal analysis,” submitted to IEEE Trans. Commun.

[87] Y. Fang, A. Noel, A. Eckford, and N. Yang, “Expected density of cooperative bacteria in

a 2D quorum sensing based molecular communication system,” in Proc. IEEE GLOBECOM

2019, Waikoloa, HI, Dec. 2019, pp. 1–6.

Chapter 5 – Molecular Information Delivery in Porous Media

In Chapter 5, we for the first time consider a PM channel in MC. We consider a binary se-

quence transmitted between a TX and a RX located at the ends of the PM channel. The main

contributions are summarized as follows:

1. Assuming that the number of molecules arrived can be approximated as a Gaussian RV,

we present numerical results for different performance metrics, i.e., throughput, mutual

information, and error probability, for the channel using fully resolved computational


fluid dynamics results for the breakthrough curves. We also numerically evaluate the di-

versity gain that is defined (as in [88]) as the exponential decrease rate of the probability

of error as the number of released molecules increases.

2. Using numerical results, we investigate the differences in channel characteristics and

performance metrics between a PM and diffusive FS channel with the flow. In particular,

we show that the tail of the PM channel response is longer than that of the FS channel,

which can significantly affect the communication performance, e.g., the ISI in the case

of concentration-modulated transmission is more severe.


reference:

[89] Y. Fang, W. Guo, M. Icardi, A. Noel, and N. Yang, “Molecular information delivery in

porous media,” IEEE Trans. Mol., Bio. Multi-Scale Commun., vol. 4, no. 4, pp. 257–262,

Dec. 2018.

Finally, Chapter 6 gives a summary of results and provides suggestions for future research

work.

20 Introduction

Chapter 2

Convex Optimization of CooperativeMC Systems

This chapter analyzes and optimizes the error performance achieved by cooperative detection

among K distributed RXs in a diffusion-based MC system. In this system, the RXs first make

local hard decisions on the transmitted symbol and then report these decisions to an FC. The

FC combines the local hard decisions to make a global decision using an N-out-of-K fusion

rule. We consider two reporting scenarios, namely, perfect reporting and noisy reporting. We

derive closed-form expressions for the expected global error probability of the system for both

reporting scenarios. We also derive new approximated expressions for the expected error prob-

ability. We then find convex constraints to make the approximated expressions jointly convex

with respect to the decision thresholds at the receivers and the FC. Based on such constraints,

we formulate and solve suboptimal convex optimization problems to determine the optimal

decision thresholds which minimize the expected error probability of the system. Numerical

and simulation results reveal that the system error performance is greatly improved by com-

bining the detection information of distributed receivers. They also reveal that the solutions

to the formulated suboptimal convex optimization problems achieve near-optimal global error

performance.

This chapter is organized as follows. In Section 2.1, we describe the system model. In

Section 2.2, we present the error performance analysis of the cooperative-RX MC system. In

Section 2.3, we formulate convex optimization problems of the cooperative-RX MC system.

Numerical and simulation results are provided in Section 2.4. In Section 2.5, we conclude and

describe future directions for this work.

2.1 System Model

In this chapter we consider a cooperative MC system in a 3D space, as depicted in Fig. 2.1,

which consists of one TX, a “cluster” of K RXs, and one device acting as an FC. The FC is

not included in the set of RXs. We assume that the RXs are generally closer to the FC than to

21

22 Convex Optimization of Cooperative MC Systems

TXFC

RX1

RX5

RX4

RX3

RX2

RX1

Figure 2.1: An example of a cooperative MC system with K = 5, where the transmission from theTX to the RXs is represented by solid arrows and the decision reporting from the RXs to the FC isrepresented by dashed arrows.

the TX to ensure reliable reporting channels from the RXs to the FC. We assume that all RXs

and the FC are spherical observers. Accordingly, we denote VRXkand rRXk

as the volume and

the radius of the kth RX, RXk, respectively, where k ∈ 1,2, . . .,K. We also denote VFC and

rFC as the volume and the radius of the FC, respectively. We also assume that the RXs and the

FC are independent passive observers such that molecules can diffuse through them without

reacting1. We further assume that all individual observations are independent of each other. In

addition, we assume that the RXs operate in the half-duplex mode such that they do not receive

information and report their local decisions at the same time.

In the considered system, the transmission of each information symbol from the TX to the

FC via RXs is completed within three phases, detailed as follows:

• In the first phase, the TX transmits one symbol of information via type A0 molecules to

the RXs through the diffusive channel. The number of the released type A0 molecules is

denoted by S0. We assume that the diffusion of all individual molecules is independent.

The type A0 molecules transmitted by the TX are detected by all RXs. In this work

we consider that the TX uses ON/OFF keying [90] to convey information. As per the

rules of ON/OFF keying, the TX releases S0 molecules of type A0 to convey information

symbol “1”, and releases no molecules to convey information symbol “0”. To enable

ON/OFF keying, the information transmitted by the TX is represented by an L-length

binary sequence where each element is “0” or “1”. The sequence is denoted by WTX =

WTX[1],WTX[2], . . .,WTX[L], where WTX[ j], j ∈ 1, . . .,L, is the jth symbol transmitted

by the TX. We assume that the probability of transmitting “1” in the jth symbol is P1 and1Although we cannot guarantee perfect independence between different RXs, the dependence between obser-

vations made at different RXs is extremely small. This is due to the fact that the time between adjacent samples atRXs is sufficiently long to ensure that observations at RXs are independent and each RX observes a small fractionof the total number of released molecules. Moreover, the validity of assuming independence will be demonstratedby the excellent agreement between analytical and simulation results depicted in Section 3.5.

§2.1 System Model 23

the probability of transmitting “0” in the jth symbol is 1−P1, i.e., Pr(WTX[ j] = 1) = P1

and Pr(WTX[ j] = 0) = 1−P1.

• In the second phase, each RX makes a local hard decision on the current transmitted

symbol. We denote WRXk [ j] as the local hard decision on the jth transmitted symbol at

RXk. Then, the RXs simultaneously report their local jth hard decisions to the FC. We

assume that RXk transmits type Ak molecules, which can be detected by the FC. The

number of the released type Ak molecules is denoted by Sk. We also assume that the

channel between each RX and the FC is diffusion-based, and each RX uses ON/OFF

keying to report its local hard decision.

• In the third phase, the FC obtains the decision at RXk by receiving type Ak molecules

over the RXk− FC link. We assume that the K RXk− FC links are independent. We

denote WFCk [ j] as the received local decision of RXk on the jth transmitted symbol at

the FC. The FC combines all WFCk [ j] using an N-out-of-K fusion rule to make a global

decision WFC[ j] on the jth symbol transmitted by the TX, where N denotes the number

of decisions of “1” received by the FC and K denotes the number of RXs. As per the

N-out-of-K fusion rule, the FC declares a global decision of “1” when it receives at least

N decisions of “1”. There are several special cases of the N-out-of-K fusion rule, such

as 1) majority rule where N = dK/2e, 2) OR rule where N = 1, and 3) AND rule where

N = K.In this work, we assume that the FC does not feed back its global decision to

RXs.

We note that the role of the RXs in our considered system appears similar to that of DF

relays in wireless systems [91]. However, the results for DF relaying cannot be used in the MC

system, due to the fact that the characteristics of the propagation channel and the methods for

recovering the received symbols in MC systems are completely different from those in wire-

less systems. In this work, we assume that the FC does not feed back its global decision to

RXs. We note that the use of a binary sequence is expected in MC between nanomachines to

exchange the amount of information required for executing complex collaborative tasks, such

as disease detection and targeted drug delivery. Thus, in this work we consider the transmis-

sion of multiple binary symbols as a sequence and take the resultant ISI into account for the

cooperative MC system.

We define WlTX = WTX[1], . . .,WTX[l] as an l-length subsequence of the information trans-

mitted by the TX, where l ≤ L. We also define WlRXk

= WRXk[1], . . .,WRXk

[l] as an l-length

subsequence of the local hard decisions at RXk. We then define WlFCk

= WFCk[1], . . .,WFCk

[l]as an l-length subsequence of the received local decision of RXk at the FC. We further define

WlFC = WFC[1], . . .,WFC[l] as an l-length subsequence of the global decisions at the FC.

We denote ttrans as the transmission interval time from the TX to the RXs and treport as

the report interval time from the RXs to the FC. Thus, the symbol interval time from the TX


to the FC is given by T = ttrans + treport. At the beginning of the jth symbol interval, i.e.,

( j−1)T , the TX transmits WTX[ j]. After this the TX keeps silent until the start of the ( j+1)th

symbol interval. We apply the weighted sum detector with equal weights[18] at the RXs and

FC for detection. Thus, the RXs and FC each take multiple samples within their corresponding

interval time, add the individual samples with equal weights, and compare the summation

with a decision threshold. The decision thresholds at RXk are denoted by ξRXk. The decision

thresholds at FC over the RXk−FC link denoted by ξFCk. Here, the assumption of equal weights

for all samples is adopted to limit the computational complexity of the detector and facilitate

its usage in MC.

We now describe the sampling schedules of the RXs and FC. The FC or RXk samples at a

certain time t by counting the number of the molecules observed. All RXs sample at the same

times2 and take MRX samples per symbol interval. The time of the mth sample for each RX in

the jth symbol interval is given by tRX( j,m) = ( j−1)T +m∆tRX, where ∆tRX is the time step

between two successive samples at each RX, m ∈ 1,2, . . .,MRX, and MRX∆tRX < ttrans. At the

time ( j− 1)T + ttrans, each RX reports its local decision for the jth interval via diffusion to

the FC. We assume that the FC takes MFC samples of each type of molecule in every reporting

interval. The time of the mth sample of type Ak molecules at the FC in the jth symbol interval

is given by tFC( j, m) = ( j− 1)T + ttrans + m∆tFC, where ∆tFC is the time step between two

successive samples at the FC and m ∈ 1,2, . . .,MFC.

2.2 Error Performance Analysis

In this section, we analyze the expected global error probability3 of the cooperative MC system.

To this end, we denote QFC[ j] as the expected global error probability in the jth symbol interval

for a given TX sequence W j−1TX . Under the assumption that there is no a priori knowledge of

WTX[ j], we express QFC[ j] as

QFC[ j] = P1Qmd[ j]+ (1−P1)Qfa[ j], (2.1)

where Qmd[ j] denotes the expected global missed detection probability (MDP) in the jth sym-

bol interval and Qfa[ j] denotes the expected global false alarm probability (FAP) in the jth

symbol interval. By averaging QFC[ j] over all possible realizations of W j−1TX and across all

symbol intervals, the expected average error probability of the cooperative MC system, QFC,

2We note that all RXs may not be synchronized perfectly in some cases. Thus, we make the assumption ofsame sampling times of all RXs to explore the best error performance achieved by the cooperative MC system,which serves as a performance bound for practical systems. We also note that various methods can be adoptedto achieve time synchronization among nanomachines, e.g., [92] and [93]. Therefore, the assumption of perfectsynchronization is widely adopted in existing MC studies, e.g., [94], [18], and [48].

3All the expected error probabilities throughout this chapter are derived for given W j−1TX , unless otherwise

specified.

§2.2 Error Performance Analysis 25

can be obtained. In the analysis, we address two different reporting scenarios, namely, perfect

reporting and noisy reporting. In the perfect reporting scenario, we assume that no error occurs

when RXk reports to the FC, i.e., WFCk [ j] = WRXk [ j]. In the noisy reporting scenario, errors can

occur when RXk reports to the FC via diffusion4.

2.2.1 Perfect Reporting

In this subsection, we start our analysis by examining the error performance of the TX−RXk link. This examination is based on the analysis in [94]. We then use the results of this

examination to analyze Qmd[ j] and Qfa[ j] in the perfect reporting scenario to obtain QFC[ j].

2.2.1.1 TX−RXk Link

We first evaluate the probability of observing a given type A0 molecule, emitted from the TX at

t = 0, inside VRXkat time t, P(TX,RXk)

ob,0 (t). Given independent molecular behavior and assuming

that the RXs are sufficiently far from the TX, we use [92, Eq. (1)] to write P(TX,RXk)ob,0 (t) as

P(TX,RXk)ob,0 (t) =

VRXk

(4πD0t)3/2 exp

(−

d2TXk

4D0t

), (2.2)

where D0 is the diffusion coefficient of type A0 molecules in m2

s and dTXkis the distance between

the TX and RXk in m.

We denote S(TX,RXk)ob,0 [ j] as the sum of the number of molecules observed within VRXk

in the

jth symbol interval, due to the emission of molecules from the current and previous symbol

intervals at the TX, W jTX. As discussed in [94], S(TX,RXk)

ob,0 [ j] can be accurately approximated by

a Poisson RV with the mean given by

S(TX,RXk)ob,0 [ j] = S0

j

∑i=1

WTX[i]MRX

∑m=1

P(TX,RXk)ob,0 (( j− i)T +m∆tRX) . (2.3)

We also denote Uz,k[ j], z ∈ 0,1, as the conditional mean of S(TX,RXk)ob,0 [ j] when the most

recent information symbol transmitted by the TX is WTX[ j] = z. Then, the decision at RXk in

the jth symbol interval is given by

WRXk[ j] =

1, if S(TX,RXk)ob,0 [ j] ≥ ξRXk

,

0, otherwise.(2.4)

4In this chapter, the notations for the symbol interval time, the number of molecules for symbol “1” released bythe TX, and the sampling schedules of the RXs in the perfect reporting scenario are the same as those in the noisyreporting scenario.


Moreover, based on [94, Eq. (9)], the expected MDP of the TX−RXk link in the jth

symbol interval for given W j−1TX is written as

Pmd,k[ j] = Pr(

S(TX,RXk)ob,0 [ j] < ξRXk

∣∣∣WTX[ j] = 1,W j−1TX

), (2.5)

and the corresponding expected FAP is written as

Pfa,k[ j] = Pr(

S(TX,RXk)ob,0 [ j] ≥ ξRXk

∣∣∣WTX[ j] = 0,W j−1TX

). (2.6)

Based on [29, Eq. 2.5], the CDF of S(TX,RXk)ob,0 [ j] is given by

Pr(

S(TX,RXk)ob,0 [ j] < ξRXk

∣∣∣W jTX

)= exp

(−S(TX,RXk)

ob,0 [ j]) ξRXk−1

∑ω=0

S(TX,RXk)ob,0 [ j]ω

ω !. (2.7)

Using (2.7) and its complementary function, we can find the closed-form expressions for

Pmd,k[ j] and Pfa,k[ j].

2.2.1.2 Global Error Probability

We consider the cooperative MC system having a symmetric topology such that the RXs

have independent and identically distributed observations. Under this consideration, we have

Uz,k[ j] =Uz[ j]. Accordingly, we assume that the decision thresholds at the RXs are the same,

i.e., ξRXk= ξRX. Thus, we have Pmd,k[ j] = Pmd[ j] and Pfa,k[ j] = Pfa[ j].

We first consider the N-out-of-K fusion rule. Using [13, Eq. (3.4.30)] and [13, Eq. (3.4.31)]

we evaluate Qmd[ j] as

Qmd[ j] = 1−K

∑n=N

(Kn

)(1−Pmd[ j])n Pmd[ j]K−n (2.8)

and evaluate Qfa[ j] as

Qfa[ j] =K

∑n=N

(Kn

)Pfa[ j]n (1−Pfa[ j])K−n . (2.9)

For the OR rule, we obtain Qmd[ j] and Qfa[ j] by substituting N = 1 into (2.8) and (2.9), leading

to

Qmd[ j] = Pmd[ j]K (2.10)


and

Qfa[ j] = 1− (1−Pfa[ j])K , (2.11)

respectively. For the AND rule, we obtain Qmd[ j] and Qfa[ j] by substituting N = K into (2.8)

and (2.9), resulting in

Qmd[ j] = 1− (1−Pmd[ j])K (2.12)

and

Qfa[ j] = Pfa[ j]K , (2.13)

respectively.

We note that the single-RX MC system, which consists of one TX, one RX, and one FC, is

a special case of the cooperative MC system. Therefore, the expected error probability of the

single-RX MC system in the jth symbol interval for a given TX sequence W j−1TX in the perfect

reporting scenario, Pe,1[ j], can be obtained by setting K = 1 in (2.8) and (2.9). Accordingly,

the expected average error probability of the single-RX MC system, Pe,1, can be obtained by

averaging Pe,1[ j] over all possible realizations of W j−1TX and across all symbol intervals.

2.2.2 Noisy Reporting

In this subsection, we first examine the error performance of the TX−RXk−FC link, based

on the analysis in [94, 95]. We then use the results of this examination to analyze Qmd[ j] and

Qfa[ j] in the noisy reporting scenario, enabling us to obtain QFC[ j].

2.2.2.1 TX−RXk−FC Link

We denote P(RXk ,FC)ob,k (t) as the probability of observing a given Ak molecule, emitted from the

RXk at t = 0, inside VFC at time t. Due to the relatively close distance between the RXs and the

FC, we find that (2.2) or [92, Eq. (1)] cannot be used to evaluate P(RXk ,FC)ob,k (t). Thus, we resort

to [95, Eq. (27)] to evaluate P(RXk ,FC)

ob,k (t) as

P(RXk ,FC)ob,k (t) =

12[erf (τ1)+ erf (τ2)]−

√Dkt

dFCk

√π

[exp(−τ

21)− exp

(−τ

22)]

, (2.14)

where τ1 =rFC+dFCk

2√

Dkt , τ2 =rFC−dFCk

2√

Dkt , Dk is the diffusion coefficient of type Ak molecules in m2

s ,

and dFCkis the distance between RXk and the FC in m.

We denote S(RXk ,FC)ob,k [ j] as the number of molecules observed within VFC in the jth symbol

interval, due to the emissions of molecules from the current and the previous symbol intervals


at RXk, W jRXk

. We note that the TX and RXk use the same modulation method and the TX−RXk and RXk− FC links are both diffusion-based. Therefore, S(RXk ,FC)

ob,k [ j] can be accurately

approximated by a Poisson RV. We denote S(RXk ,FC)ob,k [ j] as the mean of S(RXk ,FC)

ob,k [ j] and obtain it

by replacing S0, WTX[i], P(TX,RXk)ob,0 , MRX, m, and ∆tRX with Sk, WRXk

[i], P(RXk ,FC)ob,k , MFC, m, and ∆tFC in

(2.3), respectively. We define Vz,k [ j], z∈ 0,1, as the conditional mean of S(RXk ,FC)ob,k [ j] when the

most recent information symbol transmitted by RXk is WRXk[ j] = z, given previous decisions

at RXk, W j−1RXk

. Furthermore, we note that WFCk [ j] can be obtained by replacing S(TX,RXk)ob,0 [ j] and

ξRXkwith S(RXk ,FC)

ob,k [ j] and ξFCkin (2.4), respectively.

We now derive the expected MDP and FAP of the TX−RXk−FC link in the jth symbol

interval averaging over all possible realizations of W j−1RXk

for given W j−1TX , denoted by Pmd,k[ j]

and Pfa,k[ j], respectively. For a given W j−1TX , there are 2( j−1) different possible realizations of

W j−1RXk

. We defineW j as the set containing all realizations of W j−1RXk

. Considering all possible

realizations of W j−1RXk

and their likelihood of occurrence, we first derive Pmd,k[ j] and Pfa,k[ j] as

Pmd,k[ j] = ∑W j−1

RXk∈W j

[Pr(

W j−1RXk

∣∣∣W j−1TX

)×(

Pr(

S(TX,RXk)ob,0 [ j] ≥ ξRX

∣∣∣WTX[ j] = 1,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] < ξFCk

∣∣∣WRXk[ j] = 1,W j−1

RXk

)+Pr

(S(TX,RXk)

ob,0 [ j] < ξRX

∣∣∣WTX[ j] = 1,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] < ξFCk

∣∣∣WRXk[ j] = 0,W j−1

RXk

))](2.15)

and

Pfa,k[ j] = ∑W j−1

RXk∈W j

[Pr(

W j−1RXk

∣∣∣W j−1TX

)×(

Pr(


∣∣∣WTX[ j] = 0,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] ≥ ξFCk

∣∣∣WRXk[ j] = 1,W j−1

RXk

)+Pr

(S(TX,RXk)

ob,0 [ j] < ξRX

∣∣∣WTX[ j] = 0,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] ≥ ξFCk

∣∣∣WRXk[ j] = 0,W j−1

RXk

))],

(2.16)

respectively. Considering that the cooperative MC system having a symmetric topology, each

RXk has independent and identically distributed (i.i.d.) observations over each TX−RXk−FC

link. Under this consideration, we have Vz,k[ j] = Vz[ j]. Accordingly, we assume that the

decision thresholds at the FC over RXs-FC links are the same, i.e., ξFCk= ξFC. Thus, in (2.15)

and (2.16), the likelihood of occurrence of each realization of W j−1RXk

is the same for each RXk.

Also, the conditional MDP and FAP for the given realization is the same for each RXk. This

indicates that Pmd,k[ j] and Pfa,k[ j] are the same for all RXs, i.e., Pmd,k[ j] = Pmd[ j] and Pfa,k[ j] =

Pfa[ j]. We note that the high complexity caused by considering 2( j−1) possible realizations of


W j−1RXk

and their likelihood of occurrence make the evaluation of (2.15) and (2.16) cumbersome.

To facilitate the calculation of (2.15) and (2.16), we consider only one possible realization

of W j−1RXk

and refer to this considered realization as the candidate. By only considering the

candidate of W j−1RXk

, we then approximate (2.15) and (2.16) as

Pmd[ j] ≈ Pr(


∣∣∣WTX[ j] = 1,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] < ξFC

∣∣∣WRXk[ j] = 1,W j−1

RXk

)+Pr

(S(TX,RXk)

ob,0 [ j] < ξRX

∣∣∣WTX[ j] = 1,W j−1TX

)Pr(


∣∣∣WRXk[ j] = 0,W j−1

RXk

)(2.17)

and

Pfa[ j] ≈ Pr(


∣∣∣WTX[ j] = 0,W j−1TX

)Pr(

S(RXk ,FC)ob,k [ j] ≥ ξFC

∣∣∣WRXk[ j] = 1,W j−1

RXk

)+Pr

(S(TX,RXk)

ob,0 [ j] < ξRX

∣∣∣WTX[ j] = 0,W j−1TX

)Pr(


∣∣∣WRXk[ j] = 0,W j−1

RXk

),

(2.18)

where the candidate of W j−1RXk

can be obtained using a biased coin toss method. Particularly,

we model the ith decision at RXk, WRXk[i], as WRXk

[i] = |λ −WTX[i]|, where i ∈ 1, · · · , j− 1and λ ∈ 0,1 is the outcome of the coin toss with Pr(λ = 1) = Pmd,k[i] if WTX[i] = 1 and

Pr(λ = 1) = Pfa,k[i] if WTX[i] = 0. We assume that the candidate of W j−1RXk

in (2.17) and (2.18)

is the same at all RXs to ensure that Pmd,k[ j] = Pmd[ j] and Pfa,k[ j] = Pfa[ j] are still valid after

adopting the approximations of Pmd[ j]and Pfa[ j] in (2.17) and (2.18), respectively. We clarify

that the candidate is only considered for the theoretical evaluation of system error performance,

i.e., the calculation of (2.17) and (2.18). We emphasize that we do not consider the same


at all RXs in our system model. Furthermore, in our simulations, each

RX makes decisions independently and the realizations of W j−1RXk

at all RXs are not necessarily

identical. Our simulation results in Section V demonstrate the accuracy of (2.17) and (2.18).

In addition, the CDF of S(RXk ,FC)ob,k [ j] is obtained by replacing S(TX,RXk)

ob,0 [ j], ξRXk, W j

TX and

S(TX,RXk)ob,0 [ j] with S(RXk ,FC)

ob,k [ j], ξFC, W jRXk

, S(RXk ,FC)ob,k [ j] in (2.7), respectively. Using the CDFs of

S(TX,RXk)ob,0 [ j] and S(RXk ,FC)

ob,k [ j] and their complementary functions, we can find the closed-form

expressions for Pmd,k[ j] and Pfa,k[ j].

2.2.2.2 Global Error Probability

In the noisy reporting scenario, we obtain Qmd[ j] and Qfa[ j] for the N-out-of-K rule, OR rule,

and AND rule by replacing Pmd[ j] and Pfa[ j] with Pmd[ j] and Pfa[ j], respectively, in (2.8)–

(2.13). We also note that the expected error probability of the single-RX MC system in the jth

symbol interval for a given TX sequence W j−1TX in the noisy reporting scenario can be obtained


by setting K = 1.

We note that the expected error probabilities of the cooperative MC system are also tractable

if weighted sum detectors with different weights are considered at the RXs and FC for detec-

tion. Under this consideration, the mth sample at RXk or FC in the jth symbol interval can

be accurately approximated by a Poisson RV. Although the weighted sums of Poisson RVs,

S(TX,RXk)ob,0 [ j] and S(RXk ,FC)

ob,k [ j], are not Poisson RVs, the weighted sums of Gaussian approxima-

tions of the individual variables are Gaussian RVs. Thus, we can write the CDF of the Gaus-

sian RVs and complementary functions to derive Pmd,k[ j], Pfa,k[ j], Pmd,k[ j], and Pfa,k[ j]. Then

we can derive the global error probabilities using (2.8)–(2.13) for the cooperative MC system

in the perfect and noisy reporting scenarios.

2.3 Error Performance Optimization

In this section, we present a novel analysis to determine the joint optimal ξRX and ξFC that mini-

mize the global error probability of the cooperative MC system. To this end, we first derive the

convex upper bounds on QFC[ j] for the OR rule, AND rule, and N-out-of-K rule5 in the perfect

and noisy reporting scenarios, allowing us to formulate the corresponding convex optimization

problems for given W j−1TX . We then extend the formulated convex optimization problems for

given W j−1TX to the convex optimization problems for the average error performance over all

possible realizations of W j−1TX and across all symbol intervals. This extension is due to two

reasons. First, optimizing the instantaneous error performance for given W j−1TX may not be fea-

sible in practice. This optimization mandates the precise knowledge of W j−1TX at RXk, which

cannot be realized in practice. Second, the repeated optimization of the detection threshold for

each realization of W j−1TX would demand a high computational overhead for RXk.

We note that finding the optimal thresholds at the RXs and FC via exhaustive search is

time-consuming and requires relatively high complexity, compared with the adopted convex

optimization. In the symmetric topology, the distances between the TX and the RXs are identi-

cal and the distances between the RXs and the FC are also identical. This results in independent

and identically distributed observations at the RXs. We clarify that the assumption of the sym-

metric topology is to improve the tractability of our convex analysis and this assumption will

be relaxed in future work. Also, for some practical applications, such as health monitoring, we

may manually set the locations of the RXs and FC to ensure that the topology of the TX (e.g.,

monitored organism), the RXs (e.g., detectors), and the FC, is symmetric and reduce the com-

plexity of system design and performance optimization. Also, the assumption of a symmetric

topology is reasonable if the difference in distance between different TX−RX−FC links is

5We clarify that the convex upper bounds for the OR rule, AND rule, and N-out-of-K rule are derived separately.This is due to the fact that the derived convex upper bounds for the N-out-of-K rule with N = 1 and N = K are notas tight as those derived for the OR rule and AND rule, respectively.

§2.3 Error Performance Optimization 31

negligible, compared to the distance between the TX and the RXs. We note that the expected

error probability of a point-to-point MC link is minimized in [94], by deriving a closed-form

analytical expression for the optimal decision threshold at the RX. However, the derived opti-

mal decision threshold in [94] cannot be applied or extended to the cooperative MC system.


In this subsection, we formulate the convex optimization problems with respect to ξRX for the

OR rule, AND rule, and N-out-of-K rule in the perfect reporting scenario. To achieve this, we

first analyze the convexity of Pmd[ j]K and Pfa[ j]K with respect to ξRX. Since S(TX,RX)ob,0 [ j] is a Pois-

son RV with a discrete distribution, its convexity analysis with respect to ξRX is cumbersome.

To overcome this cumbersomeness, we approximate the CDF of a Poisson RV X with mean

λ by the CDF of a continuous Gaussian RV. We find that the accuracy of this approximation

becomes higher when λ increases. Thus, the tightness of the approximation can be ensured by

any method achieving large λ , such as increasing the number of molecules released, increasing

the volume (radius) of RXs and FC, and choosing the optimal sampling period. Including a

continuity correction, the CDF of the Gaussian RV is given by

Pr (X < x) =12[1+Λ (x,λ )] , (2.19)

where Λ (x,λ ) = erf((x−0.5−λ )/

√2λ

). Applying (2.19) into (2.5) and (2.6), Pmd[ j] and

Pmd[ j] are approximated as

Pmd[ j] ≈ 12[1+Λ (ξRX,U1[ j])] (2.20)

and

Pfa[ j] ≈ 1− 12[1+Λ (ξRX,U0[ j])] , (2.21)

respectively. We now present the constraints making Pmd[ j]K and Pfa[ j]K convex in the follow-

ing theorem.

Theorem 2.1. Pmd[ j]K and Pfa[ j]K are convex with respect to ξRX, if we impose the following

convex constraints:

−0.5−U1[ j]+ ξRX ≤ 0 (2.22)

and

0.5+U0[ j]−ξRX ≤ 0, (2.23)


respectively.

Proof: See Appendix A.1.

We now analyze the convexity of Qfa[ j] and Qmd[ j] for the three rules. For the OR rule, an

upper bound on Qfa[ j] is given by

Qfa[ j] ≤ KPfa[ j], (2.24)

which is obtained by applying the first degree Taylor series approximation of 1− (1−Pfa[ j])K

into (2.11) at Pfa[ j] = 0. We find that this upper bound is tight when Pfa[ j] is small. We note that

Pfa[ j] is convex with respect to ξRX, if we impose the constraint (2.23), which can be proven

by considering K = 1 in Theorem 2.1. Thus, the upper bound in (2.24) is also convex with

respect to ξRX under the same constraint, since it scales a convex function with a nonnegative

constant. Also based on Theorem 2.1, Qmd[ j] for the OR rule, Pmd[ j]K , is convex with respect

to ξRX, if we impose the constraint (2.22). Therefore, the convex optimization problem for the

cooperative MC system with the OR rule in the perfect reporting scenario is formulated as

minξRX

P1Pmd[ j]K +(1−P1)KPfa[ j]

s.t. (2.22) and (2.23).(2.25)

Due to the convexity of the objective function and the constraints, (2.25) can be quickly

solved by efficient algorithms, e.g., the interior-point method [31]. Throughout this chapter, we

refer to the optimal threshold, i.e., the threshold in the feasible set that minimizes the objective

function, as the solution to the convex optimization problem, where the feasible set is the set

containing all of the thresholds that satisfy all constraints.

Next, we focus on the AND rule. Using a similar method as in (2.24), Qmd[ j] is upper-

bounded by

Qmd[ j] ≤ KPmd[ j]. (2.26)

We note that Pmd[ j] is convex with respect to ξRX under the constraint (2.22), which can

be proven by considering K = 1 in Theorem 2.1. Thus, (2.26) is also convex with respect to

ξRX under the same constraint. Based on Theorem 2.1, Qfa[ j] for the AND rule, Pfa[ j]K , is

convex respect to ξRX, if we impose the constraint (2.23). Therefore, the convex optimization

problem for the cooperative MC system with the AND rule in the perfect reporting scenario

can be formulated asminξRX

P1KPmd[ j]+ (1−P1)Pfa[ j]K

s.t. (2.22) and (2.23).(2.27)


Finally, we consider the N-out-of-K rule. We rewrite (2.8) as

Qmd[ j] =K

∑n=K−N+1

(Kn

)Pmd[ j]n (1−Pmd[ j])K−n . (2.28)

Based on (2.28) and (2.9), we verify that

Qmd[ j] ≤K

∑n=K−N+1

(Kn

)Pmd[ j]n , Q+

md[ j] (2.29)

and

Qfa[ j] ≤K

∑n=N

(Kn

)Pfa[ j]n , Q+

fa [ j]. (2.30)

In Theorem 2.1, we showed that Pmd[ j]K and Pfa[ j]K are convex with respect to ξRX, if we

impose the convex constraints (2.22) and (2.23), respectively. We note that Pmd[ j]n and Pfa[ j]n,

where n ∈ K−N + 1, . . .,K and n ∈ N, . . .,K, are also convex with respect to ξRX, if we

impose the convex constraints (2.22) and (2.23), respectively. The convexity of Pmd[ j]n and

Pfa[ j]n with respect to ξRX can be proven by replacing K with n and n in the proof to Theorem

2.1, respectively. Since (2.29) and (2.30) are nonnegative weighted sums of convex functions,

i.e., Pmd[ j]n and Pfa[ j]n, they are also convex with respect to ξRX under the same constraints.

Therefore, the convex optimization problem for the cooperative MC system with the N-out-of-

K rule in the perfect reporting scenario is formulated as

minξRX

P1Q+md[ j]+ (1−P1)Q+

fa [ j]

s.t. (2.22) and (2.23).(2.31)

We note that the convex optimization problem for the single-RX system in the perfect

reporting scenario is a special case of problems (2.25), (2.27), and (2.31), with K = 1.


In this subsection, we first extend the formulated convex optimization problems from the per-

fect reporting scenario to the noisy reporting scenario, assuming that ξFC is fixed. We then

formulate the joint convex optimization problems with respect to both ξRX and ξFC for the OR

rule, AND rule, and N-out-of-K rule.

2.3.2.1 Optimal ξRX

We first analyze the convexity of Pmd[ j]K and Pfa[ j]K with respect to ξRX. To facilitate the

convexity analysis of Pmd[ j]K and Pfa[ j]K with respect to ξRX, we approximate (2.17) and (2.18)


using (2.19), which result in

Pmd[ j] ≈ 14(2+(1+Λ (ξRX,U1[ j]))Λ (ξFC,V0[ j]) +(1−Λ (ξRX,U1[ j]))Λ (ξFC,V1[ j]))

(2.32)

and

Pfa[ j] ≈ 14(2− (1+Λ (ξRX,U0[ j]))Λ (ξFC,V0[ j]) +(−1+Λ (ξRX,U0[ j]))Λ (ξFC,V1[ j])) ,

(2.33)

respectively. Recall that Vz,k[ j], z ∈ 0,1, denotes the conditional mean of S(RXk ,FC)ob,k [ j] when

the most recent information symbol transmitted by the RXk is z. We find that Vz[ j] depends on

W j−1RXk

and W j−1RXk

depends on ξRX. Thus, Vz[ j] depends on ξRX, which complicates the convexity

analysis of Pmd[ j]K and Pfa[ j]K with respect to ξRX. To avoid this complication, we consider

a constant Vz[ j] in the jth symbol interval, denoted by V z[ j], which is averaged over all the


, to approximate Vz[ j] in (2.32) and (2.33)6. By doing so, we obtain V z[ j]

as

V z[ j] =1|ω j| ∑

W j−1RXk∈ω j

Vz[ j], (2.34)

where ω j is the set containing all realizations of W j−1RXk

and |ω j| denotes the cardinality of ω j.

Using V z[ j], we further approximate Pmd[ j] and Pfa[ j] as 7

Pmd[ j] ≈ 14[2+(1+Λ (ξRX,U1[ j]))Λ

(ξFC,V 0[ j]

)+(1−Λ (ξRX,U1[ j]))Λ

(ξFC,V 1[ j]

)](2.35)

and

Pfa[ j] ≈ 14[2− (1+Λ (ξRX,U0[ j]))Λ

(ξFC,V 0[ j]

)+(−1+Λ (ξRX,U0[ j]))Λ

(ξFC,V 1[ j]

)],

(2.36)

6We note that the occurrence likelihood of each realization of W j−1RXk

may not be the same in practice, since it

depends on the value of ξRX. For example, when ξRX is very high, W j−1RXk

would be all “0”s, while when ξRX is very

small, W j−1RXk

would be all “1”s. In this chapter, we assume an equal occurrence likelihood to keep a low evaluationcomplexity, but this does not have a significant impact on the analytical results.

7We clarify that we approximate Vz[ j] by V z[ j] in (2.32) and (2.33) to obtain (2.35) and (2.36) for the convexityanalysis. This approximation is not for deriving the expected MDP and FAP of each TX−RXk−FC link in the jthsymbol interval, i.e., Pmd,k[ j] and Pfa,k[ j], since (2.32) and (2.33) are already the approximations of Pmd,k[ j] andPfa,k[ j], respectively.


respectively. We now present the conditions making Pmd[ j]K and Pfa[ j]K convex in the follow-

ing theorem.

Theorem 2.2. Pmd[ j]K and Pfa[ j]K are convex with respect to ξRX when ξFC is fixed, if we

impose the convex constraints (2.22) and (2.23), respectively.

Proof: See Appendix A.2.

Similar to (2.24) and (2.26), we upper-bound Qfa[ j] for the OR rule and Qmd[ j] for the

AND rule as

Qfa[ j] ≤ KPfa[ j] (2.37)

and

Qmd[ j] ≤ KPmd[ j], (2.38)

respectively. We note that Pfa[ j] and Pmd[ j] are convex with respect to ξRX, if we impose

the constraints (2.22) and (2.23), respectively, which can be proven by considering K = 1 in

Theorem 2.2. Since (2.37) and (2.38) scale a convex function with a nonnegative constant,

they are also convex with respect to ξRX under the same constraints. Next, we focus on Qmd[ j]

for the OR rule and Qfa[ j] for the AND rule. Based on Theorem 2.2, we note that Pmd[ j]K and

Pfa[ j]K are convex with respect to ξRX when ξFC is fixed, if we impose the convex constraints

(2.22) and (2.23), respectively. Then, we focus on the N-out-of-K rule. We note that Pmd[ j]n

and Pfa[ j]n are convex with respect to ξRX when ξFC is fixed, if we impose the convex constraints

(2.22) and (2.23), respectively, which can be proven by replacing K with n and n in Theorem

2.2, respectively. For the N-out-of-K rule, using a similar method to (2.28)–(2.30), we can

derive the upper bounds on Qmd[ j] and Qfa[ j] that are convex with respect to ξRX, given that

Pmd[ j]n and Pfa[ j]n are convex with respect to ξRX.

In the noisy reporting scenario, we formulate the convex optimization problems with re-

spect to ξRX given fixed ξFC for the OR rule, AND rule, and N-out-of-K rule by replacing Pmd[ j]

and Pfa[ j] with Pmd[ j] and Pfa[ j], respectively, in (2.25), (2.27), and (2.31). We note that the

convex optimization problem with respect to ξRXkfor the single-RX system in the noisy report-

ing scenario is a special case of the corresponding problem for a cooperative MC system with

K = 1.

2.3.2.2 Joint Optimal ξRX and ξFC

We first analyze the joint convexity of Pmd[ j]K and Pfa[ j]K with respect to ξRX and ξFC. To

facilitate the joint convexity analysis of Pmd[ j]K and Pfa[ j]K with respect to both ξRX and ξFC,


we consider the approximations given by

Pr(


∣∣∣WRXk[ j] = 0,W j−1

RXk

)≈ 1 (2.39)

and

Pr(


∣∣∣WRXk[ j] = 1,W j−1

RXk

)≈ 1, (2.40)

which are tight when the error probability of the RXk−FC link is low. We emphasize that we

still keep

Pr(


∣∣∣WRXk[ j] = 1,W j−1

RXk

)(2.41)

and

Pr(


∣∣∣WRXk[ j] = 0,W j−1

RXk

)(2.42)

in (2.35) and (2.36), respectively. Employing (2.39) and (2.40) into (2.35) and (2.36), respec-

tively, we further upper-bound Pmd[ j] and Pfa[ j] as

Pmdb[ j] =14[3+Λ (ξRX,U1[ j]) +(1−Λ (ξRX,U1[ j]))Λ

(ξFC,V 1[ j]

)](2.43)

and

Pfab[ j] =14[3−Λ

(ξFC,V 0[ j]

)−(1+Λ

(ξFC,V 0[ j]

))Λ (ξRX,U0[ j])

], (2.44)

respectively, where Pmdb[ j] and Pfab[ j] are the upper bounds on Pmd[ j] and Pfa[ j], respectively.

We now present the constraints making Pmdb[ j]K and Pfab[ j]K convex in the following two

theorems.

Theorem 2.3. Pmdb[ j]K is jointly convex with respect to ξRX and ξFC, if we impose the convex

constraints (2.22), and the following constraints:

−0.5−V 1[ j]+ ξFC ≤ 0, (2.45)

Φ(ξRX,ξ+

FC ,K)≤ 0, and Φ

(ξ+RX,ξFC,K

)≤ 0, (2.46)

where ξ−RX and ξ+RX are bounds on ξRX, and ξ−FC and ξ+

FC are bounds on ξFC, and Φ (µ ,ν ,K) is


given by

Φ (µ ,ν ,K) = 4Θ(ξ+RX,U1[ j]

)(−4+K +KΛ

(ξ−FC,V 1[ j]

)+KΛ

(ξ−RX,U1[ j]

)×(1+Λ

(ξ−FC,V 1[ j]

)))2− (1+Λ (ξ−RX,U1[ j]))√U1[ j]V 1[ j]

(1+Λ

(ξ−FC,V 1[ j]

))×(

2 (−1+K)√

V 1[ j](1+Λ

(ξ+RX,U1[ j]

))−

√2π

Θ(ξ+FC ,V 1[ j]

)×(0.5+V 1[ j]−ν

)(−3+Λ

(ξ+FC ,V 1[ j]

)+Λ

(ξ+RX,U1[ j]

)×(1+Λ

(ξ+FC ,V 1[ j]

))))×(

Θ(ξ+RX,U1[ j]

)(1+Λ

(ξ−FC,V 1[ j]

))× (−1+K)2

√U1[ j]−

√2π (0.5+U1[ j]−µ)

×(−3+Λ

(ξ+FC ,V 1[ j]

)+Λ

(ξ+RX,U1[ j]

)(1+Λ

(ξ+FC ,V 1[ j]

)))). (2.47)

Theorem 2.4. Pfab[ j]K is jointly convex with respect to ξRX and ξFC, if we impose the convex

constraints (2.23) and the following constraints:

0.5+V 0[ j]−ξFC ≤ 0, (2.48)

Ψ(ξRX,ξ−FC,K

)≤ 0, and Ψ

(ξ−RX,ξFC,K

)≤ 0, (2.49)

where Ψ (µ ,ν ,K) is given by

Ψ (µ ,ν ,K) = 4Θ(ξ−RX,U0[ j]

)(−4+K−KΛ

(ξ+FC ,V 0[ j]

)+KΛ

(ξ+RX,U0[ j]

)×(−1+Λ

(ξ+FC ,V 0[ j]

)))2− (1−Λ (ξ+RX,U0[ j]))√

U0[ j]V 0[ j]

(−1+Λ

(ξ−FC,V 0[ j]

))×(−2 (−1+K)

√V 0[ j]

(−1+Λ

(ξ−RX,U0[ j]

))+

√2π

Θ(ξ−FC,V 0[ j]

)×(0.5+V 0[ j]−ν

)(−3−Λ

(ξ+FC ,V 0[ j]

)+Λ

(ξ−RX,U0[ j]

)×(−1+Λ

(ξ−FC,V 0[ j]

))))(Θ(ξ−RX,U0[ j]

)(−1+Λ

(ξ−FC,V 0[ j]

))× (−1+K)2

√U0[ j]−

√2π (0.5+U0[ j]−µ)

(−3−Λ

(ξ−FC,V 0[ j]

)+Λ

(ξ−RX,U0[ j]

)(−1+Λ

(ξ+FC ,V 0[ j]

)))). (2.50)

Proof: The proof of Theorem 2.3 and Theorem 2.4 is given in Appendix A.3.

Similar to (2.24) and (2.26), we upper-bound Qfa[ j] for the OR rule and Qmd[ j] for the


AND rule as

Qfa[ j] ≤ KPfab[ j] (2.51)

and

Qmd[ j] ≤ KPmdb[ j], (2.52)

respectively. We note that Pfab[ j] is convex with respect to ξRX and ξFC, if we impose the

constraints (2.23), (2.48), Ψ (ξRX,ξ−FC,1) ≤ 0, and Ψ (ξ−RX,ξFC,1) ≤ 0, which can be proven by

considering K = 1 in Theorem 2.4. We also note that Pmdb[ j] is convex with respect to ξRX

and ξFC, if we impose the constraints (2.22), (2.45), Φ (ξRX,ξ+FC ,1)≤ 0, and Φ (ξ+

RX,ξFC,1)≤ 0,

which can be proven by considering K = 1 in Theorem 2.3. Since (2.51) and (2.52) scale

a convex function with a nonnegative constant, they are also convex with respect to ξRX and

ξFC under the same constraints. We then focus on the joint convexity analysis of Qmd[ j] for

the OR rule and Qfa[ j] for the AND rule. Based on Theorem 2.3 and Theorem 2.4, we note

that Pmdb[ j]K and Pfab[ j]K are jointly convex with respect to ξRX and ξFC, respectively. For the

N-out-of-K rule, we note that Pmdb[ j]n is jointly convex with respect to ξRX and ξFC under the

constraints (2.22), (2.45), Φ (ξRX,ξ+FC , n) ≤ 0, and Φ (ξ+

RX,ξFC, n) ≤ 0, which can be proven by

replacing K with n in Theorem 2.3. We also note that Pfab[ j]n is jointly convex with respect

to ξRX and ξFC under the constraints (2.23), (2.48), Ψ (ξRX,ξ−FC,n) ≤ 0, and Ψ (ξ−RX,ξFC,n) ≤ 0,

which can be proven by replacing K with n in Theorem 2.4. Given that Pmdb[ j]n and Pfab[ j]n

are jointly convex with respect to ξRX and ξFC and applying a similar method to (2.28)–(2.30),

we can derive the upper bounds on Qmd[ j] and Qfa[ j] which are jointly convex with respect to

ξRX and ξFC.

In the noisy reporting scenario, we formulate the convex optimization problems with re-

spect to ξRX and ξFC for the OR rule, AND rule, and N-out-of-K rule as

minξRX, ξFC

P1Pmdb[ j]K +(1− P1)KPfab[ j]

s.t. (2.22), (2.23), (2.45)− (2.48),Ψ(ξRX,ξ−FC,1

)≤ 0, and Ψ

(ξ−RX,ξFC,1

)≤ 0,

(2.53)

minξRX, ξFC

P1KPmdb[ j]+ (1−P1) Pfab[ j]K

s.t. (2.22), (2.23), (2.45), (2.48), (2.49),Φ(ξ+RX,ξFC,1

)≤ 0, and Φ

(ξRX,ξ+

FC ,1)≤ 0,(2.54)


and

minξRX, ξFC

P1Q+md[ j]+ (1−P1)Q+

fa [ j]

s.t. (2.22), (2.23), (2.45), (2.48),

Φ(ξ+RX,ξFC, n

)≤ 0, Φ

(ξRX,ξ+

FC , n)≤ 0,Ψ

(ξRX,ξ−FC,n

)≤ 0, and Ψ

(ξ−RX,ξFC,n

)≤ 0,(2.55)

respectively, where Q+md[ j] , ∑

Kn=K−N+1 (

Kn)Pmdb[ j]n and Q+

fa [ j] , ∑Kn=N (K

n)Pfab[ j]n. We em-

phasize that the constraints Φ (ξ+RX,ξFC, n) ≤ 0, Φ (ξRX,ξ+

FC , n) ≤ 0, Ψ (ξRX,ξ−FC,n) ≤ 0,

Ψ (ξ−RX,ξFC,n) ≤ 0 for each n and n are applied in (2.55), where n ∈ K−N + 1, . . .,K and

n ∈ N, . . .,K, to ensure the convexity of Q+md[ j] and Q+

fa [ j]. We note that the jointly convex

optimization problem for the single-RX system in the noisy reporting scenario is a special case

of problems (2.53), (2.54), and (2.55), with K = 1.

2.3.3 Average Error Performance Optimization

We emphasize that the solutions to the formulated optimization problems in Sections 2.3.1

and 2.3.2 are the instantaneous suboptimal thresholds which minimize the instantaneous sys-

tem error performance for given W j−1TX . As previously explained, it may not be realistic for the

RXs and FC to calculate the instantaneous suboptimal thresholds and such calculation incurs

significant computational overhead. Therefore, in this subsection we aim to obtain a single

suboptimal threshold which optimizes the average system error performance over all possible

realizations of W j−1TX and across all symbol intervals.

If we aim to optimize QFC for the OR rule in the perfect reporting scenario, based on (2.25),

then we formulate the problem as

minimizeξRX

1L

L

∑j=1

(1|ω j|∑ω j

Pmd[ j]K +KPfa[ j]

)s.t. all constraints for all considered realizations of W j−1

TX in ω j

for each symbol interval.

(2.56)

The empirical average error performance of the system is optimized in (2.56), since we

assume that the occurrence likelihoods of the realizations of W j−1TX are equal. Using a formu-

lation similar to (2.56), we can extend all convex optimization problems for optimizing the

instantaneous system error performance to those for optimizing the average system error per-

formance. Also, since all the derived inequality constraint functions are affine, the constraints

define the lower limits and upper limits on ξRX and/or ξFC. We clarify that it is reasonable to

only consider the minimum upper limit and the maximum lower limit on ξRX and/or ξFC among

all the upper and lower limits.


2.4 Numerical Results and Simulations

In this section, we present numerical and simulation results to examine the error performance of

the cooperative MC system. The simulation results are generated by a particle-based stochastic

simulator, where we track the precise locations of all individual molecules over discrete time

steps. We clarify that all the approximations in Sections III and IV are only considered for

facilitating our theoretical analysis, i.e., the theoretical evaluation and optimization of error

performance. We do not adopt these approximations in our simulations. In our simulations, we

consider a cooperative system as described in Section II. In this section, we also demonstrate

the effectiveness of the solutions to our formulated convex optimization problems, referred to

as suboptimal solutions, by comparing them with the actual optimal solutions that minimize

the expected average error probability of the system. We use the fmincon solver in MATLAB

with the interior-point algorithm to obtain the suboptimal solutions. We clarify that the actual

optimal solutions are obtained via the exhaustive search of the numerical results of the expected

average error probability. Such solutions do not require the information of W j−1TX . We denote

ξ RX and ξ FC as suboptimal solutions and denote ξ ∗RX and ξ ∗FC as actual optimal solutions. We

refer to the minimum upper bounds achieved by ξ RX and ξ FC as suboptimal error probabilities.

We refer to the expected error probability achieved by ξ RX and ξ FC as the approximated error

probabilities.

We list all the fixed environmental parameters adopted in this section in Table 2.1. The

varying parameters adopted in this section are the decision threshold at RXs, ξRX, the decision

threshold at the FC, ξFC, the number of RXs, K, the radius of RXk, rRXk, and the radius of the

FC, rFC. In particular, rRXkis set as 0.225 µm in all the figures except for Fig. 2.3 and rFC is

fixed at 0.2 µm in all the figures except for Fig. 2.6. In Fig. 2.3, we set rRXkas 0.2 µm. In the

following, we assume that the TX releases S0 = 8000 molecules for information symbol “1”

and the total number of molecules released by all RXs for symbol “1” is fixed at 2000, i.e., each

RX releases Sk = 2000/K molecules to report its decision of symbol “1”. The locations of the

TX, RXs, and FC are listed in Table 2.2. For each realization of W j−1TX , we set ξ−RX =U0[ j]+1,

ξ+RX =U1[ j], ξ−FC = V 0[ j]+ 1, and ξ+

FC = V 1[ j], since the initial convex feasible sets of ξRX and

ξFC are 0.5+U0[ j] ≤ ξRX ≤ 0.5+U1[ j] and 0.5+V 0[ j] ≤ ξFC ≤ 0.5+V 1[ j], respectively.

Throughout this section, QFC are calculated by averaging Pe,k[ j] and QFC[ j], respectively,

over all considered realizations of W j−1TX and across all symbol intervals. Here, we consider all

possible realizations of W j−1TX except for the realization of all “0” bits, i.e., when the MDP is

zero and there is no optimal threshold. Since we consider the length of the symbol sequence

from the TX is 10 bits, we consider 1023 different symbol sequences in total. The simulated er-

ror probabilities are averaged over at least 5×104 independent transmissions of the considered

symbol sequences. In Figs. 2.2–2.4, we plot the simulation for the expected error probabilities,

while in Fig. 2.6, we plot the simulation for the approximated error probabilities. Moreover,

§2.4 Numerical Results and Simulations 41

Table 2.1: Fixed Environmental Parameters Used in Section 3.5

Parameter Symbol Value

Radius of RXs rRXk0.225 µm,0.2 µm

Radius of FC rFC 0.2 µm

Time step at RXs ∆tRX 100 µs

Time step at FC ∆tFC 30 µs

Number of samples of RXs MRX 5

Number of samples of FC MFC 5

Transmission time interval ttrans 1ms

Report time interval treport 0.3ms

Bit interval time T 1.3ms

Diffusion coefficient D0 = Dk 5×10−9m2/s

Length of symbol sequence L 10

Probability of binary 1 P1 0.5

Table 2.2: Locations of TX, RXs, and FC

Devices X-axis [µm] Y-axis [µm] Z-axis [µm]

TX 0 0 0

RX1 2 0.6 0

RX2 2 −0.6 0

RX3 2 −0.3 0.5196

RX4 2 −0.3 −0.519

RX5 2 0.3 0.5196

RX6 2 0.3 −0.5196

FC 2 0 0

we clarify that ξ RX and ξ FC for the expected average error probabilities are obtained using the

optimization method in Section 2.3.3 only once for all considered realizations of W j−1TX and

across all symbol intervals, unless otherwise noted. In other words, suboptimal solutions do

not require the information of W j−1TX , unless otherwise noted. Furthermore, we clarify that the

noninteger optimization solutions are rounded to integers in Figs. 2.3, 2.5, and 2.6. Specif-

ically, the most two nearest integers around the solution are compared and the one achieving

the lower error probability is chosen.


10-3

10-2

10-1

100

AND Rule

Majority Rule

OR Rule

Expected

Gaussian Approximation

Simulated

ξRX

Upper Bound

ξRX

0 2 4 6 8 10 12 14 16 18 20

QF

C

Figure 2.2: Average global error probability QFC of different fusion rules versus the decision thresholdat RXs ξRX with K=3 in the perfect reporting scenario.


In this subsection we consider the perfect reporting scenario. In Fig. 2.2, we consider a three-

RX cooperative system and plot the average global error probability versus the decision thresh-

old at the RXs for the OR rule, AND rule, and majority rule. The expected curves for the three

rules are obtained from (2.8)–(2.13) with (2.5) and (2.6). The Gaussian approximation curves

for the three rules are obtained from (2.8)–(2.13) with (2.20) and (2.21). The upper bound

curves for the OR rule, AND rule, and majority rule are obtained from (2.10) and (2.24),

(2.13) and (2.26), and (2.29) and (2.30), respectively, with (2.20) and (2.21). The value of ξ RX

for the OR rule, AND rule, and majority rule is obtained by solving (2.25), (2.27), and (2.31),

respectively, with (2.20) and (2.21).

In Fig. 2.2, we first observe that the simulated points accurately match the expected curves,

validating our analysis of the expected results. Second, we observe that ξ RX is almost identical

to ξ ∗RX for each fusion rule, confirming the accuracy of ξ RX. Third, we observe that the Gaussian

approximation curves well approximate the expected curves. Fourth, we observe that the con-

vex upper bound curve for the OR rule is lower than its expected curve. This can be explained

as follows: In the single-RX system, the Gaussian approximations give an upper bound on

Pmd[ j] and a lower bound on Pfa[ j]. For the OR rule, Qmd[ j] is the product of Pmd[ j] and Qfa[ j]

is the sum of Pfa[ j]. Since the Gaussian approximation of Qmd[ j] is tighter than that of Qfa[ j],

the Gaussian approximation of the global error probability for the OR rule is lower than the

expected curve. Finally, observing the expected curves, we find that the majority rule outper-

forms the OR rule and the OR rule outperforms the AND rule at their corresponding optimal

decision thresholds.

In Fig. 2.3, we plot the optimal average global error probability versus the number of co-


K

65432110

-4

10-3

10-2

QF

C*

AND Rule

OR Rule

Majority RuleExpected

Approximated

Simulated

Average Approximated

Figure 2.3: Optimal average global error probability Q∗FC of different fusion rules versus the number ofcooperative RXs K in the perfect reporting scenario.

operative RXs for the OR, AND, and majority rules. The baseline case is a single TX−RX

link with K = 1, i.e., only one RX exists but no FC exists. In the baseline case, we assume

that the RX is located at (2 µm,0.6 µm,0), the TX releases 10000 molecules, the time step

between two successive samples is 100 µs, and the symbol interval time is T = 1.3ms, all of

which ensure the fairness of the error performance comparison between the baseline case and

the considered cooperative MC system. We keep the total number of molecules released by

all RXs fixed for the fairness of error performance comparison between the baseline case and

the cooperative MC system with different K. Moreover, the fixed total number of molecules

applies to realistic biological environments where the number of available molecules within

the environment may be limited. Also, for a fair comparison of different K, we consider that

all RXs sample at the same time and has the same number of samples for different K, since the

sampling time tRX( j,m) determines the mean number of molecules observed, based on (2.2)

and (2.14). The value of Q∗FC for each K in the expected curves for the three fusion rules is

the minimum QFC. For the expected curves, we consider that a single ξ ∗RX is applied to all

considered realizations of W j−1TX , which are obtained via exhaustive search of the expected ex-

pressions of (2.8)–(2.13) with (2.5) and (2.6). On the other hand, the value of Q∗FC for each K in

the approximated curves for the OR rule, AND rule, and majority rule are obtained by solving

the corresponding average error performance optimization problems given by (2.25), (2.27),

and (2.31), respectively. To this end, we use a single ξ RX for all considered realizations of

W j−1TX , and then calculate the actual values of QFC achieved by ξ RX. The value of Q∗FC for each K

in the average approximated curves are obtained by solving (2.25), (2.27), and (2.31), respec-

tively, with (2.20) and (2.21) for all considered realizations. For this purpose, we consider a

single ξ RX for each realization of W j−1TX . Hence, for average approximated curves, the informa-


tion of W j−1TX is required for suboptimal solutions. We then calculate the actual value of QFC[ j]

achieved by ξ RX for each realization of W j−1TX , and refer to it as the instantaneous approximated

error probabilities. Finally, we calculate the mean of all the instantaneous approximated error

probabilities for all realizations of W j−1TX .

In Fig. 2.3, we first observe that for the OR rule and majority rule, the approximated curves

match the expected curves, which confirms the accuracy of ξ RX. Second, we observe that for the

AND rule, the approximated curve deviates from the expected curve when K = 5 and K = 6.

This is due to the fact that ξ RX is outside the feasible set restricted by all the constraints for all

realizations of W j−1TX . Third, we observe an accurate match between the simulated points and

the expected curves. Fourth, we observe that for the three fusion rules, the error performance

clearly improves when the optimization is performed for each realization of W j−1TX . However,

as previously explained, this performance gain may not be feasible in practice and thus, we

consider the average approximated curves as the best performance bound of our considered

system. Fifth, we observe from the expected curves that the majority rule outperforms the

OR rule and AND rule, which is consistent with that in Fig. 2.2. Lastly, we observe that the

cooperative MC system outperforms the baseline case for all fusion rules, even though the

distance of the baseline case is shorter than that of the cooperative MC system. Importantly,

we see that the system error performance significantly improves as K increases. This is due to

fact that an increasing number of cooperative RXs enables more independent observations of

the transmitted information symbol. It follows that the probability that all RXs fail to detect

the transmitted information symbol is reduced.

We clarify that if we keep the total volume of all RXs fixed, then the system error perfor-

mance degrades as K increases8. We note that a single TX-RX link with one RX has the same

error performance as our simple soft fusion rule proposed in [83], since both schemes have

the same mean number of molecules observed under the assumption of uniform concentration.

In simple soft fusion, the FC adds all RXs’ observations in the jth symbol interval and then

compares it with a decision threshold ξFC to make a decision WFC[ j] (see [83]). We then note

that the simple soft fusion rule outperforms the majority rule, since local hard decisions are a

quantization that decreases the granularity of the information available to the FC. Thus, for a

fixed total volume of RXs, the single TX-RX link outperforms the majority rule.


In this subsection we focus on the noisy reporting scenario. In Fig. 2.4, we consider a three-RX

cooperative system and plot the average global error probability versus the decision threshold

at the RXs for the AND rule, OR rule, and majority rule. In this figure, we consider ξFC = 2 for

the AND rule, ξFC = 4 for the OR rule, and ξFC = 3 for the majority rule, since these thresholds

8For more details about this clarification, please refer to [96].


AND Rule

Majority Rule

OR Rule

10-3

10-2

10-1

100

Expected

Simulated

ξRX

Upper Bound

ξRX

0 2 4 6 8 10 12 14 16 18 20

QF

C

Figure 2.4: Average global error probability QFC of different fusion rules versus the decision thresholdat RXs ξRX with K = 3 in the noisy reporting scenario.

Table 2.3: Coordinates and Values of ‘’ and ‘’ in Fig. 2.5

Variable OR Rule AND rule Majority Rule

ξ ∗FC of ‘’ 4 2 3

ξ ∗RX of ‘’ 9 4 7

ξ FC of ‘’ 3 2 2

ξ RX of ‘’ 9 4 7

Value of ‘’ 2.78×10−3 8.99×10−3 2.64×10−3

Value of ‘’ 3.22×10−3 8.99×10−3 3.01×10−3

are the values obtained when the thresholds at the RXs and FC are jointly optimized for the

three fusion rules. All curves in this figure are obtained from the same expressions and the

same optimization problems as those in Fig. 2.2, except for replacing (2.5), (2.6), (2.20), and

(2.21) with (2.17), (2.18), (2.35), and (2.36), respectively. Similar to Fig. 2.2, we observe that

ξ RX is almost identical to ξ ∗RX. By comparing Fig. 2.2 with Fig. 2.4, we also observe that the

expected error probabilities in Fig. 2.2 are slightly lower than those in Fig. 2.4. We further

observe that the optimal threshold at RXs is the same in Fig. 2.2 and Fig. 2.4. This observation

is not surprising, since the relatively short distance between RXk and the FC, which leads

to a relatively low error probability in the RXk − FC link. This low error probability does

not significantly affect the error probability of the TX−RXk−FC link. In addition, we also

confirmed that increasing K significantly improves the system error performance in the noisy

reporting scenario (figure omitted for brevity).

In Fig. 2.5, we consider a three-RX cooperative system and plot the expected average


108642005

10

10-3

10-2

10-1

15

ξRX

QFC

ξFC

(a) OR Rule

10

5

005

10

10-3

10-2

10-1

15

QFC

ξRX

ξFC

(b) AND Rule

10

5

00

5

10

10-3

10-2

10-1

15

QFC

ξRX

ξFC

(c) Majority Rule

Figure 2.5: Expected average global error probability QFC versus the decision threshold at RXs ξRX

and the decision threshold at the FC ξFC with K = 3 in the noisy reporting scenario for (a) OR rule, (b)AND rule, and (c) majority rule. In (a)–(c), ‘’ is the optimal QFC achieved by ξ ∗RX and ξ ∗FC, obtained byexhaustive search, and ‘’ is the approximated QFC achieved by ξ RX and ξ FC.

global error probability versus the decision thresholds at the RXs and FC for the OR rule, AND

rule, and majority rule in Fig. 2.5(a), Fig. 2.5(b), and Fig. 2.5(c), respectively. The expected


rFC [µm]

0.1250.150.1750.20.225

QF

C

10-2

10-1

Expected

Approximated

Simulation for Approximated

AND Rule

OR Rule

Majority Rule

*

Figure 2.6: Optimal average global error probability QFC of different fusion rules versus the radius ofthe FC rFC with K = 3 in the noisy reporting scenario.

surfaces for the three fusion rules are obtained from (2.8)–(2.13) with (2.17) and (2.18). The

values of ξ RX and ξ FC, associated with ‘’, for the OR rule, AND rule, and majority rule are

obtained by solving (2.53), (2.54), and (2.55), respectively. The coordinates and values of ‘’

and ‘’ in Figs. 2.5(a), 2.5(b), and 2.5(c) are summarized in Table 2.3. Based on Table 2.3,

we quantify the accuracy loss caused by the suboptimal convex optimization for the OR rule,

AND rule, and majority rule as 15.7%, 0%, and 14%, respectively. These small losses reveal

that the joint ξ RX and ξ FC we find can achieve near-optimal error performance.

In Fig. 2.6, we consider a three-RX cooperative system and plot the average global error

probability versus the radius of the FC for the AND rule, OR rule, and majority rule. The

value of Q∗FC for each rFC in the expected curves for the three fusion rules are obtained via the

exhaustive search of (2.8)–(2.13) with (2.17) and (2.18). The value of Q∗FC for each rFC in the

approximated curves for the three fusion rules are obtained by first solving (2.53), (2.54), and

(2.55), respectively, and then searching the actual values of QFC achieved by ξ RX and ξ FC. The

simulation for approximated curves are obtained by considering ξ RX and ξ FC for each rFC. We

observe that for the AND rule and majority rule, the approximated curves well approximate

the expected curves, which confirms the accuracy of jointly optimizing ξ RX and ξ FC. We also

observe that for the OR rule, the approximated curve deviates from the expected curve when

rFC = 0.225 and rFC = 0.175. This is due to the fact that the global error probability is very

sensitive to both thresholds in the region of ξ ∗FC. Furthermore, we observe that the approximated

curves match the expected curves when rFC ≤ 0.2µm for the AND rule, rFC ≤ 0.15µm for the

OR rule, and rFC ≤ 0.175µm for the majority rule. Additionally, we observe that the expected

error performance degrades as rFC decreases for all the fusion rules. This can be explained by

the fact that the reporting from the RXs to the FC becomes less reliable when rFC decreases.


2.5 Summary

In this chapter, we optimized the error performance achieved by cooperative detection among

distributed RXs in a diffusion-based MC system. For the perfect and noisy reporting scenar-

ios, we derived closed-form expressions for the expected global error probability of the system

having a symmetric topology. We also derived approximated expressions for the expected er-

ror probability in both reporting scenarios. We then found the convex constraints under which

the approximated expressions are jointly convex with respect to the decision thresholds at the

RXs and the FC. Based on the derived convex approximations and constraints, we formulated

suboptimal convex optimization problems for the system in both reporting scenarios. Further-

more, we extended the suboptimal convex optimization problem for the instantaneous error

performance to that for the average error performance over all TX symbol sequences. Using

numerical and simulation results, we showed that the system error performance can be signif-

icantly improved by combining the detection information among distributed RXs, even when

the total number of transmitted molecules is limited. We also showed that the suboptimal de-

cision thresholds, obtained by solving our formulated convex optimization problems, achieve

near-optimal global error performance.

Chapter 3

Symbol-by-Symbol ML Detection forCooperative MC

This Chapter presents symbol-by-symbol ML detection for a cooperative diffusion-based MC

system. In this system, the TX sends a common information symbol to multiple RXs and an

FC chooses the TX symbol that is more likely, given the likelihood of its observations from all

RXs. We consider the transmission of a sequence of binary symbols and the resultant inter-

symbol interference in the cooperative MC system. We propose three ML detection variants

according to different RX behaviors and different knowledge at the FC. We derive the system

error probabilities for two ML detector variants, one of which is in closed form. We deter-

mine the optimal molecule allocation among RXs to minimize the system error probability

of one variant by solving a joint optimization problem. Also for this variant, the equal dis-

tribution of molecules among two symmetric RXs is analytically shown to achieve the local

minimal error probability. Numerical and simulation results show that the ML detection vari-

ants provide lower bounds on the error performance of simpler, non-ML cooperative variants

and demonstrate that these simpler cooperative variants have error performance comparable to

ML detectors.

This chapter is organized as follows. In Section 3.1, we describe the system model and

analytical preliminaries for ML detection design and analysis. In Section 3.2, we present the

ML detection design for the cooperative MC system with all ML detection variants. In Section

3.3, we present the error performance analysis of the cooperative MC system. Numerical and

simulation results are provided in Section 3.5. In Section 3.6, we conclude and describe future

directions for this work.

3.1 System Model and Preliminaries

In this section, we present the system model (i.e., physical environment and general behaviors

of devices) for the cooperative MC system and some preliminary results that are needed in

Section 3.2. We will describe specific behaviors of the RXs and the FC for the ML detector

49

50 Symbol-by-Symbol ML Detection for Cooperative MC

RX1

(a) First Phase

TX

RX2

FC

RX1

(b) Second Phase

RX2

TXFC

Figure 3.1: An example of a cooperative MC system with 2 RXs. The transmission from the TX tothe RXs is represented by black dashed arrows. “D” and “A” denotes the RXs making decisions andamplifying observations, respectively, and Ak denotes the type of released molecule. The transmissionfrom the RXs to the FC in MD-ML, SD-ML, and SA-ML are represented by red, blue, and green arrows,respectively.

variants in Section 3.2.

3.1.1 System Model

We consider a cooperative MC system in unbounded 3D space. An example of the system

is illustrated in Fig. 3.1. We assume that all RXs and the FC are passive spherical observers.

Accordingly, we denote VRXkand rRXk

as the volume and radius of the kth RX, RXk, respectively,

where k ∈ 1,2, . . . ,K. We also denote VFC and rFC as the volume and radius of the FC,

respectively. We use the terms “sample” and “observation” interchangeably to refer to the

number of molecules observed by a RX or the FC at some time t and assume each observation

is independent of each other1. The symbol interval time from the TX to the FC is given by

T = ttrans + treport, where ttrans is the transmission interval time from the TX to the RXs and

treport is the report interval time from the RXs to the FC.

In the following, we describe the timing schedules and general behaviors of the TX, the

RXs, and the FC. An example of the timing schedule for the system is shown in Fig. 3.2.

Various methods can be adopted to achieve time synchronization2 among nanomachines, e.g.,

[92, 93]. Since binary symbols are the easiest to transmit and detect [90], and we assume that

the TX needs to send multiple bits of information in order to execute some complex task (such

as disease localization), we consider the transmission of a sequence of binary symbols and

account for the resultant ISI due to previous symbols at the TX and the RXs in the design and

analysis of the cooperative MC system.

1Intuitively, we consider the time between samples sufficiently large and the distances between the RXs suf-ficiently large for all individual observations to be independent. The validity of assuming independence will bedemonstrated by the excellent agreement between analytical and simulation results in Section 3.5.

2All RXs may not be perfectly synchronized. We make the assumption of identical sampling times at all RXsto get a bound on the best error performance achievable by a practical cooperative MC system.

§3.1 System Model and Preliminaries 51

T

Ȇ ѡ RX Ȇ ѡ FC

TX

transmits

TX[j]

R X јtransmits ъס [j]

RX ј takes

samples

FC takes

samples

t trans

RXјъ

… …1st Symbol

Interval

2nd Symbol

Interval

jth Symbol

Interval

Lth Symbol

Interval

Ȇ ѡ RX FCȆ ѡtreport

Figure 3.2: An example of the timing schedule for the system with MRX = 5 and MFC = 5.

TX: At the beginning of the jth symbol interval, i.e., ( j− 1)T , the TX transmits WTX[ j].

The TX transmits WTX[ j] to the RXs over the diffusive channel via type A0 molecules which dif-

fuse independently. The TX uses ON/OFF keying [90] to convey information, i.e., the TX re-

leases S0 molecules of type A0 to convey information symbol “1” with probability Pr(WTX[ j] =

1) = P1, but no molecules to convey information symbol “0”. The TX then keeps silent until

the start of the ( j+ 1)th symbol interval. We denote L as the number of symbols transmitted

by the TX. We define WlTX = WTX[1], . . . ,WTX[l] as an l-length subsequence of the symbols

transmitted by the TX, where l ≤ L. Throughout the chapter, W is a single symbol and Wis a vector of symbols. We do not consider channel codes for this system since the required

encoder and the decoder may not be practical for MC systems [40, 97].

RX: Each RXk observes type A0 molecules over the TX−RXk link and takes MRX samples3

in each symbol interval at the same times. The time of the mth sample by each RX in the jth

symbol interval is given by tRX( j,m) = ( j−1)T +m∆tRX, where ∆tRX is the time step between

two successive samples by each RX, m ∈ 1,2, . . . ,MRX. The RXs operate in half-duplex

mode, such that they do not receive the information and report their decisions at the same time.

This is because half-duplex mode is more appropriate in a biological environment since it

requires lower computational complexity than full-duplex mode. At the time ( j−1)T + ttrans,

each RX transmits molecules via a diffusion-based channel to the FC. For MD-ML and SD-

ML, each RX detects with a relatively simple energy detector [18]. We denote WRXk [ j] as

RXk’s binary decision on the jth transmitted symbol. Based on the energy detector, RXk

makes decision WRXk[ j] = 1 if sk[ j] ≥ ξRXk

, otherwise WRXk[ j] = 0, where sk[ j] is the value of

the realization of SRXkob [ j] and ξRXk

is the constant detection threshold at RXk, independent of

W j−1TX . We define Wl

RXk=

WRXk[1], . . . ,WRXk

[l]

as an l-length subsequence of RXk’s binary

decisions.

3We consider multiple samples at the RXs and the FC in each symbol interval to improve the detection perfor-mance.


FC: The FC takes the mth sample in the jth symbol interval at tFC( j, m) = ( j− 1)T +

ttrans + m∆tFC, where ∆tFC is the time step between two successive samples by the FC and

m∈ 1,2, . . . ,MFC. We denote WFC[ j] as the FC’s decision on the jth symbol transmitted by the

TX. We define WlFC =

WFC[1], . . . ,WFC[l]

as an l-length subsequence of the FC’s decisions on

the symbols transmitted by the TX. We denote WFCk[ j] as the FC’s estimated binary decision

of RXk on the jth transmitted symbol. We define WlFCk

=

WFCk[1], . . . ,WFCk

[l]

as the FC’s

estimate of the first l binary decisions by RXk.

3.1.2 Preliminaries

In this subsection, we establish some preliminary results for a TX−RXk link and a RXk−FC

link. We first evaluate the probability P(TX,RXk)ob (t) of observing a given type A0 molecule,

emitted from the TX at t = 0, inside VRXkat time t. Based on [95, Eq. (27)], we write P(TX,RXk)

ob (t)

as

P(TX,RXk)ob (t) =

12[erf (τ1)+ erf (τ2)]

−√

D0tdTXk

√π

[exp(−τ

21)− exp

(−τ

22)]

, (3.1)

where τ1 =rRXk+dTXk

2√

D0t , τ2 =rRXk−dTXk

2√

D0t , D0 is the diffusion coefficient of type A0 molecules in

m2/s, dTXkis the distance between the TX and RXk in m. We denote the sum of MRX samples

by RXk in the jth symbol interval by SRXkob [ j]. As discussed in [98, 99], SRXk

ob [ j] can be accurately

approximated by a Poisson RV. The mean of SRXkob [ j] is then given by

SRXkob [ j]=

j

∑i=1

S0WTX[i]MRX

∑m=1

P(TX,RXk)ob (( j− i)T+m∆tRX) . (3.2)

We denote P(RXk ,FC)ob,k (t) as the probability of observing a given Ak molecule, emitted from

the center of RXk at t = 0, inside VFC at time t. We obtain P(RXk ,FC)

ob,k (t) by replacing rRXk, dTXk

,

and D0 with rFC, dFCk, and Dk, respectively, where Dk is the diffusion coefficient of type Ak

molecules in m2/s and dFCkis the distance between RXk and the FC in m.

3.2 ML Detection Design and Derivation

In this section, we design and derive three symbol-by-symbol ML detectors, i.e., MD-ML, SD-

ML, and SA-ML. We summarize these variants in Table 3.1. We design the detectors according

to different relaying modes and numbers of types of molecules available at RXs. These variants

use either DF relaying or AF relaying and multi-type or single-type molecules. Generally, DF

outperforms AF [100] and multi-type outperforms single-type molecules, but assumptions of

§3.2 ML Detection Design and Derivation 53

Table 3.1: Variants of ML Detectors

Acronym Relayingat RXs

MoleculeType used

in RXs

Behaviorat FC

ComplexityComparison

MD-ML DF MultipleML

DetectionMD-ML>SD-ML>SA-MLSD-ML DF Single

MLDetection

SA-ML AF SingleML

Detection

Table 3.2: Illustration of the FC’s local history

Interval The FC’s decisions The FC’s local history

1 WFC[1] and WFCk[1] No History

2 WFC[2] and WFCk[2] WFC[1] and WFCk

[1]...

......

L WFC[L] and WFCk[L]

WFC[L−1], . . . ,WFC[1]

and WFCk[L−1], . . . ,WFCk

[1]

AF and single-type molecules are more realistic in biological environments.

Throughout this section, the FC uses its local history to choose the current symbol, i.e., the

FC evaluates the likelihood of the observations W j−1FC and W j−1

FCk(W j−1

FCkis not needed for SA-

ML) in the jth symbol interval, as shown in Table 3.2, where k ∈ 1,2, . . . ,K. Using the local

history at the FC, we formulate the general decision rule of ML detection in the jth interval as

WFC[ j] = argmaxWTX[ j]∈0,1

L[

j|WTX[ j],W j−1FC

](3.3)

or

WFC[ j] = argmaxWTX[ j]∈0,1

L[

j|WTX[ j],W j−1FC ,W j−1

FCk

], (3.4)

where we define L [ j|·] , Pr (FC’s observations in jth interval|·). Eq. (3.3) applies to SA-ML

and (3.4) applies to SD-ML and MD-ML. For simplicity, we also write the likelihoods in (3.3)

and (3.4) as L [ j]. In the following, we present the specific behaviors of the RXs and the

FC of each ML detector, derive the corresponding L [ j], and compare the complexities of the

detectors.

Although the ML detection requires high complexity, our system could be implemented


in a practical scenario for the following reasons: 1) We consider relatively simple RXs with

an energy detector or a signal amplifier. The computations required at the RXs can be imple-

mented at the molecular level [14]; 2) We keep the relatively high complexity required for ML

detection at the FC since it could have a direct interface with the macroscopic world and easier

access to computational resources; 3) The memory required at the FC may be implemented by

synthesizing a memory unit inside the FC [101]; 4) A modified cell and a synthetic oscillator

can be introduced into devices to release specific molecules and control the timing of molecule

release [102].

3.2.1 MD-ML

Each RXk in MD-ML transmits type Ak molecules, which can be independently detected by the

FC, to report WRXk [ j] to the FC. Similar to the TX, each RX uses ON/OFF keying to report its

decision to the FC and the RX releases Sk molecules of type Ak to convey information symbol

“1”. The FC receives type Ak molecules over the RXk−FC link and takes MFC samples of each

of the K types of molecules transmitted by all RXs in every reporting interval. The FC adds MFC

observations for each RXk−FC link in the jth symbol interval. We denote SFC,Dob,k[ j] as the total

number of Ak molecules observed within VFC in the jth symbol interval, due to both current and

previous emissions of molecules by RXk. The TX and RXk use the same modulation method

and the TX−RXk and RXk−FC links are both diffusion-based. Therefore, like SRXkob [ j], SFC,D

ob,k[ j]

can also be accurately approximated as a Poisson RV. We denote SFC,Dob,k[ j] as the mean of SFC,D

ob,k[ j].

Values of realizations of SFC,Dob,k[ j] are labeled sk[ j]. We assume that the K RXk−FC links are

independent, so the FC has K independent sums sk[ j] from the K RXk− FC links. The FC

chooses the symbol WFC[ j] that is more likely, given the joint likelihood of the K sums sk[ j] in

the jth interval. We obtain L [ j] by

L [ j] =K

∏k=1

[Pr(

WRXk[ j] = 1|WTX[ j],W j−1

FC

)Pr(

SFC,Dob,k[ j] = sk[ j]|WRXk

[ j] = 1,W j−1FCk

)+Pr

(WRXk

[ j] = 0|WTX[ j],W j−1FC

)Pr(

SFC,Dob,k[ j] = sk[ j]|WRXk

[ j] = 0,W j−1FCk

)]. (3.5)

For the evaluation of the likelihood in all future intervals, i.e., L [ j+ 1] , . . . ,L [L], the FC

also chooses the symbol WFCk[ j] in the jth interval given the likelihood of the sum sk[ j] from

the RXk−FC link in the jth interval. By doing so, WFCk[ j] is obtained by

WFCk[ j]= argmax

WRXk [ j]∈0,1Pr(SFC,D

ob,k[ j]= sk[ j]|WRXk[ j],W j−1

FCk

). (3.6)

Eqs. (3.5) and (3.6) can be evaluated by applying the conditional CDF of the Poisson RV

SRXkob [ j] and the conditional PMF of the Poisson RV SFC,D

ob,k[ j]. The conditional means SFC,Dob,k[ j]

§3.2 ML Detection Design and Derivation 55

given W j−1FCk

are obtained by replacing S0, WTX[i], P(TX,RXk)ob , MRX, m, and ∆tRX in (3.2) with Sk,

WFCk[i], P(RXk ,FC)

ob,k , MFC, m, and ∆tFC, respectively.

3.2.2 SD-ML

The behavior of each RXk in SD-ML is the same as that in MD-ML, except we assume that

each RXk transmits type A1 molecules to report WRXk [ j] to the FC. This is because it may not

be realistic for each RX to release a unique type of molecule. For simplicity, the number of

released type A1 molecules for each RXk in SD-ML is also denoted by Sk. The FC receives

type A1 molecules over all K RXk − FC links and takes MFC samples of type A1 molecules

in each symbol interval. The FC adds MFC observations for all RXk − FC links in the jth

symbol interval. We denote SFC,Dob [ j] as the total number of A1 molecules observed within

VFC in the jth symbol interval, due to both current and previous emissions of molecules by

all RXs. We note that SFC,Dob [ j] = ∑

Kk=1 SFC,D

ob,k[ j] is also a Poisson RV whose mean is given by

SFC,Dob [ j] = ∑

Kk=1 SFC,D

ob,k[ j]. Values of realizations of SFC,Dob [ j] are labeled s[ j]. The FC chooses the

symbol WFC[ j] that is more likely, given the likelihood of s[ j] in the jth interval. To facilitate

the evaluation of L [ j] for SD-ML, we define WFCl = WFC1 [l], . . . ,WFCK [l]. Using the notation

WFCl , we derive L [ j] as

L [ j] =2K

∑h=1

[Pr(WRX

j,h|WTX[ j],W j−1FC

)Pr(SFC,D

ob [ j] = s[ j]|WRXj,h,WFC

j−1, . . . ,WFC1)]

, (3.7)

where WRXj,h is the hth realization of the vector WRX1 [ j], . . . ,WRXK [ j], h ∈ 1,2, . . . ,2K. For

(3.7), we need to consider each WRXj,h and the corresponding probability leading to s[ j]. For the

evaluation of the likelihood in all future intervals, the FC chooses WFCj that gives the maximum

likelihood of s[ j]. By doing so, WFCj is obtained by

WFCj =argmax

WRXj,h

Pr(SFC,D

ob [ j]= s[ j]|WRXj,h,WFC

j−1, . . . ,WFC1). (3.8)

We now derive the conditional mean of SFC,Dob [ j] given WRX

j,h and WFCj−1, . . . ,WFC

1 . To this end,

we evaluate SFC,Dob [ j] as

SFC,Dob [ j] =

K

∑k=1

MFC

∑m=1

Sk

(WRXk

[ j]P(RXk ,FC)ob,k (m∆tFC)+

j−1

∑i=1

WFCk[i]P(RXk ,FC)

ob,k (( j− i)T + m∆tFC))

.

(3.9)

3.2.3 SA-ML

For SA-ML, each RX amplifies the number of molecules observed in the jth symbol interval,

i.e., SAk [ j] = αkSRXk

ob [ j], where SAk [ j] denotes the number of molecules released by RXk in the


jth symbol interval and αk is the constant amplification factor at RXk. The RXs retransmit

SAk [ j] molecules of type A1 to the FC at the same time. Since all RXs in both SA-ML and

SD-ML release molecules of the same type A1, the description of the behavior of the FC in SA-

ML is analogous to that in SD-ML. We denote SFC,Aob,k[ j] as the number of molecules observed

within VFC in the jth symbol interval, due to the emissions of molecules from the current and

the previous intervals by RXk. The TX−RXk and RXk−FC links are both diffusion-based.

Therefore, SFC,Aob,k[ j] can be accurately approximated as a Poisson RV. We denote SFC,A

ob,k[ j] as the

mean of SFC,Aob,k[ j]. The FC adds MFC observations for all RXk − FC links in the jth symbol

interval and this sum is denoted by the RV SFC,Aob [ j]. We note that SFC,A

ob [ j] = ∑Kk=1 SFC,A

ob,k[ j] is also

a Poisson RV whose mean is given by SFC,Aob [ j] = ∑

Kk=1 SFC,A

ob,k[ j]. Values of realizations of SFC,Aob [ j]

are labeled s[ j]. The FC chooses the symbol WFC[ j] that is more likely given the likelihood of

s[ j] in the jth interval and L [ j] is given as

L [ j] =S0

∑s1[1]=0

. . .S0

∑s1[ j]=0

. . .S0

∑sK [1]=0

. . .S0

∑sK [ j]=0

×Pr(

SRX1ob [1] = s1[1], . . . ,S

RX1ob [ j] = s1[ j], . . . ,SRXK

ob [1] = sK [1], . . . ,SRXKob [ j] = sK [ j]|WTX[ j],W j−1

FC

)×Pr

(SFC,A

ob [ j] = s[ j]|SRX1ob [1] = s1[1], . . . ,S

RX1ob [ j] = s1[ j], . . . ,SRXK

ob [1] = sK [1], . . . ,SRXKob [ j] = sK [ j]

),

(3.10)

where SRXkob [i] and sk[i], i ∈ 1, . . . , j and k ∈ 1, . . . ,K, are defined in Section 3.1.1.

Theoretically, any number of molecules between 0 and S0 can be observed at each RX.

Thus, there is a large number of realizations for each Poisson RV SRXkob [i] in (3.10), which

makes the complete evaluation of (3.10) cumbersome. To simplify the evaluation of (3.10),

we consider finitely many random realizations of each Poisson RV SRXkob [i]4. For example,

we generate 5000 random realizations of each SRXkob [i] for a given W j−1

FC , which is sufficient

to ensure the accuracy of (3.10). It is shown that (3.10) can be evaluated by applying the

conditional PMF of the Poisson RV SFC,Aob [ j]. We obtain the conditional mean of SFC,A

ob,k[ j] by

replacing S0WTX[i], P(TX,RXk)ob , MRX, m, and ∆tRX in (3.2) with SA

k [ j], P(RXk ,FC)ob,k , MFC, m, and ∆tFC,

respectively. Based on SFC,Aob [ j] = ∑

Kk=1 SFC,A

ob,k[ j], we can then obtain the conditional mean of

SFC,Aob [ j].

3.2.4 Comparison of Complexity

We summarize the complexity comparison in Table 3.1. MD-ML requires higher complexity

than SD-ML. This is because each RX releases a unique type of molecule in MD-ML, whereas

in SD-ML the RXs release a single type of molecule. SD-ML requires higher complexity

4We assume that the FC may have sufficiently high computational capabilities such that it can generate randomrealizations. This assumption is because the FC could have a direct interface to additional computational resources.


than SA-ML. This is because the RXs need to decode the TX’s symbols and the FC needs

to estimate the RXs’ decisions in SD-ML, but in SA-ML the RXs only need to amplify the

received signal and the FC does not need to estimate the RXs’ decisions.

3.3 Error Performance Analysis

In this section, we derive the error probability of SD-ML and SA-ML using the genie-aided

history, which leads to tractable expressions. Also, the error probability with genie-aided his-

tory provides a lower bound on that with local history. We denote QFC[ j] as the error probability

of the system in the jth symbol interval for a TX sequence W j−1TX . The closed-form expressions

of QFC[ j] for SD-ML with K = 1 and SA-ML are mathematically tractable.

To derive QFC[ j], we first derive equivalent decision rules with lower-complexity than (3.3)

and (3.4) for SD-ML and SA-ML in Theorems 3.1 and 3.2, respectively. The decision rules

when not all previously-transmitted symbols are “0” cannot be directly applied to the case

where all previously-transmitted symbols are “0”. Based on these theorems, the general forms

of these lower-complexity decision rules are that the FC compares the observation with adap-

tive thresholds when not all previously-transmitted symbols are “0” and the FC compares the

observation with 0 when all previously-transmitted symbols are “0”. Notably, these adaptive

thresholds adapt to different ISI in different symbol intervals.

3.3.1 SD-ML

We now derive QFC[ j] for the SD-ML variant. To this end, we first define λ DI [ j] as the expected

ISI at the FC in the jth symbol interval due to the previous symbols transmitted by all RXs,

W j−1RX1

,W j−1RX2

, . . . ,W j−1RXK

, i.e.,

λDI [ j] =

K

∑k=1

Sk

j−1

∑i=1

WRXk[i]

MFC

∑m=1

P(RXk ,FC)ob,k (( j− i)T + m∆tFC) . (3.11)

If not all previous symbols transmitted by all RXs are “0”. i.e.,j−1∑

i=1

K∑

k=1WRXk

[i] 6= 0, we have

λ DI [ j]> 0; otherwise, we have λ D

I [ j] = 0. We then define λD,tots,h [ j] as the total number of signal

molecules at the FC in the jth symbol interval due to the hth realization of currently-transmitted

RX symbols WRXj,h, i.e.,

λD,tots,h [ j] =

K

∑k=1

SkWRXk[ j]

MFC

∑m=1

P(RXk ,FC)ob,k (m∆tFC) . (3.12)


For the sake of brevity, for SD-ML, we define L[

j|WTX[ j] = 1,W j−1TX ,W j−1

RXk

], LSD

1 [ j] and

L[

j|WTX[ j] = 0,W j−1TX ,W j−1

RXk

], LSD

0 [ j]. Applying the conditional PMF of SFC,Dob [ j] to (3.7),

we write LSDb [ j] as

LSDb [ j] =

2K

∑h=1

[Pr(WRX

j,h|WTX[ j] = b,W j−1TX

)(s[ j]!)−1

×exp(−λ

DI [ j]− λ

D,tots,h [ j]

)(λ

DI [ j]+ λ

D,tots,h [ j]

)s[ j]]

, (3.13)

where b ∈ 0,1. Based on (3.13), we rederive the decision rule of SD-ML in (3.4) as a lower-

complexity decision rule in the following theorem.

Theorem 3.1. When λ DI [ j] > 0, the decision rule of SD-ML is WFC[ j] = 1 if s[ j] ≥ ξ

ad,SDFC [ j];

otherwise, WFC[ j] = 0, where ξad,SDFC [ j] is the solution to LSD

1 [ j] = LSD0 [ j] in terms of s[ j]. We

note that LSD1 [ j] = LSD

0 [ j] has a solution only when λ DI [ j] > 0. When λ D

I [ j] = 0, the decision

rule for SD-ML is WFC[ j] = 1 if s[ j] > 0 and WFC[ j] = 0 if s[ j] = 0,

Proof: See Appendix B.1.

Based on Theorem 3.1, when λ DI [ j] > 0, we evaluate the conditional QFC as

QFC

[j|λ D

I [ j] > 0]= P1Pr

(SFC,D

ob [ j] < ξad,SDFC [ j]|WTX[ j] = 1, λ D

I [ j] > 0)

+(1−P1)Pr(SFC,D

ob [ j]≥ξad,SDFC [ j]|WTX[ j]= 0, λ D

I [ j]>0)

, (3.14)

where the conditional CDF of the Poisson RV SFC,Dob [ j] can be evaluated by

Pr(

SFC,Dob [ j] < ξ

ad,SDFC [ j]|WTX[ j], λ D

I [ j])

=2K

∑h=1

[Pr(WRX

j,h|WTX[ j],W j−1TX

)exp(−λ

DI [ j]− λ

D,tots,h [ j]

)

×ξ

ad,SDFC [ j]−1

∑η=0

(λ

DI [ j]+ λ

D,tots,h [ j]

)η

/(η !)

], (3.15)

where λ DI [ j] can be evaluated by (3.11) via the approximated W j−1

RXk, k ∈ 1,2, . . . ,K. The ap-

proximated W j−1RXk

can be obtained using the biased coin toss method introduced in [94]. Specif-

ically, we model the ith decision at RXk, WRXk[i], as WRXk

[i] = |λ−WTX[i]|, where i∈ 1, . . . , j−1 and λ ∈ 0,1 is the outcome of the coin toss with Pr(λ = 1) = Pr

(WRXk

[i] = 0|WTX[i] = 1)

if WTX[i] = 1 and Pr(λ = 1) = Pr(WRXk

[i] = 1|WTX[i] = 0)

if WTX[i] = 0. When λ DI [ j] = 0, we


evaluate the conditional QFC as

QFC

[j|λ D

I [ j] = 0]= P1Pr

(SFC,D

ob [ j] = 0|WTX[ j] = 1, λ DI [ j] = 0

)+(1−P1)Pr

(SFC,D

ob [ j] > 0|WTX[ j] = 0, λ DI [ j] = 0

), (3.16)

where the conditional CDF of the Poisson RV SFC,Dob [ j] can be evaluated analogously to (3.15).

Combining (3.16) and (3.14), we obtain QFC[ j] for SD-ML as

QFC[ j] = Pr(

λDI [ j] > 0|W j−1

TX

)QFC

[j|λ D

I [ j] > 0]+Pr

(λ

DI [ j] = 0|W j−1

TX

)QFC

[j|λ D

I [ j] = 0]

.

(3.17)

Finally, we derive the closed-form expression for QFC[ j] for SD-ML with K = 1. To this

end, we first rewrite LSD1 [ j] and LSD

0 [ j] using (3.13) with K = 1. We then solve LSD1 [ j] =

LSD0 [ j] in terms of ξ

ad,SDFC [ j] and obtain the closed-form expression for ξ

ad,SDFC [ j] when K = 1 as

ξad,SDFC [ j] =

⌊λ D

s [ j]/log(

λ DI [ j]+ λ D

s [ j]/λ DI [ j]

)⌉. We note that QFC[ j] for SD-ML with K = 1

can be obtained using (3.17) via this expression.

3.3.2 SA-ML

We now derive QFC[ j] for SA-ML. In (3.10), multiple possible realizations of each Poisson

RV SRXkob [i] make the analytical error performance analysis cumbersome. To facilitate the error

performance analysis, we consider only one random realization of SRXkob [i] with the mean SRXk

ob [i]

for the given previous symbols transmitted by the TX, W j−1TX . We define λ A

I [ j] as the expected

ISI at the FC in the jth symbol interval due to W j−1TX . We define λ A

s [ j] as the number of the

signal molecules at the FC in the jth symbol interval due to WTX[ j] = 1. By modeling the

realization of SRXkob [i] as its mean SRXk

ob [i], we write λ AI [ j] and λ A

s [ j] as

λAI [ j] =

K

∑k=1

(j−1

∑i=1

αkSRXkob [i]

MFC

∑m=1

P(RXk ,FC)ob,k (( j− i)T + m∆tFC)

+αkS0

j−1

∑i=1

WTX[i]MRX

∑m=1

P(TX,RX)ob (( j− i)T +m∆tRX)

MFC

∑m=1

P(RXk ,FC)ob,k (m∆tFC)

)(3.18)

and

λAs [ j] =

K

∑k=1

αkS0

MRX

∑m=1

P(TX,RX)ob (m∆tRX)

MFC

∑m=1

P(RXk ,FC)ob,k (m∆tFC) , (3.19)

respectively. λ AI [ j] in (3.18) consists of two components. The first summation over i is the

expected ISI at the FC in the jth symbol interval due to the molecules released by the RXs but

without the amplification of the RXs’ ISI from the TX. The second summation over i accounts


for the amplification of the ISI in the jth symbol interval at all RXs due to W j−1TX . We note that

the conditional mean of SFC,Aob [ j] is λ A

s [ j] + λ AI [ j] when WTX[ j] = 1, and the conditional mean

of SFC,Aob [ j] is λ A

I [ j] when WTX[ j] = 0. If not all previous symbols transmitted by the TX are

“0”. i.e., W j−1TX 6= 0, then we have λ A

I [ j]> 0. If all previous symbols transmitted by the TX are

“0”, i.e., W j−1TX = 0, then we have λ A

I [ j] = 0. For the sake of brevity, for SA-ML, we define

L[

j|WTX[ j] = 1,W j−1TX

], LSA

1 and L[

j|WTX[ j] = 0,W j−1TX

], LSA

0 . Applying the conditional

PMF of the Poisson RV SFC,Aob [ j] to (3.10), we derive LSA

1 and LSA0 as

LSA1 =

(λ A

s [ j]+ λ AI [ j]

)s[ j]exp(−(

λ As [ j]+ λ A

I [ j]))

s[ j]!(3.20)

and

LSA0 =

(λ A

I [ j])s[ j]

exp(−λ A

I [ j])

(s[ j]!), (3.21)

respectively. Based on (3.20) and (3.21), we rewrite the general decision rule of SA-ML in

(3.3) as a lower-complexity decision rule in the following theorem.

Theorem 3.2. When λ AI [ j] > 0, the decision rule of SA-ML is WFC[ j] = 1 if s[ j] ≥ ξ

ad,SAFC [ j];

otherwise, WFC[ j] = 0, where ξad,SAFC [ j] =

⌊λ A

s [ j]/log(

λ AI [ j]+ λ A

s [ j]/λ AI [ j]

)⌉. When λ A

I [ j] =

0, the decision rule is WFC[ j] = 1 if s[ j] > 0 and WFC[ j] = 0 if s[ j] = 0.


Based on Theorem 3.2, when W j−1TX 6= 0, we evaluate QFC[ j] for SA-ML as

QFC[ j] = (1−P1)Pr(

SFC,Aob [ j] ≥ ξ

ad,SAFC [ j]|WTX[ j] = 0,W j−1

TX

)+P1Pr

(SFC,A

ob [ j] < ξad,SAFC [ j]|WTX[ j] = 1,W j−1

TX

), (3.22)

where Pr(

SFC,Aob [ j] < ξ

ad,SAFC [ j]|W j

TX

)can be evaluated by replacing W j−1

FC and SFC,Aob [ j] = s[ j]

in (3.10) with W j−1TX and SFC,A

ob [ j] < ξad,SAFC [ j], respectively. Similar to the evaluation of (3.10),

we consider finitely many random realizations of SRXkob [i] in (3.22). When W j−1

TX = 0, QFC[ j]

for SA-ML can be obtained by replacing ≥, <, and ξad,SAFC [ j] with >, =, and 0 in (3.22),

respectively.

3.4 Error Performance Optimization

In this section, we determine the optimal molecule distribution among RXs that minimizes

the error probability of SD-ML using the genie-aided history, inspired by the fact the quantity


of any type of molecule is usually constrained in practical biological environments. We also

analytically prove that the equal allocation of molecules among two symmetric RXs achieves

the local minimal error probability of SD-ML.

To this end, we first formulate the optimization problem as follows:

minS

QFC[ j] in (3.17)

s.t. S1 + S2 + · · ·+ SK−N = 0,

Sk ≥ 0,

(3.23)

where S = S1,S2, . . . ,SK, k ∈ 1,2, . . . ,K, and N is the total number of molecules released

by K RXs for symbol “1”. Combining (3.14) and (3.17), we note that ξad,SDFC [ j] is required to

evaluate QFC[ j]. Based on Theorem 3.1, the adaptive threshold ξad,SDFC [ j] is obtained by numer-

ically solving LSD1 [ j] = LSD

0 [ j] in terms of s[ j], while the closed-form expression for ξad,SDFC [ j]

is mathematically intractable. Therefore, there is no closed-form expression for QFC[ j], which

makes it very hard to optimize QFC[ j] in (3.17). To tackle this challenge, we find a closed-form

approximation for QFC[ j] in (3.17) by considering a constant threshold ξ in (3.14). By doing

so, we find the approximation of QFC[ j] as

Q]FC[ j] = P1

2K

∑h=1

[Pr(WRX

j,h|WTX[ j] = 1,W j−1TX

)Λ]

+(1−P1)2K

∑h=1

[Pr(WRX

j,h|WTX[ j] = 0,W j−1TX

)(1−Λ)

], (3.24)

where Q]FC[ j] is the approximation of QFC[ j], Λ is given by

Λ =ξ−1

∑η=0

exp(−λ

DI [ j]− λ

D,tots,h [ j]

) (λ DI [ j]+ λ

D,tots,h [ j]

)η

(η !), (3.25)

and ξ is a constant. In (3.25), λ DI [ j] and λ

D,tots,h [ j] are the functions of S based on (3.11) and

(3.12).

Lemma 3.1. The approximation of QFC[ j] by Q]FC[ j] is tight when ξ = ξ

ad,SDFC [ j].


Lemma 3.2. Since the adaptive threshold ξad,SDFC [ j] adapts to different ISI for different sym-

bol intervals, ξad,SDFC [ j] is the optimal ξ that minimizes Q]

FC[ j] if P1 = 12 , i.e., ξ

ad,SDFC [ j] =

argminξ

Q]FC[ j].



Based on Lemma 3.1 and Lemma 3.2, the approximation of QFC[ j] by Q]FC[ j] is tight when

ξ = ξad,SDFC [ j] and ξ

ad,SDFC [ j] is the optimal ξ which minimizes Q]

FC[ j]. Therefore, the optimal

S that minimizes QFC[ j] in (3.17) can be obtained by finding the jointly optimal S and ξ to

minimize Q]FC[ j] in (3.24), i.e., the approximate solution to the problem (3.23) can be obtained

by solving the optimization problem given by:

minS, ξ

Q]FC[ j]

s.t. S1 + S2 + · · ·+ SK−N = 0,

Sk ≥ 0.

(3.26)

To solve (3.26), we examine its convexity. The convexity of an optimization problem can

be proven by showing that its objective function and constraints are convex with respect to

the optimization variables. Since the constraints in (3.26) are affine, they are convex. The

convexity of the objective function, i.e., Q]FC[ j], can be proven by showing that its Hessian is

positive semidefinite with respect to its optimization variables. For the convexity of Q]FC[ j], we

have the following proposition:

Proposition 3.1. The Hessian of Q]FC[ j] is not positive semidefinite with respect to S and ξ .


Based on Proposition 3.1, the multi-dimensional optimization problem (3.26) is not a con-

vex optimization problem. To overcome this challenge, we use GlobalSearch in MATLAB to

repeatedly run a local solver with the sequential quadratic programming (SQP) algorithm until

convergence is achieved (i.e., the global minimum is found) to solve the problem (3.26). Our

numerical results in Section 3.5 confirm the effectiveness of this optimization method.

To obtain additional analytical insights in molecule distribution, we discuss the optimal

distribution of the number of molecules in a symmetric topology. Intuitively, we expect that an

equal distribution of molecules among symmetric RXs is the optimal allocation to minimize

the error probability. To confirm this conjecture, we first find that the equal distribution locally

minimizes Q]FC[ j] under certain conditions. We derive such conditions in the following Lemma:

Lemma 3.3. In the symmetric topology with K = 2, if Υ(ξ ) > 0, Q]FC[ j] achieves a local

minimum when S1 =N2 ; otherwise, it achieves a local maximum, where Υ(ξ ) is given by

Υ(ξ ) = (α(P1−1)+βP1) (2+N(ν + 2σ)−2dξe) , (3.27)


where

σ1 = σ2 = σ , ν1 = ν2 = ν ,

α(1,0) = α(0,1) = α , and β (1,0) = β (0,1) = β , (3.28)

σk =j−1

∑i=1

WRXk[i]

MFC

∑m=1

P(RXk ,FC)ob,k (( j− i)T + m∆tFC) , (3.29)

νk =MFC

∑m=1

P(RXk ,FC)ob,k (m∆tFC) , (3.30)

α(a1,a2) = Pr(

WRX1 [ j] = a1|WTX[ j] = 0,W j−1TX

)Pr(

WRX2 [ j] = a2|WTX[ j] = 0,W j−1TX

),

(3.31)

and5

β (a1,a2) = Pr(

WRX1 [ j] = a1|WTX[ j] = 1,W j−1TX

)Pr(

WRX2 [ j] = a2|WTX[ j] = 1,W j−1TX

).

(3.32)


Using Lemma 3.1, Lemma 3.2, and Lemma 3.3, we find that the equal distribution of

molecules always achieves the local minimal error probability for SD-ML in a two-RX system,

as stated in the following theorem:

Theorem 3.3. In the symmetric topology with two RXs, QFC[ j] achieves a local minimal value

when S1 =N2 if P1 =

12 .


Remark 3.1. If all channel parameters are available offline and approximately constant during

the whole transmission, then the optimization problem can be solved offline and the solution

can be used to set the optimal molecule allocation among RXs for the entire transmission,

as discussed in [103]. If some channel parameters are not available offline and may change,

(e.g., distances between devices may change due to mobility, and the diffusion coefficient may

5In the symmetric topology, σ1 = σ2 is valid because the observations at symmetric RXs are independently andidentically distributed (even though symmetric RXs may not necessarily make the same decisions). We need toconsider all possible realizations of W j−1

RXkat each RXk to evaluate Q]

FC[ j], but this requires high complexity. To

facilitate the calculation, we only consider one realization of W j−1RXk

at each RX and it is sufficiently accurate for the

evaluation of Q]FC[ j] to assume that this realization is the same for all RXs.


Table 3.3: Environmental Parameters

Parameter Symbol ValueVolume of each RX VRXk

43 ×π×0.23 µm3

Radius of FC rFC 0.2 µmTime step at RXs ∆tRX 100 µsTime step at FC ∆tFC 30 µs

Number of samples by RXs MRX 5Number of samples by FC MFC 10Transmission time interval ttrans 1ms

Report time interval treport 0.3msBit interval time T 1.3ms

Diffusion coefficient D0 = Dk 5×10−9m2/sLength of symbol sequence L 20

Probability of binary 1 P1 0.5

vary over time and space), then the optimization problem could be solved by a controller device

having higher computational capability than the RXs, as discussed in [104, 105]. The controller

device can use estimation methods discussed in [106, 107] to obtain the channel parameters

in the current symbol interval. It is reasonable to assume channel parameters remain constant

during each symbol interval if the interval duration is sufficiently small. Once the controller

device obtains the solution, it can be shared with the RXs to set the optimal molecule allocation

for the emission in the current symbol interval.


In this section, we present numerical and simulation results to examine the error performance

of the ML detectors. We simulate using a particle-based method considered in [8], where we

track the precise locations of all individual molecules. Unless otherwise noted, we consider

the environmental parameters in Table 3.3. Throughout this section, we keep the TX and the

FC fixed at (0µm,0µm,0µm) and (2µm,0µm,0µm), respectively. To clearly demonstrate

the impact of the number of samples and the number of RXs on the error probability of the

system, we consider a symmetric topology in Section 3.5.1. To clearly show the impact of

asymmetric RX location on the error probability of the system and the corresponding optimal

molecule distribution, we consider an asymmetric topology in Section 3.5.2.

We assume that the TX releases 104 molecules for symbol “1”. We also assume the total

number of molecules released by all RXs for symbol “1” is fixed at 2000 throughout this section

to ensure the fairness of error performance comparison for different K. For MD-ML and SD-

ML, in Figs. 3.3-3.5, each RX releases SD = b2000/Ke molecules to report a decision of “1”.

For SA-ML, in Figs. 3.3–3.5, each RX uses an amplification factor to ensure that the average


Table 3.4: Summary of Considered Variants

Variants Relayingat RXs

MoleculeType Used

in RXs

Behaviorat FC

Majority Rule [1, 83] DF Multiple Constant ThresholdMD-ML DF Multiple ML Detection

SD-Constant[2] DF Single Constant ThresholdSD-ML DF Single ML Detection

SA-Constant AF Single Constant ThresholdSA-ML AF Single ML Detection

number of molecules released by all RXs for transmission of one symbol is 1000 for the fair

comparison among SA-ML, SD-ML, and MD-ML. QFC is obtained by averaging QFC[ j] over

all symbol intervals and 50000 random-generated realizations of W j−1TX , and then the value of

Q∗FC is the minimum QFC found by numerically optimizing the corresponding constant decision

thresholds via exhaustive search. To decrease the complexity of exhaustive search, we consider

the same decision threshold at all RXs such that ξRXk= ξRX,∀k.

In Figs. 3.3-3.5, for each ML detection variant, we plot the error probability with the local

history and genie-aided history. We observe that the error performance using the local history

has a very small degradation from that using the genie-aided history. This demonstrates the

effectiveness of our proposed method to estimate the previous symbols. We also observe that

the simulations have very strong agreement with the analytical results, thereby validating our

analytical results. In Figs. 3.3-3.5, we observe that the error performance degradation with

the local history compared to the genie-aided history for SA-ML is more noticeable than that

for SD-ML and MD-ML. This is because in SD-ML and MD-ML, the FC directly estimates

previous RX symbols from the RX-FC links. However, for SA-ML, the FC does not directly

estimate the previous RX emissions from the RX-FC links and the error in the estimation of

previous TX symbols propagates to the estimated previous RX emissions.

3.5.1 Symmetric Topology

We consider at most 6 RXs in this subsection and the specific locations of RXs are:

(2µm,±0.6µm,0) and (2µm,±0.3µm,±0.5196µm), where the RXs are placed on a circle

perpendicular to the line passing from the TX, the FC, and the center of the circle.

In order to provide trade-offs between the performance versus the information available,

we compare the error performance of the ML detectors with the majority rule [1, 83] and SD-

Constant [2]. Notably, we also propose a new variant for comparison, namely, SA-Constant.

In SA-Constant, the behavior of each RX is the same as that in SA-ML, but the FC makes

a decision WFC[ j] by comparing s[ j] with a constant threshold ξFC, independent of W j−1TX . It


2 4 6 8 10

10-2

2 4 6 8 10

10-2

1 2 3 4 5 6

10-1

2 4 6 8 10

10-2

Simulation, Local HistorySimulation, Genie-aided HistoryAnalysisSimulation

SA-MLSD-MLMD-ML

MajorityRule

SA-ML

SA-ConstantSD-Constant

SD-ML

(a) (b)

(c) (d)

Figure 3.3: Optimal average error probability Q∗FC versus the number of samples by FC MFC for (a)SD-ML and SA-ML, (b) MD-ML and the majority rule, (c) SD-ML and SD-Constant, and (d) SA-MLand SA-Constant. The analytical error performance of the majority rule and SD-Constant is presentedin [1] and [2], respectively.

can be shown that QFC[ j] for SA-Constant with any realization of W j−1TX can be obtained by

replacing ξad,SAFC [ j] with the threshold ξFC in (3.22). We summarize all variants considered in

this subsection in Table 3.4. For these variants, we consider the same parameters as the ML

detectors for the fairness of our comparisons.

In Fig. 3.3, we plot the optimal average global error probability Q∗FC of different variants

versus the number MFC of samples by the FC. In Fig. 3.3, the report time interval is fixed at

treport = 0.3ms as in Table 3.3 and the time step at the FC for each MFC is ∆tFC = 0.3ms/MFC.

We observe that the system error performance improves as MFC increases. This is because when

MFC increases, the number of molecules expected to be observed at each RX increases.

In Fig. 3.3(a), we consider a single-RX system (which is analogous to the two-hop environ-

ment considered in [94]). We observe that SD-ML outperforms SA-ML. In Fig. 3.3(b)-(d), we

consider a three-RX system. We observe that MD-ML, SD-ML, and SA-ML outperform the

majority rule, SD-Constant, and SA-Constant, respectively. However, the error performance

degradation with these simpler cooperative variants are all within an order of magnitude for

the range of MFC considered. This demonstrates the relatively good performance of the simpler

variants.

In Fig. 3.4, we plot the optimal average global error probability versus the number K of

cooperative RXs for different variants. We see that the system error performance improves as

K increases, even though the total number of molecules is constrained. The same observation

of error performance improvement may be observed in a channel with additive signal depen-

dent noise if our results can be well approximated by the Gaussian signal dependent noise

model[108]. The system error performance does not always improve as K increases. This is


2 3 4 5 6

10-3

10-2

Simulation, Local HistoryAnalysisSimulation, Genie-aided HistorySimulation

2 3 4 5 6

10-3

10-2

Opt

imal

Ave

rage

Err

or P

roba

bilit

y

MD-ML

SD-ML

MajorityRule

SA-ML

SD-Constant

SD-ML

SA-ML

SA-Constant

(a) (b)

Figure 3.4: Optimal average error probability Q∗FC of different variants versus the number of RXsK. The analytical error performance of SD-Constant and the majority rule is presented in [2] and [1],respectively.

because if we keep increasing K, the number of released molecules for each RXk decreases,

which leads to the RXk−FC link becoming unreliable. The system error performance would

improve as the volume of the FC increases for the fixed K, since the FC can observe more

molecules, but the volume of microorganisms cannot be easily altered.

In Fig. 3.4(a), we observe that SD-Constant and SA-ML using the local history achieve

similar error performance. In Fig. 3.4(b), we observe that the majority rule has similar error

performance with SD-ML and the majority rule outperforms SA-ML using the local history.

These observations demonstrate the good performance of the majority rule, relative to SD-ML

and SA-ML. Importantly, we observe that MD-ML outperforms SD-ML and SD-ML outper-

forms SA-ML. This is because the knowledge of individual sk[ j] for each RXk−FC link in

MD-ML improves detection performance over only knowing the sum s[ j] in SD-ML. Compar-

ing to RXk making a binary decision in the current symbol interval in SD-ML, RXk in SA-ML

amplifies the ISI at RXk in the current symbol interval due to the previous TX symbols.

The system error performance in the subsection would degrade relative to the independent

case if any of the links become dependent. This can be explained by a special case where all

RXs overlap each other and thus have the same observations. Then, the error performance of

this case would be the same as that of a cooperative system with K = 1.

3.5.2 Asymmetric Topology

In Fig. 3.5, we consider a three-RX system and plot the optimal average error probability of

different variants versus the distance between the TX and RX3. We keep the positions of RX1

and RX2 fixed and move RX3 along the line segment between the symmetric position and the


0.418 (5)0.835 (4)1.253 (3)1.67 (2)2.088 (1)10

-3

10-2

AnalysisSimulation, Genie-aided HistorySimulation, Local History

MD-ML

SD-ML

SA-ML

Figure 3.5: Optimal average error probability Q∗FC of different variants versus the distance dTX3 betweenthe TX and RX3. RX1 and RX2 are fixed at (2µm,0,0.6µm) and (2µm,0,−0.6µm), respectively.The locations of RX3 are (1) (2µm,0.6µm,0), (2) (1.6µm,0.48µm,0), (3) (1.2µm,0.36µm,0), (4)(0.8µm,0.24µm,0), (5) (0.4µm,0.12µm,0).

TX, as indicated in the caption. We observe for our three variants that the error performance

first improves and then decreases as RX3 moves toward the TX. This is because both the TX-

RX3 link and the RX3-FC link contribute to the error performance of the system. When dTX3

is relatively large, the system error performance is dominated by the TX-RX3 link and this

link becomes more reliable as dTX3 decreases. For dTX3 is relatively small, the system error

performance is dominated by the RX3-FC link, which becomes weaker when dTX3 decreases.

We also observe that MD-ML outperforms SD-ML and SD-ML outperforms SA-ML, which

is consistent with our observations in Fig. 3.4(b).

In the following figures, we present results to assess the accuracy of our proposed optimiza-

tion method in Section 3.4. We denote the solution to problem (3.26) by S† = S†1,S†

2, . . . ,S†K.

We denote the optimal solution via exhaustive search by S? = S?1,S?2, . . . ,S?K.In Fig. 3.6, we consider a two-RX system and plot the error probability of SD-ML versus

the number of molecules released by RX1 for different location of RX1, where we keep RX2

fixed at (2µm,0.6µm,0µm) and move RX1 along the line segment between the symmetric

position and the TX, as indicated in the caption. The x-axis coordinate of is the solution S†1

to (3.26) and the corresponding y-axis coordinate is the Q]FC[ j] achieved at S†. We observe that

S†1 and Q]

FC[ j]|S=S† are almost identical to S?1 and QFC[ j]|S=S? , respectively, which confirms the

validity of Lemmas 3.1 and 3.2, the effectiveness of (3.26), and the accuracy of our method to

solve (3.26). In Fig. 3.6(a), we observe that S1 = 1000 achieves the minimal QFC[ j], which

verifies Theorem 3.3. Interestingly, we observe that from Figs. 3.6(a)–(d), when we move RX1

towards the TX, the optimal molecule allocation for RX1 first increases and then decreases.

This is because, when RX1 approaches to the TX, the TX−RX1− FC link becomes more

§3.6 Summary 69

0 1000 2000

0.01

0.02

0.03

0.040.05

0 1000 2000

0.01

0.02

0.03

0.04

0.05

0 1000 200010

-3

10-2

10-1

0 500 1000 1500 200010

-4

10-2

Solution to (34)Simulation,Genie-aided HistoryAnalysis

(a) (b)

(c) (d)

Figure 3.6: Error probability QFC[ j] of SD-ML versus the number of molecules released byRX1, S1, for different locations of RX1: (a) (2µm,0.6µm,0µm), (b) (1.5µm,0.45µm,0µm),(c) (1µm,0.3µm,0µm), (d) (0.5µm,0.15µm,0µm). The location of RX2 is fixed at(2µm,−0.6µm,0µm).

reliable, so increasing the number of molecules for RX1 optimizes the whole system; and

when RX1 is very close to the TX, the TX−RX1−FC link becomes less reliable due to a weak

RX1−FC link. In particular, in Fig. 3.6(c), the optimal solution is to allocate all molecules to

RX1. This is because when RX1 is at (1µm,0.3µm,0µm), RX1 is very close to the optimal

relay location, i.e., the midpoint between the TX and the FC, thus the TX−RX1−FC link is

much more reliable than the TX−RX2−FC link and allocating all molecules to RX1 optimizes

the whole system.

In Fig. 3.7, we consider a three-RX system and plot the error probability of SD-ML versus

the number of molecules released by RX1, RX2, and RX3. The locations of the three RXs are

generated randomly, as indicated in the caption. The x-axis, y-axis, and z-axis coordinates of

‘’ are the solutions S†1, S†

2, and S†3 to problem (3.26), respectively. The corresponding 4th

coordinate (i.e., color bar) is the Q]FC[ j] achieved at S†. We observe that S† and Q]

FC[ j]|S=S† are

almost identical to S? and QFC[ j]|S=S? , respectively, which again verifies Lemma 3.1, Lemma

3.2, and the effectiveness of problem (3.26).

3.6 Summary

Combined with our initial work in [84], we presented for the first time symbol-by-symbol ML

detection for the cooperative diffusion-based MC system with multiple communication phases.

We considered the transmission of a sequence of binary symbols and accounted for the resultant

ISI in the design and analysis of the system. We presented three ML detectors, i.e., MD-ML,

SD-ML, and SA-ML. For practicality, the FC chooses the current symbol using its own local


000

500

1000

1500

1000

2000

1000

20002000

AnalysisSolution to (34)

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Err

or

Pro

bab

ilit

y

Figure 3.7: Error probability QFC[ j] of SD-ML versus the number of molecules released by RX1, S1,the number of molecules released by RX2, S2, and the number of molecules released by RX3, S3. TheX-axis, Y-axis, and Z-axis coordinates of ‘’ are the solutions to problem (3.26). RX1, RX2, and RX3

are at (1.915µm,0.58µm,0), (1.827µm,0.579µm,0), and (1.265µm,0.328µm,0), respectively. Thex-axis and y-axis coordinates of the locations of the RXs are randomly generated.

history. For tractability, we derived the system error probabilities for SD-ML and SA-ML

using the genie-aided history. We formulated and solved a multi-dimensional optimization

problem to find the optimal molecule allocation among RXs that minimizes the system error

probability of SD-ML. We analytically proved that the equal distribution of molecules among

two symmetric RXs obtains the local minimal error probability of SD-ML. Using numerical

and simulation results, we corroborated the accuracy of these analytical expressions and the

effectiveness of the formulated optimization problem. Our results revealed trade-offs between

the performance, knowledge of previous symbols, the types of molecule, relaying modes, and

computational complexity.

The results of this chapter could be applied to health and environmental monitoring and

drug delivery scenarios. In these scenarios, the TX can be a nanomachine that transmits en-

vironmental sensor values, e.g., concentration, blood pressure, and temperature, or broadcasts

the location of a target site. Although the system topology design and general communication

processes can be adapted for traditional cooperative communications, our results cannot be

directly applied to traditional cooperative communications due to unique ISI, the propagation

channels, and the signal types in this work.

Chapter 4

Characterization of Cooperators inQS with 2D Molecular Signal Analysis

This Chapter analytically models a QS-based MC system by characterizing the diffusion and

degradation of signaling molecules. Different from prior studies, the motion of molecules un-

dergoing independent diffusion and degradation is taken into consideration. Microorganisms

are randomly distributed in a 2D environment where each one continuously releases molecules

at random times. We derive the 2D channel response at an observer due to the continuous

emission of molecules from one bacterium or randomly-distributed bacteria. Using the chan-

nel response, we then derive the exact and approximate expressions for the expected prob-

ability of cooperation at the bacterium at a fixed location due to the emission of molecules

from randomly-distributed bacteria. Based on such a probability, we derive the approximate

expressions for the MGF and different statistics of the number of cooperators. The analytical

results agree with simulation results where the Brownian motion of molecules is simulated by

a particle-based method. In addition, the Poisson and Gaussian distributions are compared to

approximate the PDF and CDF of the number of cooperators. Our results show that the Poisson

distribution provides the overall best approximation, especially when the population density is

low. Although we simplify the QS process for tractability, our model captures the basic fea-

tures of QS and accounts for the diffusion and degradation of molecules. The derived channel

response can be generally applied to any molecular communication model where single or mul-

tiple transmitters continuously release molecules into a 2D environment. The derived statistics

of the number of cooperative bacteria can be used to predict and control the QS process, e.g.,

predicting and the likelihood of biofilm formation in a particular environment and decreasing

this likelihood by adjusting environmental parameters such as the diffusion coefficient.

This chapter is organized as follows. In Section 4.1, we describe the system model. In

Section 4.2, we derive the channel response due to the continuous emission of molecules from

a point TX or randomly distributed TXs. These analyses lay the foundations for our derivations

of the observations at bacteria and the expected density of cooperators in the next subsection.

In Section 4.3, we derive the expected probability of cooperation at a bacterium at a fixed

71

72 Characterization of Cooperators in QS with 2D Molecular Signal Analysis

Figure 4.1: A population of bacteria randomly distributed on a circle according to a 2D spatial pointprocess, where each bacterium acts as a point TX and as a circular passive RX. The molecules diffuseinto and out of the bacteria.

location. Based on this probability, in Section 4.4, we derive the statistics of the number of

cooperative bacteria and the expected number of pairs of two nearest nodes both cooperating.

In Section 4.5, we finally present numerical and simulation results to validate the accuracy of

analytical results derived in Sections 4.2-4.4. In Section 4.6, we conclude and describe future

directions for this work.

4.1 System Model

We consider an unbounded 2D environment since a 2D environment facilitates future exper-

imental validation of our current theoretical work. Biological experiments, especially with

bacteria, are usually conducted in a 2D environment, e.g., bacteria residing on a petri dish and

the formation of biofilms [109]. A population of bacteria is spatially distributed on a bounded

circle S1 with radius R1 centered at (0,0) according to a 2D PPP with constant density λ , as

shown in Fig. 4.1. PPPs are commonly used to model randomly-distributed locations, e.g.,

[110] used a homogeneous PPP to model locations of nano-TXs and [111] used a PPP to

model locations of bacterial populations. We denote ~xi as the location of the center of the ith

bacterium. We denote Φ (λ ) as the set of random bacterial locations. We consider bacteria

behavior analogous to QS as shown in Fig. 1.3, i.e., 1) emit signaling molecules; 2) detect the

concentration of signaling molecules; and 3) decide to cooperate if the concentration exceeds

a threshold. In the following, we detail the emission, propagation, and reception of signaling

molecules, and decision-making by the bacteria.

Emission: We model bacteria as point TXs. The ith bacterium continuously emits A

molecules from ~xi at random times according to an independent one-dimensional PPP with

constant rate q molecule/s, as shown in Fig. 4.2.

Propagation: All A molecules diffuse independently with a constant diffusion coefficient


Figure 4.2: An example of release times due to continuous emission of molecules at a bacteriumaccording to a random process.

0 0.2 0.4 0.6 0.8 1 1.2Time [s]

5

10

15

Num

ber

of

Mole

cule

s

Simulation

Figure 4.3: The time-varying expected number of molecules observed N†agg (~xi, t|λ ) versus time t.

R1 = 20 µm, λ = 7.9×10−2/µm2, and ~xi = (10 µm,10 µm). For other simulation details, please seeSec. 3.5.

D and they can degrade into a form that cannot be detected by the bacteria, i.e., A k→ /0, where

k is the reaction rate constant in s−1. If k = 0, this degradation is negligible. Since we consider

a single type of molecules, we only mention “the molecules”, instead of “A molecules”, in the

remainder of this chapter.

Reception: We model the ith bacterium as a circular passive RX with radius R0 and

area S0 centered at ~xi. Bacteria perfectly count molecules if they are within S0. Since the

molecules released from all bacteria may be observed by the ith bacterium, the number of

molecules observed at the ith bacterium at time t, N†agg (~xi, t|λ ), is given by N†

agg (~xi, t|λ ) =∑~x j∈Φ(λ ) N (~xi, t|~x j), where N (~xi, t|~x j) is the number of molecules observed at the ith bacterium

at time t due to the jth bacterium. The means of N†agg (~xi, t|λ ) and N (~xi, t|~x j) are denoted by

N†agg (~xi, t|λ ) and N (~xi, t|~x j), respectively. We assume that the expected number of molecules

observed at the ith bacterium is constant after some time. To demonstrate the suitability of this

assumption, see Fig. 4.3 (and Remark 4.1 will discuss the validation of this assumption). In

Fig. 4.3, N†agg (~xi, t|λ ) is independent of t after time t = 0.5s. We denote time t?i as the time

after which N†agg (~xi, t|λ ) is constant, i.e.,

N†agg(~xi, t|λ )|t>t?i = lim

t→∞N†

agg(~xi, t|λ )=N†agg(~xi,∞|λ ) . (4.1)

Decision-Making: We assume that the ith bacterium can make observations after t?i to

make a decision, when the expected number of observed molecules becomes stable. Hence,


we refer to the stage when after t?i as the steady stage in the following of the chapter. This

assumption is reasonable since t?i is very small, e.g., t = 0.5s in Fig. 4.3, and bacteria can

reach the steady state very quickly, especially relative to the timescale of gene regulation to

coordinate behavior1. Also, bacteria can wait until there are enough molecules to trigger be-

havior change. Therefore, bacteria do not need to explicitly know whether the steady state

has been reached and the precise synchronization of emission and detection is not needed. In-

spired by QS, we consider a threshold-based strategy at bacteria to decide to cooperate or not.

We consider bacteria compare N†agg (~xi,∞|λ ) with a threshold η . If N†

agg (~xi,∞|λ ) ≥ η , the ith

bacterium decides to be a cooperator, otherwise the ith bacterium is noncooperative.

Our model captures the basic features of QS by adopting the following assumptions:

1. We consider bacteria that are randomly spatially distributed on a bounded circle since

the location of bacteria cannot be manually controlled in reality.

2. We consider bacteria that continuously2 emit molecules at random times since bacteria

emit molecules sporadically in reality, which captures the randomness of the stochastic

molecule release process.

3. We consider that each bacterium is both a TX and a RX which captures the features of

emission and reception of molecules at bacteria.

4. We adopt the same decision strategy at bacteria as QS, i.e., the concentration threshold-

based strategy.

5. Our model accounts for the random propagation of signaling molecules based on reaction-

diffusion equations.

We acknowledge the major simplifying assumptions that are made for our analysis to iden-

tify the applicability of our work and areas for future study. These assumptions are as follows:

1. We ignore the mobility of bacteria. This assumption is appropriate when bacteria swim

very slowly or for some non-motile bacteria, e.g., coliform and streptococci.

2. We assume that bacteria are passive observers since the observations at multiple bacteria

are correlated for reactive RXs, which makes analysis much more cumbersome.

3. We do not consider the death or birth of bacteria during the QS process.

1In practice, we are not able to show the value of t?i in every environmental setup, but t?i = 0.5s is a typicalvalue of t? since the values of environmental parameters used in Fig. 4.3 is chosen to be on the same orders ofthose used in the biological experiments, e.g., [112, 113, 114]. Also, based on [112, 113, 114], the cooperation ofbacteria is observed after the signaling molecules diffuse for at least tens of minutes.

2Note that continuous emission does not mean there is no time interval between two successive emissions ofmolecules. Instead, we assume the time interval is a RV and the expected interval length is inversely proportionalto the emission rate.

§4.2 2D Channel Response 75

4. We simplify bacteria as a point source emitting molecules and molecules can diffuse in

any directions in the environment. Considering emitting molecules from imperfect TXs

to a certain direction is a future work for MC.

5. We assume the average emission rate of molecules is constant. We acknowledge that in

real QS process, bacteria may increase the emission rate when they observe the higher

concentration of molecules or when they change from being selfish to being cooperat-

ing. This assumption is appropriate for the scenarios where bacteria goes from being

selfish to going to ramp up the molecule production before the emission rate has not

been changed.

6. Each bacterium makes one decision based on one sample of the observed signal. We

acknowledge that bacteria usually make decisions to cooperate multiple times in their

life. Modeling evolutionary or repeat behavior coordination over time with noisy signal

propagation is interesting for future work, as identified in [69].

4.2 2D Channel Response

In this section, we derive the channel response, i.e., the expected number of molecules observed

at a RX, due to continuous emission of molecules from TX(s), in the following cases: 1) a point

TX and 2) randomly distributed TXs. We assume that the RX is a circular passive observer S0

centered at~b with radius R0 throughout this section, unless specified otherwise. These analyses

lay the foundations for our derivations of the observations at bacteria and the expected density

of cooperators in Sec. IV.

4.2.1 One Point TX

In this subsection, we present the channel response due to one point TX. We also include

the special case when the TX is at the center of the RX, since each bacterium receives the

molecules released from not only other bacteria but also itself.

We recall that the emission times of molecules at each bacterium is distributed according

to an independent one-dimensional PPP with constant rate q molecule/s. Thus, the asymptotic

channel response of continuous emission can be obtained by calculating the expected sum of all

impulse emissions over the one-dimensional PPP, i.e., Nct

(~b,∞

)= E

∑τ∈(0,∞) Nim

(~b,τ)

.

Using Campbell’s Theorem, E

∑τ∈(0,∞) Nim

(~b,τ)

can be obtained by multiplying the chan-

nel response of an impulse emission by the emission rate q and then integrating it over all time

to infinity. By doing so, the asymptotic channel response Nct

(~b,∞

)at~b, due to continuous


emission with rate q from the point (0,0) since time t = 0, is given by

Nct

(~b,∞

)= E

∑

τ∈(0,∞)

Nim

(~b,τ)

(4.2)

= q∫

∞

τ=0Nim

(~b,τ)

dτ .

where Nim

(~b,τ)

is the channel response at ~b at time τ due to an impulse emission of one

molecule at time t = 0 from the point (0,0). To evaluate Nct

(~b,∞

), we first derive Nim

(~b,τ)

for any~b and~b = 0.

We solve Nim

(~b,τ)

for a circular passive observer S0 and for a square observer in the

following theorem.

Theorem 4.1 (Impulse Emission for Any~b). The channel response Nim

(~b,τ)

for a circular

passive observer S0 centered at any~b with radius R0 is given by

Nim

(~b,τ)=

4

∑i=1 1

2Dexp(−kτ)

∫ R0

r=0

rτ

exp(−|~b|2 + r2

4Dτ)αi exp

(βi|~b|r2Dτ

)dr

=4

∑i=11αi exp(−R0

2 + |~b|2

4Dτ− kτ)[exp(

R02

4Dτ)− exp(

R0|~b|βi

2Dτ)]

+qαi|~b|βi

√Dπ

2D√

τexp(−|

~b|2(1−βi2)

4Dτ− kτ)[erf(

|~b|βi

2√

Dτ)+ erf(

R0−|~b|βi

2√

Dτ)],

(4.3)

where the exact values of αi and βi for different ranges of z are given in [115]. Due to space

limitation, we do not present these exact values in this chapter. Please refer to [115] for these

values.

Proof: See Appendix C.1.

In addition, the channel response Nim

(~b,τ)

when the RX is a square passive observer

centered at~b with the length l is given by

Nim

(~b,τ)=∫ |~b|+ l

2

x=|~b|− l2

∫ l2

y=− l2

C (~r1,τ)dxdy,

=∫ |~b|+ l

2

x=|~b|− l2

∫ l2

y=− l2

exp(−x2 + y2

4Dτ− kτ

)dxdy,

=exp(−kτ)

2erf[

l4√

Dτ]

(−erf[

2|~b|− l4√

Dτ]+ erf[

2|~b|+ l4√

Dτ]

)(4.4)

We next consider the special case when the TX is at the center of the circular RX S0,


i.e., |~b| = 0. We denote Nim,self (τ) as the channel response at ~b = (0,0) at time τ due to

an impulse emission of one molecule at time t = 0 from the point (0,0), i.e., Nim,self (τ) =

lim~b→0 Nim

(~b,τ)

. We derive Nim,self (τ) in the following theorem.

Theorem 4.2 (Impulse Emission for |~b|= 0). The channel response due to an impulse emission

from itself is given by

Nim,self (τ) = exp(−kτ)

(1− exp

(−R2

04Dτ

)). (4.5)


We then evaluate the asymptotic channel response due to continuous emission for any~b in

the following theorem.

Theorem 4.3 (Continuous Emission for Any~b). The asymptotic channel response Nct

(~b,∞

)for the circular passive RX S0 centered at any~b, due to continuous emission with rate q from

the point (0,0) since time t = 0, is given by

Nct

(~b,∞

)≈

qR20

2DK0

(|~b|√

kD

). (4.6)


Remark 4.1. We have analytically found that N†agg (~xi|λ ) converges as time t → ∞, since

N†agg (~xi, t|λ ) = ∑~x j∈Φ(λ ) N (~xi, t|~x j) and N (~xi,∞|~x j) is a constant based on (4.6). Although it

takes infinite time to reach the steady state in theory, it is sufficiently accurate to consider a

finite but sufficiently large time. The accuracy will be validated by simulation results in Section

4.5.

The accuracy of uniform concentration assumption used in (4.6) will be verified in Sec.

3.5. Furthermore, we also evaluate the time-varying channel response Nct

(~b, t)

at time t with

no molecule degradation, i.e., k = 0, which is given by

Nct

(~b, t)∣∣∣

k=0≈ πR2

0

∫ t

τ=0

q(4πDτ)

exp

(− |

~b|2

4Dτ

)dτ ,

≈Γ(

0, |~b|24Dt

)qR2

0

4D. (4.7)

We finally evaluate the asymptotic channel response at the circular RX due to continuous

emission from itself in the following theorem.

Theorem 4.4 (Continuous Emission for |~b| = 0). The asymptotic channel response at the

circular RX S0, due to continuous emission with rate q from the center of this RX since time


t = 0, Nct,self

(~b,∞

), is given by

Nct,self (∞) =∫

∞

τ=0Nim,self (τ)dτ .

=qk

(1−√

kR0√D

K1

(√kD

R0

)). (4.8)


4.2.2 Randomly Distributed TXs

In this subsection, we consider that many point TXs are randomly distributed on a circle S1

according to a point process with a constant density λ . The circle S1 is centered at (0,0) with

radius R1. We represent ~a as the location of an arbitrary point TX a and Φ (λ ) as the random

set of TXs’ locations. We denote the asymptotic channel response at the circular RX S0 cen-

tered at~b with radius R0, due to continuous emission with rate q since time t = 0 from TX a by

Nct

(~b,∞|~a

)and the corresponding aggregate channel response at the RX due to all randomly

distributed TXs on circle S1 with density λ by Nagg,ct

(~b,∞|λ

)= ∑~a∈Φ(λ ) Nct

(~b,∞|~a

). We

denote E

Nagg,ct

(~b,∞|λ

)as the expected Nagg,ct

(~b,∞|λ

)over the point process Φ (λ ).

For compactness, we remove ∞ and subscript “ct” in all notation in the remainder of this chap-

ter since we assume that bacteria use asymptotic observations due to the continuous emission at

TXs to make decisions. We next derive E

Nagg

(~b|λ

)in the following theorem and simplify

it in different special cases.

Theorem 4.5. The expected aggregate channel response at the RX due to all randomly dis-

tributed TXs on circle S1 with density λ over the point process Φ (λ ) is given by

E

Nagg

(~b|λ

)=∫ R1

|~r|=0

∫ 2π

ϕ=0N(~b|~r)λ |~r|dϕ d|~r|

= λ

∫ R1

|~r|=0

∫ 2π

ϕ=0

∫ R0

|~r0|=0

∫ 2π

θ=0K0

(√kD

Υ(~b)

)q

2Dπ|~r0||~r|dθ d|~r0|dϕ d|~r|, (4.9)

where Υ(~b) is given in .

Υ(~b) =

√Ω(~b)+ |~r0|2 + 2

√Ω(~b)|~r0|cosθ , (4.10)

and Ω(~b) = |~b|2 + |~r|2 + 2|~b||~r|cosϕ .



Although we consider a point TX, a circular TX is also of interest. We discuss the channel

response due to a circular TX in the following remark:

Remark 4.2. It can be shown that the asymptotic channel response at the circular RX with

radius R0 centered at~b, due to continuous emission with rate q from a circular TX centered at

(0,0) with radius R1 since time t = 0, can be obtained by removing density λ in (4.9).

We note that the evaluation (4.9) requires very high computation complexity, since it in-

volves four integrals. Therefore, we simplify (4.9) in the following cases.

1) Uniform Concentration Assumption within RX: We assume the concentration within

the RX S0 is uniform. Using this assumption, we have

N(~b|~r) ≈(∫

∞

τ=0qC(~l,τ)

dτ

)πR0

2,

≈∫

∞

τ=0

qπR02

(4πDτ)exp

(−|~b|2 + |~r|2 + 2|~b||~r|cosϕ

4Dτ− kτ

)dτ ,

≈ qR02

2DK0

(√kD

Ω(~b)

)(4.11)

We then substitute (4.11) into (C.11), we obtain

E

Nagg

(~b|λ

)≈ λ

∫ R1

|~r|=0

∫ 2π

ϕ=0

qR02

2DK0

(√kD

Ω(~b)

)|~r|dϕ d|~r|. (4.12)

The numerical results in Sec. 3.5 will demonstrate the accuracy of uniform concentration

assumption used in (4.11).

2) RX at Environment Circle Center: When the RX is at the center of the circle S1 where

TXs are randomly distributed, we have |~b|= 0. Apply |~b|= 0 to (4.12), we obtain

ENagg(~b|λ )∣∣∣|~b|=0

= λ

∫ R1

|~r|=0

∫ R0

|~r0|=0

∫ 2π

θ=0

qD

K0

(√kD

√|~r|2 + |~r0|2 + 2|~r||~r0|cosθ

)|~r0||~r|dθ d|~r0|d|~r| (4.13)

3) Uniform Concentration and Environment Circle Center: We assume that the con-

centration within the RX S0 is uniform and the RX is at the center of the circle S1. Under these


assumptions, we apply |~b|= 0 to (4.12) to rewrite (4.12) as

ENagg(~b|λ )∣∣∣|~b|=0

≈∫ R1

|~r|=0

∫ 2π

ϕ=0

qR02

2DK0

(√kD|~r|

)λ |~r|dϕ d|~r|

≈∫ R1

|~r|=0

qπR02

DK0

(√kD|~r|

)λ |~r|d|~r|

≈ λqπR02

k

1−

√kR1K1

(√kD R1

)√

D

. (4.14)

4.3 Cooperating Probability at a Fixed-Located Bacterium

In this section, we derive the expected probability of cooperation (i.e., received molecules

from itself and other PPP distributed bacteria is larger than the threshold η) at a bacterium at a

fixed location ~xi over the spatial random point process Φ(λ ). We denote such a probability by

Pr(N†

agg(~xi|λ ) ≥ η). Please note that in this section ~xi is a fixed location and does not change

in each instantaneous realization of the spatial random point process Φ(λ ). In the following,

we first derive the exact expression of Pr(N†

agg(~xi|λ ) ≥ η)

and then derive its approximate

expression. We emphasize that the deriving such a probability is a challenging problem since

we need to take into account the random observation at ~xi caused by the randomness both in

locations of all bacteria and molecular propagation.

4.3.1 Exact Cooperating Probability

In this subsection, we derive the exact expression of Pr(N†

agg(~xi|λ ) ≥ η), i.e.,

Pr(N†


= EΦ

Pr(

N†agg(~xi|λ ) ≥ η |N†

agg(~xi|λ ))

, (4.15)

where EΦ denotes the expectation over the spatial random point process Φ(λ ). N†agg(~xi|λ ) is

the instantaneous observation at the ith bacterium and N†agg(~xi|λ ) is its expected observation in

a given instantaneous realization of random bacterial locations. Based on [116], it is accurate

to model the instantaneous number of received molecules due to random walk of molecules

as a Poisson RV. By assuming N†agg(~xi|λ ) as a Poisson RV with mean N†

agg(~xi|λ ), we rewrite


to ρ using Faà di Bruno’s formula, which is given by

∂ n f (g(ρ))∂ρn = ∑

n!∏

nj=1 m j! j!m j

f (m1+···+mn)(g(ρ))n

∏j=1

(g( j)(ρ))m j , (4.19)

where the sum is over all n-tuples of nonnegative integers (m1, . . . ,mn) satisfying the constraint

1m1 + 2m2 + 3m3 + · · ·+ nmn = n. By assuming LN†agg(~xi|λ )

(−ρ) = f (g(ρ)) = expg(ρ)where g(ρ) is given by

g(ρ) = ρNself− λ

∫ R1

|~r|=0

∫ 2π

ϕ=0

(1− exp

(ρN(~xi|~r)

))|~r|dϕ d|~r|, (4.20)

we obtain f (m1+···+mn)(g(ρ)) and g( j)(ρ) in (4.19) as

f (m1+···+mn)(g(ρ)) = exp(m1+···+mn)(g(ρ)) = exp(g(ρ)) = LN†agg(~xi|λ )

(−ρ) (4.21)

and

g( j)(ρ) = λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~xi|~r)

j exp(ρN(~xi|~r)

)|~r|dϕ d|~r|+Nself, j = 1

g( j)(ρ) = λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~xi|~r)

j exp(ρN(~xi|~r)

)|~r|dϕ d|~r|, j ≥ 2. (4.22)

Combining (4.17), (4.19), (4.21), and (4.22), we obtain

Pr(N†


= 1−η−1

∑n=0

1n!

∂ nLN†agg(~xi|λ )

(−ρ)

∂ρn

∣∣∣∣∣ρ=−1

= 1−LN†agg(~xi|λ )

(1)η−1

∑n=0

∑1

∏nj=1 m j! j!m j(

×λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~xi|~r)

j exp(ρN(~xi|~r)

)|~r|dϕ d|~r|+Nself

)m1

×n

∏j=2

(λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~xi|~r)

j exp(ρN(~xi|~r)

)|~r|dϕ d|~r|

)m j

(4.23)

4.3.2 Approximate Cooperating Probability

In this subsection, we derive the approximate expression of Pr(N†


which has

lower computational complexity than the exact expression of that derived in (4.23).

We recall that in (4.15), we consider the actual number of molecules observed and its PDF,

which makes the evaluation of (4.15) very complicated. To ease the computational burden,

§4.4 Characterization of Number of Cooperative Bacteria 83

we approximate the actual number of molecules observed, i.e., the instantaneous realization

of N†agg(~xi|λ ), by expection over the spatial random process Φ(λ ), EΦ

N†

agg(~xi|λ )

and as-

sume that N†agg(~xi|λ ) is a Gaussian/Poisson RV with mean EΦ

N†

agg(~xi|λ )

. By doing so, we

approximate (4.15) as follows:

Pr(N†


= EΦ

Pr(


agg(~xi|λ ))

,

≈ Pr(

N†agg(~xi|λ ) ≥ η |EΦ

N†

agg(~xi|λ ))

(4.24)

By assuming that N†agg(~xi|λ ) is a Poisson RV, we further rewrite (4.24) as

Pr(N†

agg(~xi|λ ) ≥ η)= 1−

Γ(

η ,EΦ

N†

agg(~xi|λ ))

Γ (η), (4.25)

where EΦ

N†

agg(~xi|λ )

is given by

EΦ

N†

agg(~xi|λ )

= EΦ

∑

~x j∈Φ(λ )

N (~xi|~x j)

= EΦ

N (~xi|~xi)+ ∑

~x j∈Φ(λ )/~xi

N (~xi|~x j)

= Nself +EΦ ∑~a∈Φ(λ )

N(~xi|~a),

= Nself +EΦNagg(~xi|λ ), (4.26)

where ENagg(~xi|λ ) can be obtained by replacing |~b| with |~xi| and λ with λ in (4.9) or (4.12).

4.4 Characterization of Number of Cooperative Bacteria

In this section, we characterize the distribution of the number of cooperators. To this end,

we first derive the MGF of the number of cooperators. Using the derived MGF, we then

derive the expressions for the moments and cumulants of the number of cooperators. Using the

derived moments and cumulants, we study the convergence of the distribution of the number

of cooperators to a Gaussian distribution. Furthermore, we derive the expected number of pairs

of two nearest nodes both cooperating, which can be used to study the neighboring cooperative

bacteria in a QS system. The problem addressed in this section is challenging since we need to


consider the random received signal at each bacterium in a random location due to the random

motion of molecules released from a population of randomly-distributed bacteria.

4.4.1 Moment and Cumulant Generating Functions

We denote the decision of cooperation and noncooperation of the ith bacterium by B(~xi,Φ) = 1

and B(~xi,Φ) = 0, respectively. We note that B(~xi,Φ) is a Bernoulli RV with mean B(~xi,Φ).

We denote the number of all cooperators by K, i.e., K = ∑~xi∈Φ(λ ) B(~xi,Φ). We first derive

the exact expression of MGF of K and then provide its approximated expression which can be

easily used to derive the nth moment and the nth cumulant of K.

Using the definition of MGF [29], the MGF of K is given by

MK(t) = EKexp(tK). (4.27)

We substitute K = ∑~xi∈Φ(λ ) B(~xi,Φ) into (4.27) to rewrite (4.27) as

MK(t) = EΦexp(t ∑~xi∈Φ(λ )

B(~xi,Φ))

= EΦ ∏~xi∈Φ(λ )

exp(tB(~xi,Φ)). (4.28)

Since B(~xi,Φ) is a Bernoulli RV with mean Pr (B(~xi,Φ) = 1), we rewrite (4.28) as

MK(t) = EΦ ∏~xi∈Φ(λ )

EBexp(tB(~xi,Φ))


exp(t)Pr (B(~xi,Φ) = 1)+ (1−Pr (B(~xi,Φ) = 1))


1+(exp(t)−1)Pr (B(~xi,Φ) = 1) (4.29)

We recall that the ith bacterium is a cooperator, i.e., B(~xi,Φ) = 1, if N†agg (~xi|λ ) is larger

than η . Thus, we derive Pr (B(~xi,Φ) = 1) as

Pr (B(~xi,Φ) = 1) = Pr(


agg (~xi|λ ))

, (4.30)

where Pr(


agg (~xi|λ ))

is the conditional cooperating probability for a given

instantaneous realization of the spatial random point process Φ. Analogously to Sec. 4.3.1,

we assume N†agg(~xi|λ ) is a Poisson RV and apply N†

agg (~xi|λ ) = ∑~x j∈Φ(λ ) N (~xi|~x j) to rewrite


(4.30) as

Pr (B(~xi,Φ) = 1) = 1−

(η−1

∑n=0

1n!

exp−N†agg(~xi|λ )

(N†

agg(~xi|λ ))n)

= 1−

η−1

∑n=0

1n!

exp− ∑~x j∈Φ(λ )

N(~xi|~x j)

∑~x j∈Φ(λ )

N(~xi|~x j)

n (4.31)

We finally substitute (4.31) into (4.29), we derive the exact expression of MK(t) as

MK(t) = EΦ ∏~xi∈Φ(λ )

h(~xi,Φ), (4.32)

where h(~xi,Φ) is given by

h(~xi,Φ) = 1+(exp(t)−1)

1−

η−1

∑n=0

1n!

exp− ∑~x j∈Φ(λ )

N(~xi|~x j)

∑~x j∈Φ(λ )

N(~xi|~x j)

n .

(4.33)

We note that h(~xi,Φ) not only depends on ~xi but also depends on the location of other

bacteria in Φ. Hence, it is mathematically intractable to write E∏x∈Φ h(x,Φ) as a form that

only includes addition, multiplication, or integrals using existing tools in Stochastic Geome-

try, which makes deriving moments or cumulants based on (4.32) cumbersome. To tackle this

problem, we next derive the approximated expression of MK(t). To this end, we use the ex-

pected cooperating probability over the spatial point process Φ to approximate the conditional

cooperating probability for a given instantaneous realization of this point process Φ. By doing

so, we approximate (4.30) as

Pr (B(~xi,Φ) = 1) ≈EΦ

Pr (B(~xi,Φ) = 1)

, (4.34)

where

EΦ

Pr (B(~xi,Φ) = 1)

= EΦ

Pr(


agg(~xi|λ ))

= Pr(N†


, (4.35)

where Pr(N†


is evaluated in Sec. 4.3. The approximated Pr (B(~xi,Φ) = 1) in

(4.34) only depends on the location~xi and do not depend on the position of other bacteria in Φ.

We discuss the accuracy of approximation in (4.34) and its appropriateness in the application

of QS in the following remark:


Remark 4.3. Intuitively, the approximation of Pr (B(~xi,Φ) = 1) in (4.34) is more accurate

when the density of the bacterial population, λ , is lower. This is because when the density is

lower, the instantaneous number of received molecules from other bacteria is more closed to

the expected number of that over the spatial point process Φ. It leads to that the cooperating

probability for a given instantaneous realization of Φ is more closed to the expected one over

Φ, thus the approximation is more accurate. The numerical results in Sec. 4.5 will verify this

conjecture. Based on this conjecture, the results obtained via the approximation in (4.34) is

appropriate to model the number of cooperators in the early stage of QS since at such stage the

population density is usually very low.

We then substitute (4.34) into (4.29), we obtain the approximated MK(t) as

MK(t) ≈EΦ ∏~xi∈Φ(λ )

1+(exp(t)−1)Pr(N†

agg(~xi|λ ) ≥ η). (4.36)

Using pgfl [38, eq. (4.8)] for a PPP, we derive (4.36) as

MK(t) ≈ exp(−λ

∫ R1

|~r1|=0(1− exp(t))Pr

(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|)

, (4.37)

where Pr(N†

agg(~r1|λ ) ≥ η)

can be obtained by replacing ~xi with ~r1 in (4.23) or (4.25). Based

on (4.37), we derive the approximated CGF of K, KK(t), as

KK(t) = logEΦexp(tK)

≈ −λ

∫ R1

|~r1|=0(1− exp(t))Pr

(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|. (4.38)

4.4.2 Moments and Cumulants

Based on [29], the nth moment of K is related to the MGF of K by

E(K)n= ∂ nMK(t)∂ tn

∣∣∣∣t=0

. (4.39)

Using Faà di Bruno’s formula given in (4.19) and MK(t) derived in (4.37), we derive the

nth derivative of MK(t) with respect to t as

∂ nMK(t)∂ tn ≈ ∑

n!∏

nj=1 m j! j!m j

exp(−λ

∫ R1

|~r1|=0(1− exp(t))Pr

(N†

agg(~r1|λ ) ≥ η)

× 2π|~r1|,d|~r1|)n

∏j=1

(λ

∫ R1

|~r1|=0exp(t)Pr

(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|)m j ,

(4.40)


where the sum is over all n-tuples of nonnegative integers (m1, . . . ,mn) satisfying the constraint

1m1 + 2m2 + 3m3 + · · ·+ nmn = n. Applying t = 0 to (4.40), we derive the nth moment of K

as

E(K)n ≈ ∑n!

∏nj=1 m j! j!m j

n

∏j=1

(λ

∫ R1

|~r1|=0Pr(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|)m j

. (4.41)

Based on (4.41), we have the following remarks about the moments of K:

Remark 4.4. The approximation of the first moment of K, EK, given by (4.41) is tight, i.e.,

EK= λ

∫ R1

|~r1|=0Pr(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|. (4.42)


Remark 4.5. When the density of the bacterial population, λ , is relatively low, the variance of

K, denoted by VarK, can be well approximated by its mean EK, i.e.,

VarK ≈EK. (4.43)


Using (4.38), we evaluate the nth cumulant of K, denoted by κ(n) as

κ(n) =∂ nKK(t)

∂ tn |t=0

≈ λ

∫ R1

|~r1|=0Pr(N†

agg(~r1|λ ) ≥ η)

2π|~r1|,d|~r1|. (4.44)

Interestingly, combining (4.41), (4.42), and (4.44), we obtain the relation between E(K)n,EK, and κ(n), as follows:

E(K)n ≈∑n!

∏nj=1 m j! j!m j

n

∏j=1

(EK)m j ; (4.45)

κ(n) ≈EK (4.46)

Thus, once EK is determined, E(K)n and κ(n) can be easily determined via (4.45)

and (4.46). Combining (4.42), (4.23), (4.18), (4.11), and (4.8), we write the full expression of

EK as


EK=∫ R1

|~r1|=0

1−

LN†

agg(~r1|λ )(1)

η−1

∑n=0

∑1

∏nj=1 m j! j!m j(

×λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~r1|~r)

j exp(ρN(~r1|~r)

)|~r|dϕ d|~r|+Nsel f

)m1

×n

∏j=2

(λ

∫ R1

|~r|=0

∫ 2π

ϕ=0N(~r1|~r)

j exp(ρN(~r1|~r)

)|~r|dϕ d|~r|

)m j

λ2π|~r1|d|~r1|,

(4.47)

where

LN†agg(~r1|λ )

(1) = exp

−Nself− λ

∫ R1

|~r|=0

∫ 2π

ϕ=0

(1− exp

(−N(~r1|~r)

)), |~r|dϕ d|~r|

, (4.48)

N(~r1|~r) ≈qR0

2

2DK0

(√kD×√|~r1|2 + |~r|2 + 2|~r1||~r|cosϕ

), (4.49)

and

Nself =qk

1−

√kR0K1

(√kD R0

)√

D

. (4.50)

4.4.3 Distribution

The skewness and kurtosis describe the symmetry and peakedness of the distribution of a RV,

respectively. Using (4.44) and [117], we derive the skewness, β1 and kurtosis, β2, of K as

β1 =κ(3)

κ(2)3/2 ≈ (EK)−12 (4.51)

and

β2 =κ(4)κ(2)2 ≈ (EK)−1 . (4.52)

Based on [118], the skewness and kurtosis together can be employed to assess the normality

of a distribution. For a Gaussian distribution, β1 = β2 = 0. Thus, if both β1→ 0 and β2→ 0,

we can say that the RV is closely approximated by a Gaussian distribution [119]. Based on

(4.51) and (4.52), we can approximate K by Gaussian distribution if EK→ 0. Using EKand VarK derived in Sec. 4.4.2, we can use the well-known closed-form distributions (e.g.,


Poisson and Gaussian distributions) to approximate the PDF and CDF of K. In Sec. 3.5, we

will use Poisson and Gaussian distributions with derived mean and variance to fit the PDF and

CDF of the number of cooperators.

4.4.4 Pairs of Two Nearest Nodes Both Cooperating

In this subsection, we evaluate the expected number of pairs of one node and its nth nearest

node both to be cooperators in the mth round, denoted by P(n). We first write P(n) as

P(n) = E ∑~xi∈Φ(λ )

Pr (B(~xi,Φ) = 1)Pr (B(~x j,Φ) = 1), (4.53)

where ~x j is the nth nearest node to node ~xi. For any node ~xi, we evaluate Pr (B(~x j) = 1) as

Pr (B(~x j,Φ) = 1) =∫ R1

|~r2|=0

∫ 2π

ψ=0Pr (B(~r2,Φ) = 1)

gn(r(~xi))

2πr(~xi)|~r2|d|~r2|dψ , (4.54)

where ~r2 is a vector from (0,0) to a point within the environment circle S1 and ψ is the

supplementary angle of the angle between ~r2 and ~xi, r(~xi) is the distance between ~r2 and ~xi,

i.e., r(~xi) =√|~r2|2 + |~xi|2 + 2|~r2||~xi|cosψ , and gn(r) is the PDF of distance r given by [38,

eq. (2.12)]

gn(r) =2

Γ(n)(λπ)nr2n−1 exp(−λπr2). (4.55)

Substituting (4.54) into (4.53), we rewrite P(n) as

P(n) = E ∑~xi∈Φ(λ )

Pr (B(~xi,Φ) = 1)∫ R1

|~r2|=0

∫ 2π

ψ=0Pr (B(~r2,Φ) = 1)

gn(r(~xi))

2πr(~xi)|~r2|d|~r2|dψ.

(4.56)

Using (C.21) and (4.35), we rewrite (4.56) as

P(n)

= λ

∫ R1

|~r1|=0

Pr(N†

agg(~r1|λ ) ≥ η)

×∫ R1

|~r2|=0

∫ 2π

ψ=0Pr(N†

agg(~r2|λ ) ≥ η) gn(r(~r1))

2πr(~r1)|~r2|d|~r2|dψ

2π|~r1|d|~r1|, (4.57)

where Pr(N†

agg(~x|λ ) ≥ η)

can by obtained by replacing ~xi with~x in (4.23) or (4.25).



In this section, we present simulation and numerical results to assess the accuracy of our de-

rived analytical results and reveal the impact of environmental parameters on the number of

molecules observed, the cooperating probability, and the statistics and the distribution of the

number of cooperators derived in Sections 4.2–4.4.

The simulation details are as follows: The simulation environment is unbounded. We vary

density bacteria community radius R1 and threshold η . Unless specified otherwise, we consider

molecule degradation with rate k = 1× 101/s in the environment, a circular RX with R0 =

0.757 µm, and emission rate q= 1×103 molecule/s. The value of environmental parameters is

chosen to be on the same orders of those used in [112, 113, 114, 120]. In particular, the chosen

value of D is the diffusion coefficient of the 3OC6-HSL in the water at room temperature

[120]. The volume of a sphere with the chosen radius is approximately equal to the volume of

V. fischeri. We simulate the Brownian motion of molecules using a particle-based method as

described in [8]. The molecules are initialized at the center of bacteria. The location of each

molecule is updated every time step ∆t, where diffusion along each dimension is simulated

by generating a normal RV with variance 2D∆t. Every molecule has a chance of degrading

in every time step with the probability exp(−k∆t). In simulations, the locations of bacteria

are distributed according to a 2D PPP. Each bacterium releases molecules according to an

independent Poisson process, thus the times between the release of consecutive molecules at

different bacteria are simulated as i.i.d exponential RVs. In Fig. 4.4, there is one TX at a fixed

location and for each realization we randomly generate molecule release times at the TX. In

Figs. 4.5–4.10, for each realization we randomly generate both the locations and molecule

release times for all TXs (bacteria).

In Fig. 4.4, we plot the expected number of molecules observed at the RX due to one

TX’s impulse emission with 105 molecules in Fig. 4.4(a) and one TX’s continuous emission

in Fig. 4.4(b). The analytical curves in Case a)–Case f) are obtained by (4.3), (4.4), (4.5),

(4.6), (4.7), and (4.8), respectively. In Fig. 4.4(a), we see that there is an optimal time at which

channel response is maximal when the RX is not at the TX, while the channel response always

decreases with time when the RX is at the TX. This is not surprising since the molecules

diffuse away once released. In Fig. 4.4(b), we see that the channel response with molecular

degradation converges as time goes to infinity, while the channel response without molecular

degradation always increases with time.

In the following figures, we consider the average number of TXs is 100. In Fig. 4.5, we

plot the expected number of molecules observed at the RX in Fig. 4.5(a) and the corresponding

cooperating probability at the RX in Fig. 4.5(b) due to continuous emission at randomly-

distributed TXs for different environmental radii.

We first discuss the results in Fig. 4.5(a). The asymptotic curves when the RX is at (0,0)


0.02 0.04 0.06 0.08 0.1Time[s]

103

104

105

Num

ber

of M

olec

ules

Obs

erve

d

SimulationAnalytical

(a) Impulse Emission

0 0.2 0.4 0.6 0.8 1Time [s]

10-1

100

Num

ber

of M

olec

ules

Obs

erve

d


(b) Continuous Emission

Figure 4.4: The expected number of molecules observed at the RX N(~b, t)

versus time due to the

emission of one TX located at (0,0). In Fig. 4.4(a), we consider one impulse emission with 105

molecules and molecular degradation is considered. We consider three cases of the RX in Fig. 4.4(a):Case a) the circular RX located at (0,5 µm), Case b) the square RX located at (0,5 µm), and Case c)the circular RX located at (0,0). In Fig. 4.4(b), we consider continuous emission and the circular RX isconsidered. We also consider three cases of the RX in Fig. 4.4(b): Case d) the RX located at (0,5 µm)

with molecular degradation, Case e) the RX located at (0,5 µm) without molecular degradation, andCase f) the RX located at (0,0) with molecular degradation.

with uniform concentration assumption (UCA) and without UCA are obtained by (4.14) and

(4.13), respectively. The asymptotic curves when the RX is at (R12 , R1

2 ) with UCA and without

UCA are obtained by (4.12) and (4.9), respectively. As observed in Fig. 4.4(b), we see the

expected number of molecules observed in Fig. 4.5(a) first increases as time increases and

then becomes stable after some time. We then see that the asymptotic curves with UCA and


0 0.2 0.4 0.6 0.8 1Time[s]

10-1

100

Num

ber

of M

olec

ules

Obs

erve

d

Analytical, UniformSimulationAnalytical, Nonuniform

(a) The Expected Number of Molecules Observed

1 2 3 4 5 6 7 8 9 10Threshold

10-4

10-3

10-2

10-1

100

Prob

abili

ty o

f C

oope

ratin

g

SimulationAnalytical, ExactAnalytical, Approximate

(b) The Cooperating Probability

Figure 4.5: The expected number of molecules observed at the RX, E

Nagg

(~b|λ

), in Fig. 4.5(a)

and the corresponding cooperating probability at the RX, Pr(

N†agg(~xi|λ ) ≥ η

), in Fig. 4.5(b) due to

continuous emission at randomly-distributed TXs. For different environmental radii R1 = 50 µm, R1 =

100 µm, and R1 = 150 µm, the RX’s location is (R12 , R1

2 ). For R1 = 50 µm, we also consider the RXlocated at the center of environment, i.e., (0,0).

without UCA almost overlap with each other. This demonstrates the accuracy of the UCA in

the derivation of the asymptotic channel response where a circular field of TXs continuously

emit molecules.

We then discuss the results in Fig. 4.5(b). The exact and approximate analytical curves are

obtained by (4.23) via (4.11) and (4.25) via (4.12), respectively. We see that (4.23) is always

accurate while (4.25) is only accurate when the probability of cooperation is relatively high,

e.g., Pr(N†

agg(~xi|λ ) ≥ η)≥ 10−1. We note that the computational complexity of (4.25) is much

lower than that of (4.25). Thus, in the circumstances of limited computational capabilities


1 2 3 4 5 6 7 8 9 10Threshold

10-2

10-1

100

101

102

Exp

ecte

d N

umbe

r of

Coo

pera

tors

SimulationAnalytical, ExactAnalytical, Approximate

Figure 4.6: The expected number of cooperators over spatial PPP EK versus threshold η for differ-ent population radii R1.

and high probability of cooperation, (4.25) is a good method to estimate the probability of

cooperation. Finally, we note that when R1 decreases, the expected number of molecules and

the probability of cooperation increase. This is because the density of TXs is higher when R1

is smaller.

In Fig. 4.6, we plot the first moment (i.e, the mean) of the number of cooperative bacteria

versus threshold for different population radii. The exact analytical curves are obtained by

(4.42) via (4.23) and (4.11) and the approximate analytical curves are obtained by (4.42) via

(4.25) and (4.12). We see that the curves obtained by (4.42) via (4.25) is only accurate when

EK ≥ 10. This is because (4.25) is only accurate when Pr(N†

agg(~xi|λ ) ≥ η)≥ 10−1, as

observed in Fig. 4.5(b). We also see that the analytical mean obtained obtained by (4.42) via

(4.23) exactly match with simulations. This observation numerically validates Remark 4.5,

i.e., the approximation in (4.34) is tight for the first moment of the number of cooperators.

We also see that the expected number of cooperators decreases when the threshold increases,

because the probability of cooperation is smaller when the threshold is higher, as observed in

Fig. 4.5(b).

In Fig. 4.7, we plot the variance and moments of number of cooperators versus threshold

for different population radii. The analytical variances are obtained by (4.43) and the analytical

moments are obtained by (4.45) via (4.47). We first see that when the population density is

smaller (i.e., R1 is larger), the accuracy of the analytical variances and the moments improves,

thereby validating Remark 4.5.

In Fig. 4.8, we use the Poisson and Gaussian distributions with analytical mean EK and

variance VarK shown in Figs. 4.6 and 4.7 to fit the PDF of the simulated number of cooper-

ators. To assess the accuracy of Poisson and Gaussian approximations quantitatively, we calcu-


Table 4.1: Deviation between Simulation and Analytical Values at Peak PDF

(a) (b) (c) (d) (e) (f)

Poisson Approximation 6.01% 5.09% 6.02% 45.42% 9.65% 1.77%

Gaussian Approximation 6.01% 5.09% 6.04% 48.23% 10.20% 4.01%

late the deviation between the simulation and analytical curves by |analysis− simulation|/simulation.

We are interested in the deviation at the peak PDF since such deviation is the largest difference

across the whole range of PDF. The calculated deviation for different cases in Fig. 4.8 is listed

in Table 4.1. Based on Table 4.1, we see that the distribution of the number of cooperators can

generally be well approximated by the Poisson and Gaussian distributions, especially when

the expected number is relatively large, which meets our expectations discussed in Sec. 4.4.3.

When the number of cooperators is relatively small, e.g., K < 15, the Poisson approximation

has better accuracy than the Gaussian approximation. The deviation between the Poisson and

Gaussian distributions and simulated distribution for R1 = 50 µm and η = 5 is caused by the

deviation between the analytical variance and simulated variance, as observed in Fig. 4.7(a).

In Fig. 4.9, we plot the complementary CDF (CCDF) of the number of cooperators versus

threshold for different population radii. The analytical curves are obtained by the CCDF of the

Poisson distribution with analytical mean EK and variance VarK shown in Figs. 4.6 and

4.7, respectively. We see that the CCDF of the number of cooperators can be well approximated

by that of the Poisson distributions. We also see that the CCDF of the number of cooperators

decreases as the threshold increases.

In Fig. 4.10, we plot the number of pairs of any node and its nth nearest node both coop-

erating versus the population radius R1 for different thresholds η . The analytical curves are

obtained by (4.57). We first see that for the same threshold η , the curves of P1(n) with different

n almost overlap. This is because bacteria are randomly distributed and the observations among

different bacteria are independent. Second, we see that the curves of P1(n) first decrease and

then converge to a constant number as the population radius R1 increases. This because when

the population radius increases, the number of molecules observed by the bacteria decrease,

but the population radius approaches to infinity, the molecules received by any bacterium is

dominated by the molecules released from itself and the number of molecules received by any

bacterium converges converges to a constant number. Thirdly, we see that when the threshold

η decreases, for the same the population radius R1, P1(n) decreases.

4.6 Summary

In this work, we provided an analytically tractable model for predicting the concentration of

molecules observed by bacteria and the statistics of the number of responsive cooperative bac-

§4.6 Summary 95

teria, by taking the motion of molecules undergoing independent diffusion and degradation

into consideration. We derived the 2D channel response and the expected probability of co-

operation at a bacterium due to the continuous emission of molecules at randomly-distributed

bacteria. We also derived the different order moments and cumulants, CDF, and PDF of the

number of cooperators. We validate the accuracy of our analytical results via a particle-based

simulation method where we track the random walk of each signaling molecule over time.

We highlight that the channel response can generally be applied to any context where a

TX (TXs) is (are) impulsively or continuously releasing molecules into a 2D environment.

The statistical moments can help predict and control the QS process, which can lead to an

improvement in our medical and healthcare outcomes. For example, biofilm formation via

QS is an important mechanism for bacteria to resist antibiotics. However, a biofilm could be

prevented from forming if the density of cooperators is too small. Our derived statistics help

to answer the following questions for decreasing the antibiotic resistance and optimizing the

performance of an antibiotic drug: “how many cooperators would there be?”, “how likely is

the density of cooperators to be below a certain range?”, and “how to prevent stable biofilm

formation by changing the environmental parameters?”

We note that our results could be readily extended to a 3D environment by changing the

2D area integrations to 3D volume integrations. The technical methodologies adopted in the

chapter, e.g., Campbell’s theorem and the PGFL for the PPP, can be borrowed by other works

that study the group behavior of a randomly-distributed decision system using a consensus

algorithm and broadcast channel models.


1 2 3 4 5 6 7 8 9 10Threshold

100

105

1010

Var

ianc

e/ M

omen

ts


Variance

5th Moment 4th Moment,3rd Moment,2nd Moment

(a) R1 = 50 µm

1 2 3 4 5 6 7 8 9 10

100

105

1010

Var

ianc

e/ M

omen

ts

AnalyticalSimulation


Variance

(b) R1 = 100 µm

1 2 3 4 5 6 7 8 9 10Threshold

100

105

1010

Var

ianc

e/M

omen

ts

AnalyticalSimulation


Variance

(c) R1 = 150 µm

Figure 4.7: The variance VarK and different orders of moments E(K)n of number of cooperatorsversus threshold η for different population radii R1.

§4.6 Summary 97

(a) R1 = 50 µm,η = 1 (b) R1 = 100 µm,η = 1

(c) R1 = 150 µm,η = 1 (d) R1 = 50 µm,η = 5

(e) R1 = 100 µm,η = 5 (f) R1 = 150 µm,η = 5

Figure 4.8: The PDF of number of cooperators for different population radii R1 and different thresholdsη .


0

0.5

1

1 2 3 4 5 6 7 8 9 10Threshold

0

0.5

10

0.5

1

CC

DF

of N

umbe

r of

Coo

pera

tors

> 10%

Simulation

>20%

>30%

Figure 4.9: The probability when the number of cooperators is larger than 10%, 20%, 30% of allbacteria versus threshold η for different population radii R1.

101 102 103

Radius of Environment

0

20

40

60

80

100

Num

ber

of P

airs

of

Tw

o

N

eare

st N

odes

Coo

pera

ting

Simulation, n = 1Simulation, n = 2Simulation, n = 3Analytical, n = 1Analytical, n = 2Analytical, n = 3

Figure 4.10: The number of Pairs of any node and its nth nearest node both cooperating P1(n) versusthe population radius R1 for different thresholds η .

Chapter 5

Molecular Information Delivery inPorous Media

This chapter considers a PM channel in MC. Many natural chemical and biological environ-

ments, such as cell membranes, soils, catalyst beds, can be classified as PM materials. PM

is a solid with non-uniformly distributed pores [25]. It is important to investigate the unique

characteristics of molecular information delivery over such realistic channels. We investigate

different performance metrics, i.e., throughput, mutual information, and error probability, of a

PM channel in MC. We also investigate the differences in channel characteristics and perfor-

mance metrics between a PM and diffusive FS channel with the flow.

This chapter is organized as follows. The system model is presented in Section 5.1. The

performance metrics are derived in Section 5.2. Numerical results are presented in Section 5.3.

The chapter is concluded in Section 5.4.

5.1 System Model

We consider an MC system via the PM in a 3D environment where the TX and the RX are

located at the inlet and the outlet of the PM, respectively. A 2D sketch of the considered

system is given in Fig. 5.1(a) and a 3D sample of a PM is shown in Fig. 5.1(b). In a PM,

pores and grains refer to its void and solid components, respectively. Grain size distribution

and porosity (i.e., the ratio of the volume of voids over the total volume) affect the transport

behavior in the PM. In the following, we detail the key steps of the considered system.

Modulation and Emission: A sequence of binary symbols is transmitted with Pr(Xn = 1) =

P1, where Xn is the nth transmitted symbol. We consider the on-off keying modulation scheme

with a fixed symbol slot length T , which is commonly adopted in MC literature, i.e., at the be-

ginning of the nth symbol slot, the TX releases N molecules if Xn = 1; otherwise, no molecule

is released. The TX uniformly releases the molecules over the cross section at the inlet of the

PM. We note that the use of a binary sequence is expected in MC between nanomachines to

exchange the amount of information required for executing complex collaborative tasks, e.g.,

99

100 Molecular Information Delivery in Porous Media

L

Grain Pore Molecule

Inlet Outlet

Pore Length

TX RX

(a)

(b) (c)

Figure 5.1: (a): A 2D sketch of the considered system model, where L is the distance between theTX and RX. (b): A 3D sample of a PM [3]. (c): Illustration of molecular transport through a PM withheterogeneous advection [4], where the red lines represent streamlines of the laminar flow; the shadingof the background denotes the flow velocity which decreases from light to dark; the horizontal arrowdenotes transport of molecules over the length of a pore in streamwise direction; and the vertical arrowindicates transport of molecules across streamlines into low velocity zones in the wake of the solidgrains. In (b) and (c), the grains are represented in grey and black, respectively.

disease detection [121], and binary symbols are easier to transmit than symbols that carry more

bits of information.

Transport through the PM: We consider the PM filled with an incompressible fluid of vis-

cosity µ , moving with a mean velocity ~vm oriented from the TX to the RX. Due to the small

pore sizes, the flow is laminar (Reynolds number of the flow is negligible) and governed by the

Stokes equation µ∇2~v(a) = ∇p(a) together with the incompressibility condition ∇~v(a) = 0,

where ∇ is the nabla operator, a denotes location, ~v is the velocity, and p(a) is the pressure.

The boundary conditions are of zero velocity (no-slip) on the surface of the solid grains, and

periodic on the external boundaries, with a fixed pressure gradient along the mean flow di-

rection. The resulting velocity field ~v is characterized by a chaotic heterogeneous structure.


Mechanical entrapment of molecules may occur in PM when the molecules are too large to en-

ter small pores [122]. Small molecules such as water and salt molecules can travel through PM,

but large molecules such as polymer molecules will be trapped and accumulate in these small

pores. Although these effects are not explicitly modelled here (they would, in fact, require a

Lagrangian description of molecules as rigid bodies), a similar effect is here included when

the flow velocity is high compared to molecular diffusion. The few molecules that diffuse into

stagnant regions can get trapped for relatively long times before diffusing back into the main

flow channels.

The molecular transport in the pores is due to molecular diffusion and the complex hetero-

geneous advection around solid grains, as shown in Fig. 5.1(c). The molecular concentration

c(a, t) is modeled by an advection-diffusion equation [123]:

∂c(a, t)/∂ t +~v(a)∇c(a, t)−D∇2c(a, t) = 0, (5.1)

where D is the constant diffusion coefficient, with a constant flux of molecules on the inlet and

zero diffusive flux on all other boundaries. Although these equations are linear and relatively

easy to solve, the complexity of the geometry makes the discretization and solution particularly

cumbersome[27].

The Péclet number (Pe), which compares advective and diffusive transport over the whole

PM length L, is given by Pe = |~vm|L/D. Thanks to the interplay of these two phenomena,

molecules not only are transported along the streamlines but also travel across streamlines,

experiencing therefore a wide range of velocities, and possibly reaching stagnant zones in

the wake of the solid grains. Molecules that enter these zones can remain there for some

time before they escape and return into the mobile portion of the medium. The transport

of molecules through PM may also be affected by electro-chemical effects. For example,

molecules may contain polar groups, which will attach to the available polar points on the

PM surface [122]. Depending on the PM surface net ionic charge, electrostatic attraction

or repulsion would occur for ionic molecules, which enhance or reduce the ionic molecular

adsorption on the surface of PM. For the tractability of the distribution of first arrival time of

molecules, we do not consider electrical effects on molecular propagation.

Reception and Demodulation: We consider a RX that is mounted on the cross section at

the outlet of the PM and is able to count the number of molecules that arrive. To decrease the

complexity, we consider a fixed threshold-based demodulation rule at the RX: Yn = 1 if Nobn ≥

ξ ; otherwise, Yn = 0, where Yn is the nth received symbol, Nobn is the number of molecules

that arrive during the nth slot, and ξ is a fixed threshold. The transmission and reception

of multiple symbols is possible. The encoding function at the TX can be implemented by

synthesizing logic gates [124]. A metabolic pathway of a biological cell can be synthesized

into the TX to release specific molecules [125]. The computational processing at the RXs can


be implemented based on [14, 15]. The time synchronization between the TX and the RXs

can be implemented using various methods, e.g., a blind synchronization algorithm [92] and

QS-based method [93].

5.2 Performance Metrics

In this section, we present the analytical results of system performance metrics. To this end,

we first analyze the (cumulative) breakthrough curve, i.e., the CDF of the first arrival time at

the outlet of any molecule released from the inlet, which is used for characterizing molecular

transport in the PM. This is given by [4]

F(t)=∫ ∫

c(a1=L,a2,a3, t)|~v(a1 = L,a2,a3)|da2da3∫ ∫|~v(a1 = L,a2,a3)|da2da3

, (5.2)

where a = a1,a2,a3 denotes location in Cartesian coordinates. The analytical expression for

F(t) is mathematically intractable, so we will rely on a numerical solution obtained by the full

discretization of (5.1) and (5.2). For more details about numerical solvers, we refer the readers

to [27].

If the TX and RX only partially cover the media inlet and outlet, then the breakthrough

curve needs to be re-computed since the boundary conditions change and the dimension of

the problem effectively increases (since the whole coverage case is effectively a 1D system).

More generally, when the TX and the RX are located arbitrarily in an open three-dimensional

domain, one would need to consider a full non-diagonal and anisotropic dispersion tensor [25]

and not only the longitudinal dispersion studied here. We expect that the difference between

breakthrough curves with full and partial coverage, to resembles the difference between diffu-

sion processes in one and more dimensions.

Remark 5.1. Assuming that the number of molecules arrived can be approximated as a Gaus-

sian RV, we derive the mutual information I, throughput C, and error probability Q. Using

particle-based simulation methods, [126, 127] have verified the accuracy of Gaussian approx-

imation. According to the central limit theorem, the accuracy of this approximation improves

as N increases. Due to the space limitation, we present the derivation of statistical distributions

of molecules arrived and system performance metrics in Appendix D.1.

We next discuss the diversity gain. Each molecule behaves independently and experiences

different propagation paths. Thus, the channel can be seen as a multiple-input and multiple-

output channel and the RX achieves diversity when N molecules are released. Also, there is

an optimal ξ that minimizes error probability Q, i.e., Q∗ = minξ

Q, where Q∗ is the optimal

error probability. We define the diversity gain as the exponentially decreasing rate of Q∗ as a

function of increasing N. That is to say, if we can well approximate Q∗ with a form of Q∗ ≈

§5.3 Numerical Results 103

exp(−αN +β ), then α is the diversity gain. The assessment of the diversity gain of different

channels indicates which channel is more sensitive to the increase in the number of molecules

released, without the need for explicitly calculating the probability of error. Specifically, if

a higher diversity gain is achieved, the channel is more sensitive. Thus, the evaluation of

diversity gain provides information regarding the fundamental properties of different channels,

which for example facilities the appropriate selection of the number of molecules released for

MC system design. Since an explicit expression for α is mathematically intractable, we use

a data-fitting method to obtain α . The method will be detailed in Sec. 5.3. We note that a

similar definition of α was studied in [88] for timing channels, but our method for evaluating

α is different from [88].

For P1 =12 , we have following corollaries on Q and I:

Corollary 5.1. The optimal error probability converges to zero when the released number of

molecules for symbol “1” tends to infinity, i.e., limN→∞ Q∗ = 0.

Proof: See Appendix D.2.

Corollary 5.2. The mutual information is bounded by I ≤ 1bits/slot and I = 1bits/slot is

obtained if and only if Q→ 0.

Proof: See Appendix D.3.

5.3 Numerical Results

In this section, we present numerical results to investigate the channel response and com-

munication performance of MC via the PM. We consider the 3D sand-like PM described in

[4, 27]. The medium was generated according to the characteristics of standard sand samples.

Specifically, the PM is a cube of size L = 2mm, which is of the size of a representative ele-

mentary volume in terms of the definition of volumetric porosity [25]. The typical porosity

of many kinds of soils, e.g., gravel, sand, silt, and clay, is between 20% and 50%, based on

[128, 129, 130]. The typical grain diameter for medium sand is between 0.25mm and 0.5mm

[131]. We consider the porosity of 35% and the grain diameter of 0.277mm, which are within

the normal range for sand samples. The grain size distribution follows a Weibull distribution

with Weibull parameter k = 7. We also consider n = 10 symbols are transmitted with P1 =12 .

The other parameters are given in Table 5.1. With these parameters, [4] numerically solved

(5.1) and (5.2), obtaining the values of F(t) for Pe = 3,30,300,1000. The results in the fol-

lowing figures are obtained based on this simulation data. In Fig 5.4, Fig 5.6, and Table 5.2,

we assume that Nobn is a Poisson RV since we consider N ≤ 100. In Fig 5.5, we assume Nob

n is

a Gaussian RV since we consider N = 105.


Table 5.1: Environmental Parameters

Parameter Symbol ValueLength of PM L 2mm

Number of grains φ 2×103

Average grain diameter d 0.277mmCharacteristic pore length (estimated) `0 0.277mm

Mean velocity |~vm| 5.73×10−6m/s

0

0.2

0.4

0.6

0.8

1

CD

F

Pe = 3 Pe = 30 Pe = 300 Pe = 1000

0 100 200 300 400 500 600 700 800 900 1000

Time [s]

0

2

4

PDF

10-3

Figure 5.2: The CDF and PDF f (t) of the arrival time of the molecule versus time t in the PM channelfor different Pe.

In order to provide more insights, we compare with a 1D diffusive FS channel with a flow

oriented from the TX to the RX, which is referred to as the “FS channel” in the following for

brevity. This is because the PM channel is effectively a 1D channel due to the TX and RX

covering the entire inlet and outlet. The probability density function (PDF) of the first arrival

time at a1 = L in the FS channel is given by f (t) = L√4πDt3 exp(− (|~vm|t−L)2

4Dt ) [132]. For this FS

channel, we consider the same parameter values as those for the PM channel for the fairness

of our comparison.

5.3.1 Channel Response

In Fig. 5.2, we show the arrival time distribution in the PM channel. The PDF curves are

obtained by numerically evaluating the derivative of F(t). Firstly, for all Pe, F(t)→ 1 as

t→∞, which means that all molecules released will eventually arrive at the RX. This is because

no flow is going out of the lateral directions and no molecule can escape from the lateral

directions by advection nor by diffusion. Secondly, when Pe is smaller, the CDF converges

more quickly to 1, meaning that less molecules stay trapped in the PM.

In Fig. 5.3, we compare the arrival time PDF in the PM channel with that in the FS

channel. Interestingly, when Pe is 3, the PDF curve for the PM is similar to that for FS. This is

§5.3 Numerical Results 105

0 500 1000

10-4

10-2

0 500 1000

10-4

10-2

Porous MediaFree Space

0 500 1000

10-4

10-2

PDF

of A

rriv

al T

ime

0 500 1000

Time [s]

10-4

10-2

(d) Pe = 1000

(b) Pe = 30(a) Pe = 3

(c) Pe = 300

Figure 5.3: The PDF f (t) of the arrival time of the molecule versus time in the PM and FS channelsfor different Pe.

because the fact that molecular diffusion is fairly large, causing particles to uniformly sample

the velocity space, and resulting in an overall transport that can be conveniently described as a

single advection-diffusion channel. Secondly, PM channel behavior is much less sensitive to Pe

than in the FS channel. This is due to molecules entering dead-end pores or stagnant regions,

and taking a long time to escape in the PM. For FS, when Pe is larger, since there are no

such regions, the only effect is a more dominant advection than the diffusion, thus FS channel

behavior is more sensitive to larger Pe. Importantly, as Pe increases (e.g., larger molecules with

smaller diffusion coefficient), the peak value of the PDF curve for the FS channel increases,

while that for the PM model decreases (as seen in Fig. 5.2), i.e., the PDF curve for the FS

channel becomes narrower but the PDF curve for the PM becomes longer. This is because

for the PM, the particles travel in all directions through the complex network of pores, thus

generating a much larger “longitudinal dispersivit”, i.e, a higher equivalent diffusion in the

longitudinal direction, proportional to Pe[27]. This means that, as Pe increases, the ISI of

the PM channel increases but ISI of the FS channel decreases. Based on this, for the PM

channel we expect the error performance and mutual information would become worse when

Pe increases, which will be verified by the observations in Fig. 5.4.

5.3.2 Performance Evaluation

In Fig. 5.4, we show the average mutual information and the average error probability of the

PM channel. Firstly, when ξ = 45, I is maximal (i.e., I = 1bits/slot) and Q is minimal, which

numerically validates Corollary 5.2. Secondly, the average mutual information is smaller and

the error probability is higher as Pe increases. This is because when Pe is higher, the tail of the

channel response of the PM is longer1, i.e., larger ISI, as we observed in Figs. 5.2 and 5.3.

1The long tails in the arrival time distribution do not necessarily mean the existence of ISI. For example, whentiming-based modulation is considered, the long tail of channel response leads to transposition errors [133].


0.5

0.6

0.7

0.8

0.9

Mut

ual I

nfor

mat

ion

[bits

/slo

t]

Pe = 3

Pe = 30Pe = 300Pe = 1000

10 20 30 40 50 60 70 80 90 100

Threshold

10-5

100

Err

or P

roba

bilit

y

Figure 5.4: The average mutual information I and the average error probability Q of the MC systemvia the PM versus the threshold ξ for different Pe: Pe = 3,30,300,1000. N = 100 and T = 400s.

100 200 300 400 5000

0.5

1

Cap

acity

[bi

ts/in

terv

al]

100 200 300 400 5000

0.5

1Porous MediumFree Space

100 200 300 400 500

Symbol Interval [s]

0

0.5

1

100 200 300 400 5000

0.5

1

(d) Pe = 1000

(b) Pe = 30(a) Pe = 3

(c) Pe = 300

Figure 5.5: The throughput C of the MC system via the PM and FS channels versus the symbol slot Twith N = 105 for different Pe: (a) Pe = 3, (b) Pe = 30, (c) Pe = 300, and (d) Pe = 1000.

In Fig. 5.5, we show the throughput of the PM and FS channels. Firstly, for both channels

and all Pe, C increases as T increases and C = 1bits/slot is achieved when T ≥ 400s. This

is because of a very small probability that a molecule arrives at t ≥ 400s, as observed in Fig.

5.3. Secondly, the difference of C between the PM and FS channels when T ≤ 300s becomes

larger as Pe increases. This is because in Fig. 5.3, when Pe increases, the PM and FS channels

diverge.

In Fig. 5.6, we plot the optimal average error probability versus the number of molecules

released for bit “1” for different symbol slots. The considered symbol slots are around the

detection time that maximizes the PM and FS channel responses based on Fig. 5.3. Firstly,

Q∗ decreases when N increases. We then see that error probability curves can be well approx-

imated by the fitted curves, Q∗ ≈ exp(−αN + β ), where α and β are obtained by solving

Q∗|N=10 = exp(−α10+β ) and Q∗|N=100 = exp(−α100+β ). Thus, we can use the diversity

gain α to quantify the decrease rate of Q∗ as N increases. We present α for different T and

Pe in Table 5.2. We find that the PM achieves higher α than the FS channel for any Pe with

§5.4 Summary 107

10 20 30 40 50 60 70 80 90 100

Number of Molecules for bit 1

10-5

10-4

10-3

10-2

10-1

100

Opt

imal

Ave

rage

Err

or P

roba

bilit

yPorous MediaFree SpaceFitted Curve

T = 300s

T = 350s

T = 400s

Figure 5.6: The optimal average error probability Q∗ of the MC system versus the number of moleculesN released for bit “1” for different symbol slots: T = 300s, T = 350s, and T = 400s with Pe = 3.

Table 5.2: Diversity Gain

Diversity Gain α Pe = 3 Pe = 30 Pe = 300 Pe = 1000

T = 300sFS 0.0013 0.0055 0 0PM 0.0009 0.0013 0.0010 0.0008

T = 350sFS 0.0089 0.0032 0.0048 0.0094PM 0.0186 0.0109 0.0144 0.0184

T = 400sFS 0.0651 0.2689 0.8334 0.9487PM 0.0822 0.0651 0.0569 0.0585

T = 350s. This is because the decrease rate of Q∗ is affected by ISI. The PM has less ISI than

the FS channel for these parameter values, based on the tails of the PDF curves of arrival time

shown in Fig. 5.3.

5.4 Summary

We for the first time considered MC via a realistic PM channel, modeled as a 3D complex pore

structure. Using fully resolved computational fluid dynamics results for the arrival time dis-

tribution, we explored the differences in channel characteristics between PM and FS channels

and their impact on communication performance metrics (i.e., throughput, mutual information,

error probability, and diversity gain) in both channels. Our results suggest that the reliability

of a PM channel can be improved by decreasing Pe, while opposite trends for an FS channel.

Although the parameters (e.g., porosity, size, and topology) of different types of natural PM

vary widely, their fundamental channel characteristics, i.e., the changing trends in the molecu-

lar arrival time distribution as Pe changes, are the same. This is because the key characteristic

of molecular transport through the PM channel is that molecules may become trapped in the


vicinity of solid grains, therefore taking some time to exit and causing non-trivial anomalous

transport phenomena, such as long tails in the arrival time distributions. Our results reveal such

changing trends in the molecular arrival time distribution and its impact on the different per-

formance metrics of PM as Pe changes. These results provide useful guidelines for designing

the optimal MC system through PM and predicting the system communication performance in

a practical biological environment where Pe may change due to the instability of temperature

and diffusion coefficients.

Chapter 6

Conclusions

In this chapter, we first summarize the general conclusions drawn from the thesis, and then

outline some future research directions arising from this work.

6.1 Thesis Conclusions

This thesis focused on mathematical modeling, analysis, optimization, and simulation vali-

dation of cooperative and large-scale MC systems. In particular, we focused on cooperative

detection in MC systems in Chapters 2 and 3, large-scale bacterial MC systems in Chapter 4,

and a realistic MC environment in Chapter 5. The key contribution and research impact are

summarized in Table 6.1. The detailed contributions and research impact are given as follows:

Cooperative detection in MC systems: In Chapters 2 and 3, we quantified and maximized

the benefits brought by cooperative detection among K distributed RXs in a diffusion-based

MC system. We for the first time considered all of the following factors: i) multiple-symbol

transmission and resultant ISI, ii) multiple noisy communication phases from the TX to the

FC via the RXs, iii) cooperative detection among multiple RXs, in a cooperative MC system.

Based on this novel system model, we considered a number of variants according to differ-

ent relaying modes, the number of types of molecules available at RX, and different detection

methods at the FC. We derived error probabilities of these variants, optimized molecule allo-

cation among RXs for the SD-ML variant, optimized thresholds at the RXs and FC for some

other variants. All of the results are validated using particle-based simulations. Numerical

and simulation results revealed that i) the system error performance is greatly improved by

combining the detection information of distributed RXs, ii) the solutions to the formulated

suboptimal convex optimization problems achieve near-optimal global error performance, iii)

the ML detection variants provide lower bounds on the error performance of simpler, non-

ML cooperative variants and demonstrate that these simpler cooperative variants have an error

performance comparable to ML detectors.

The presented work serves as the first step to explore the fundamental benefits of the co-

operative detection among multiple RXs in MC, which has huge potential to improve the re-

109

110 Conclusions

Table 6.1: Summary of Work in Thesis

Key Contributions/Novelty Research Impact

Cooperative

detection

in MC systems

First to consider

1) multiple-RX cooperation

2) multiple-phase communication

3) multiple-symbol transmission

• Derive error probability of a

number of detection variants

• Optimize molecule allocation at RXs;

Optimize thresholds at RXs and FC

• Particle-based simulation validation

• Improve the reliability

and best achievable

error performance

• Enable high-accuracy

disease detection and

health monitoring

Large-scale

bacterial

MC systems

First to consider

1) random location of bacteria

2) random walk and degradation of molecules

3) random releasing time of molecules

• Derive the channel response due to

one TX or randomly-distributed TXs

• Derive statistics of number of cooperators

• Particle-based simulation validation

• Predict and control

cooperative bacterial

behaviors

• Prevent undesirable

bacterial infections

and new environmental

remediation

Realistic

MC

environment

First investigate MC performance via

a PM channel

Practical propagation

environments

§6.2 Future Work 111

liability performance of MC networks and enables high-accuracy disease detection and health

monitoring.

Large-scale bacterial MC systems: In Chapter 4, we presented an analytically tractable

model for characterizing i) QS signal propagation within randomly-distributed microorganisms

and ii) the distribution of the number of responsive cooperative microorganisms. Different

from prior studies, the motion of molecules undergoing independent diffusion and degradation

is taken into consideration. Microorganisms are randomly distributed in a 2D environment

where each one continuously releases molecules at random times. We derived the 2D channel

response at an observer due to one bacterium or randomly-distributed bacteria. We then derived

the expected probability of cooperation at the bacterium at a fixed location. We finally derived

the approximate expressions for the MGF and different statistics of the number of cooperators.

The analytical results agree with simulation results where the Brownian motion of molecules

is simulated by a particle-based method. Our results showed that the Poisson distribution

provides the overall best approximation of the PDF and CDF of the number of cooperators,

especially when the population density is low. Our model captures the basic features of QS

and accounts for the diffusion of molecules.

The presented work serves as the first step to prevent undesirable bacterial infections and

lead to new environmental remediation, since cooperative behaviors of microscopic popula-

tions, e.g. the formation of biofilms and the production of antibiotics, play a crucial role in

bacterial infections, environmental remediation, and wastewater treatment.

Realistic MC environment: In Chapter 5, we for the first time investigated communica-

tion through realistic porous channels via statistical breakthrough curves. Assuming that the

number of arrived molecules can be approximated as a Gaussian RV and using fully resolved

computational fluid dynamics results for the breakthrough curves, we presented the numeri-

cal results for the throughput, mutual information, error probability, and information diversity

gain. Using these numerical results, we revealed the unique characteristics of the PM channel.

The presented work provides useful guidelines for designing the optimal MC system through

porous media and predicting the system communication performance in a practical biological

environment.

6.2 Future Work

The research of MC is still at a very early stage. In the future, the following work can be

conducted based on the contributions presented in this thesis:

112 Conclusions

6.2.1 Theoretical Modeling

• Since the work in Chapters 2 and 3 adopted many unrealistic assumptions, interesting

future work includes relaxing these assumptions:

–Modeling an imperfect TX: It is interesting to develop a mathematical model for an

imperfect TX by considering the volume of the TX and the molecules that are randomly

generated within the TX. This consideration is motivated by the fact that in real biologi-

cal environments, the cells are spherical, rod-shaped or spiral-shaped and the molecules

are generated in different sections of cells. The chemical reactions that occur in the real-

istic generation and emission processes of molecules in cells can also be considered. As

a starting point, the results in Chapter 4 can be used to establish the model for an imper-

fect TX. For this model, stochastic processes such as the PPP will be used to model the

random location of generated molecules.

–Modeling flow-aided propagation in a bounded environment: This modeling is in-

spired by the fact that in blood vessels, the propagation of molecules is driven by the

blood flow in a bounded channel. Considering laminar flows with non-uniform flow ve-

locity, the mechanical dispersion coefficient of molecules caused by the interaction of

diffusion and non-uniform flow can be derived. Using this coefficient, the PDF of the

first arrival time of a molecule in the bounded environment can be derived by jointly

solving (i) the Stokes equation for the flow velocity and (ii) the reaction-diffusion differ-

ential equation for chemical reactions.

–Modeling channel response at multiple reactive RXs: Chapters 2 and 3 assumed

transparent RXs for tractability due to the independence among observations at multiple

transparent RXs. However, many practical RX surfaces may interact with the molecules

of interest, e.g., by providing binding sites for absorption or other reactions [134]. In

an environment where multiple non-transparent RXs co-exist, one non-transparent RX

may impact the molecules received by other non-transparent RXs. Derivation of channel

response at RXs when multiple reactive RXs co-exist, by taking into account the mutual

influence between RXs, is interesting.

• Cooperative localization and channel estimation: Most of MC studies considered the

estimation of location, distance, and channel response using the observation at one RX.

Our results in Chapters 2 and 3 focus on the estimation of transmitted symbols. Cooper-

ative localization or channel estimation using multiple RXs has not been exploited in the

MC area. Our work in cooperative detection provides useful guidelines for exploiting

benefits of combining observations at multiple distributed RXs.

• Predicting dynamic behaviors of microorganisms: We can use game theory to un-

derstand noisy real-time signaling and the resulting behavioral dynamics in microscopic

§6.2 Future Work 113

populations such as bacteria and other cells, as identified in [69]. In Chapter 4, we

presented an analytically tractable model for predicting the distribution of the number

of cooperative microorganisms, but did not consider evolutionary behavior coordination

over time and did not apply game theory to our current model with elaborated payoffs

and strategies. Based on Chapter 4, we can develop a model for predicting dynamic be-

havior of a population of randomly-distributed microorganisms over time by considering

real-time noisy signaling among microorganisms and applying game theory.

6.2.2 Validation of Theoretical Work

The theoretical results must be validated. There are three main methods, i.e., simulation,

testbed, and experiments, to validate the theoretical results. To enable the future application of

MC, the following advancements in simulation, testbed, and experiments should be done:

• Simulation validation: All of the theoretical results in Chapters 2–5 are validated by

particle-based simulations where the location of each particle over time is tracked. Al-

though simulation is cost-effective, but is still time-consuming, especially simulating

multiple-TX, multiple-RX, and relay-aided MC systems. Developing more efficient sim-

ulation methods to decrease the required time for simulation speeds up research progress

of MC.

• Testbed development: York University, University of Warwick, Yonsei University, and

Australian National University have developed macro-scale MC testbeds, e.g., multiple-

input multiple-output macro-scale MC systems done by Yonsei University. Development

of macro-scale testbeds for our cooperative MC systems considered in Chapters 2 and 3

and MC systems over porous media considered in Chapter 5 provides practical insights

on reliability improvement brought by cooperative MC and unique characteristics of

molecular information delivery over porous media.

• Biological experiments: Our work in 4 considered a 2D environment since a 2D envi-

ronment facilitates experimental validation of our theoretical work. Biological experi-

ments, especially with bacteria, are usually conducted in a 2D environment. Collabora-

tion with biologists to conduct lab experiments to validate our theoretical results is an

essential and promising direction of future work.

114 Conclusions

Appendix A

Appendix A

A.1 Proof of Theorem 2.1

The convexity of Pmd[ j]K can be proven by showing that its second derivative with respect to

ξRX is nonnegative [31]. We derive the second derivative of Pmd[ j]K as

∂ 2Pmd[ j]K

∂ξRX2 =

12K

(2(−1+K)K

πΞ (−2+K,2,1) +

√2π

K (0.5+U1[ j]−ξRX)

Ξ(−1+K,1,

32

)), (A.1)

where

Ξ (α ,β ,γ) =(1+Λ (ξRX,U1[ j]))α Θ (ξRX,U1[ j])β

U1[ j]γ(A.2)

and Θ (x,λ ) , exp(− (0.5+λ − x)2 /2λ

). Due to the fact that the value of Λ (x,λ ) is be-

tween −1 and 1 and the value of Θ (x,λ ) is always greater than zero, (A.1) is always nonneg-

ative if we impose the constraint (2.22). Following a similar procedure, we prove that Pfa[ j]K

is also convex with respect to ξRX, if we impose the constraint (2.23).

A.2 Proof of Theorem 2.2

We derive the second derivative of Pmd[ j]K as

∂ 2Pmd[ j]K

∂ξRX2 = (−0.5−U1[ j]+ ξRX)Υ (K−1,1,3/2)+ (K−1)Υ (K−2,2,1) (A.3)

115

116 Appendix A

where

Υ (α ,β ,γ) =[(1+Λ (ξRX,U1[ j]))

(1+Λ

(ξFC,V 0[ j]

))+(1−Λ (ξRX,U1[ j]))

(1+Λ

(ξFC,V 1[ j]

))]α

×(Λ(ξFC,V 1[ j]

)−Λ

(ξFC,V 0[ j]

))β KΘ (ξRX,U1[ j])β

U1[ j]γ4α(2√

2π)β. (A.4)

We next examine the monotonicity of Λ (ξFC,V ) with respect to V , V ∈ V 1,k[ j],V 0,k[ j].We derive the first derivative of Λ (ξFC,λ ) with respect to λ as

∂ Λ (ξFC,λ )∂λ

=2Θ (ξFC,λ ) (−ξFC + 0.5−λ )

2√

2πλ32

. (A.5)

Since Θ (ξFC,λ ) > 0 and (−ξFC + 0.5−λ ) < 0, we find that Λ (ξFC,λ ) is a monotonically

decreasing function with respect to λ . Therefore, we have(Λ(ξFC,V 1[ j]

)−Λ

(ξFC,V 0[ j]

))≤

0. It follows that (A.3) is always nonnegative if we impose the constraint (2.22), and thus

Pmd[ j]K is convex with respect to ξRX. Similarly, we prove that Pfa[ j]K is convex with respect

to ξRX, if we impose the convex constraint (2.23).

A.3 Proofs of Theorem 2.3 and Theorem 2.4

The convexity of Pmdb[ j]K can be proven by showing that its Hessian is PSD [31]. Although

the Hessian of Pmdb[ j]K is not always PSD, we can show that the Hessian of Pmdb[ j]K is PSD

over a convex region if we impose a set of additional constraints. Recall that a matrix is PSD if

and only if all of its principal minors are nonnegative [32]. Thus, we prove the joint convexity

of Pmdb[ j]K with respective to ξRX and ξFC by finding when ∂ 2Pmdb[ j]K

∂ξRX2 ≥ 0, ∂ 2Pmdb[ j]K

∂ξFC2 ≥ 0, and(

∂ 2Pmdb[ j]K

∂ξRX2

)(∂ 2Pmdb[ j]K

∂ξFC2

)−(

∂ 2Pmdb[ j]K

∂ξRXξFC

)2≥ 0.

We derive the second partial derivatives of Pmdb[ j]K with respect to ξRX and ξFC as

∂ 2Pmdb[ j]K

∂ξRX2 = Γ

(ξRX,ξFC,U1[ j],V 1[ j]

)= (−0.5−U1[ j]+ ξRX) Υ (K−1,1,3/2)+ (K−1)Υ (K−2,2,1) (A.6)

and

∂ 2Pmdb[ j]K

∂ξFC2 = Γ

(ξFC,ξRX,V 1[ j],U1[ j]

), (A.7)

§A.3 Proofs of Theorem 2.3 and Theorem 2.4 117

respectively, where

Υ (α ,β ,γ) =((1−Λ (ξRX,U1[ j]))

(1+Λ

(ξFC,V 1[ j]

))+ 2 (1+Λ (ξRX,U1[ j]))

)α (Λ(ξFC,V 1[ j]

)−1)β KΘ (ξRX,U1[ j])β

U1[ j]γ4α(2√

2π)β. (A.8)

Since Λ (x,λ ) is between −1 and 1 and Θ (x,λ ) is greater than zero, (A.6) and (A.7) are

always nonnegative if we impose the convex constraints (2.22) and (2.45), respectively.

Finally, we show how the third condition of the joint convexity is satisfied. To this end, we

derive the second mixed derivative of Pmdb[ j]K with respect to ξRX and ξFC as

∂ 2Pmdb[ j]K

∂ξRXξFC

=21−2KK

π

√U1[ j]V 1[ j]

Θ (ξRX,U1[ j])Θ(ξFC,V 1[ j]

)(3−Λ (ξRX,U1[ j])

×(1+Λ

(ξFC,V 1[ j]

))−Λ

(ξFC,V 1[ j]

))−2+K(−4+K +KΛ

(ξFC,V 1[ j]

)+KΛ (ξRX,U1[ j])

(1+Λ

(ξFC,V 1[ j]

))). (A.9)

Combining (A.6), (A.7), and (A.9), and performing some algebraic manipulations, we have(∂ 2Pmdb[ j]K

∂ξRX2

)(∂ 2Pmdb[ j]K

∂ξFC2

)−(

∂ 2Pmdb[ j]K

∂ξRXξFC

)2

= Ω (ξRX,ξFC)K22−4K

π2U1[ j]V 1[ j]Θ(ξFC,V 1[ j]

)Θ (ξRX,U1[ j])

×(

3−Λ(ξFC,V 1[ j]

)−Λ (ξRX,U1[ j])

(1+Λ

(ξFC,V 1[ j]

)))(−4+2K), (A.10)

where Ω (ξRX,ξFC) is given by

Ω (ξRX,ξFC) = −4Θ (ξRX,U1[ j])(−4+K +KΛ

(ξFC,V 1[ j]

)+KΛ (ξRX,U1[ j])

×(1+Λ

(ξFC,V 1[ j]

)))2+

(1+Λ (ξRX,U1[ j]))√U1[ j]V 1[ j]

(1+Λ

(ξFC,V 1[ j]

))×(

2 (−1+K)√

V 1[ j] (1+Λ (ξRX,U1[ j]))−√

2π

Θ(ξFC,V 1[ j]

)×(0.5+V 1[ j]−ξFC

)(−3+Λ

(ξFC,V 1[ j]

)+Λ (ξRX,U1[ j])

×(1+Λ

(ξFC,V 1[ j]

))))(Θ (ξRX,U1[ j]) (−1+K)

(1+Λ

(ξFC,V 1[ j]

))×2√

U1[ j]−√

2π (0.5+U1[ j]−ξRX)(−3+Λ

(ξFC,V 1[ j]

)+Λ (ξRX,U1[ j])

(1+Λ

(ξFC,V 1[ j]

))))(A.11)

118 Appendix A

We note that (A.10) is always nonnegative if the following constraint is satisfied:

Ω (ξRX,ξFC) ≥ 0. (A.12)

The constraint (A.12) is not convex, and ξRX and ξFC in the exponential and error functions

make joint convexity analysis with respect to ξRX and ξFC cumbersome. To tackle this cum-

bersomeness, we can bound ξRX with ξ−RX or ξ+RX, and bound ξFC with ξ−FC or ξ+

FC to lower the

value of the left-hand side of (A.12). Thus, we obtain (2.46) to ensure that (A.10) is always

nonnegative. Under the constraints (2.22), (2.45), and (2.46), we define a convex region where

Pmdb[ j]K is jointly convex with respect to ξFC and ξRX.

Similar to the proof of the joint convexity of Pmdb[ j]K , it can be proven that Pfab[ j]K is also

jointly convex with respect to ξRX and ξFC under the constraints (2.23), (2.48), and (2.49).

Appendix B

Appendix B

B.1 Proof of Theorem 3.1

We first prove the decision rule for SD-ML when λ DI [ j]> 0. To this end, based on (3.4), we first

rewrite the general decision rule for SD-ML as WFC[ j] = 1 if LSD1 [ j]

LSD0 [ j]≥ 1, otherwise WFC[ j] = 0.

Thus, if LSD1 [ j]

LSD0 [ j]

is a monotonically increasing function with respect to s[ j], then we can obtain

the decision rule. We can prove that LSD1 [ j]

LSD0 [ j]

is a monotonically increasing function with respect

to s[ j] by proving that(LSD

1 [ j]LSD

0 [ j]

)′> 0. Based on (3.13), we first rewrite LSD

1 [ j] and LSD0 [ j] as

LSD1 [ j] =

2K

∑h1=1

[Pr (h1|1)exp

(−λ

DI [ j]− λ

D,tots,h1

[ j])(

λDI [ j]+ λ

D,tots,h1

[ j])s[ j]

(s[ j]!)−1]

(B.1)

and

LSD0 [ j] =

2K

∑h0=1

[Pr (h0|0)exp

(−λ

DI [ j]− λ

D,tots,h0

[ j])(

λDI [ j]+ λ

D,tots,h0

[ j])s[ j]

(s[ j]!)−1]

, (B.2)

respectively, where Pr (h|b) = Pr(WRX


). Based on (B.1) and (B.2), we find

the first derivative of LSD1 [ j]

LSD0 [ j]

with respect to s[ j] as

(LSD

1 [ j]LSD

0 [ j]

)′=

2K

∑h1=1

2K

∑h0=1

[Pr (h1|1)Pr (h0|0)Π(h1,h0)]

(LSD0 [ j] (s[ j]!))2 , (B.3)

where

Π(h1,h0) = exp(−2λ

DI [ j]− λ

D,tots,h1

[ j]− λD,tots,h0

[ j])(

λDI [ j]+ λ

D,tots,h0

[ j])s[ j](

λDI [ j]+ λ

D,tots,h1

[ j])s[ j]

× log

(λ D

I [ j]+ λD,tots,h1

[ j]

λ DI [ j]+ λ

D,tots,h0

[ j]

). (B.4)

119

120 Appendix B

We observe that in (B.3), all terms are positive except for the log(·) term. Since log(x)> 0

when x > 1, we separate (B.3) into two parts: log(·) > 0 and log(·) < 0. By doing so, we

rewrite (B.3) as the sum of A and B, i.e.,

A =2K

∑h1=1

2K

∑h0=1, λ D,tot

s,h1[ j]>λ

D,tots,h0

[ j]

[Pr (h1|1)Pr (h0|0)Π(h1,h0)]

(LSD0 [ j] (s[ j]!))2 (B.5)

and

B =2K

∑h1=1

2K

∑h0=1, λ D,tot

s,h1[ j]<λ

D,tots,h0

[ j]

[Pr (h1|1) Pr (h0|0)Π(h1,h0)]

(LSD0 [ j] (s[ j]!))2 . (B.6)

We further rearrange the summation orders and exchange h1 and h0 in (B.6) to rewrite B as

B =2K

∑h1=1

2K

∑h0=1, λ D,tot

s,h1[ j]>λ

D,tots,h0

[ j]

[Pr (h0|1) Pr (h1|0)Π(h0,h1)]

(LSD0 [ j] (s[ j]!))2 . (B.7)

Combining (B.5) and (B.7) and applying Π(h1,h0) = −Π(h0,h1), we have

(LSD

1 [ j]LSD

0 [ j]

)′=

2K

∑h1=1

2K

∑h0=1, λ D,tot

s,h1[ j]>λ

D,tots,h0

[ j]

[ϑ (h1,h0)Π(h1,h0)]

(LSD0 [ j] (s[ j]!))2 , (B.8)

where ϑ (h1,h0) = Pr (h1|1)Pr (h0|0)−Pr (h0|1)Pr (h1|0). We find that (B.8) > 0 holds when

ϑ (h1,h0)> 0 is valid, i.e., where λD,tots,h1

[ j]> λD,tots,h0

[ j]. We note that λD,tots,h1

[ j]> λD,tots,h0

[ j] leads to

‖ WRXj,h1‖1>‖ WRX

j,h0‖1, where ‖ x ‖1 is the 1-norm of the vector x. When ‖ WRX

j,h1‖1>‖ WRX

j,h0‖1

holds, we have Pr (h1|1) > Pr (h0|1) and Pr (h0|0) > Pr (h1|0), which leads to ϑ (h1,h0) > 0.

Thus, ϑ (h1,h0)> 0 holds if λD,tots,h1

[ j]> λD,tots,h0

[ j]. This proves that(LSD

1 [ j]LSD

0 [ j]

)′> 0 and thus proves

the decision rule for SD-ML when λ DI [ j] > 0.

We finally prove the decision rule when λ DI [ j] = 0. We recall that λ D

I [ j] = 0 means all

previous RX symbols are “0”. It probably occurs when all previous TX symbols are “0”

(i.e., no ISI at RXk) if the error probability of the first phase is small. Hence, there is no

likelihood that “1” is detected at RXk when “0” is transmitted by the TX, which leads to

Pr(WRX

j,h = 0|WTX[ j] = 0,W j−1TX

)≈ 1 and Pr

(WRX

j,h 6= 0|WTX[ j] = 0,W j−1TX

)≈ 0. Using these

approximations and λ DI [ j] = 0, we have LSD

0 [ j]≈ exp (0) (0)s[ j]. When λ AI [ j] = 0 and s[ j] = 0,

LSD0 [ j] is 1, thus the decision at the FC is always WFC[ j] = 0 since LSD

1 [ j]< 1. When λ AI [ j] = 0

and s[ j] > 0, LSD0 [ j] is 0, thus the decision at the FC is always WFC[ j] = 1 since LSD

1 [ j] > 0.

§B.2 Proof of Theorem 3.2 121


Applying (3.20) and (3.21) to (3.3), we rewrite the decision rule for SA-ML as

(λ

As [ j]+ λ

AI [ j]

)s[ j]exp(−(

λAs [ j]+ λ

AI [ j]

)) WFC[ j]=1

RWFC[ j]=0

(λ

AI [ j]

)s[ j]exp(−λ

AI [ j])

). (B.9)

We then discuss the cases when W j−1TX = 0 and W j−1

TX 6= 0. When W j−1TX 6= 0, then we have

λ AI [ j] > 0 and we rewrite (B.9) as

(λ A

s [ j]

λ AI [ j]

+ 1

)s[ j] WFC[ j]=1

RWFC[ j]=0

exp(

λAs [ j]

). (B.10)

We rearrange (B.10) and obtain the decision rule for SA-ML when W j−1TX 6= 0. We next discuss

the case W j−1TX = 0, which leads to λ A

I [ j] = 0. If λ AI [ j] = 0 and s[ j] = 0, we write (B.9) as

exp(−λ

As [ j]

) WFC[ j]=1

RWFC[ j]=0

1, (B.11)

where the decision at the FC is always WFC[ j] = 0 since < always holds. If λ AI [ j] = 0 and any

s[ j] > 0, we write (B.9) as

(λ

As [ j]

)s[ j]exp(−(λ A

s [ j]))

/s[ j]!WFC[ j]=1

RWFC[ j]=0

0, (B.12)

where the decision at the FC is always WFC[ j] = 1 since > always holds. Thus, we obtain the

decision rule obtain the decision rule for SA-ML when W j−1TX = 0

B.3 Proof of Lemma 3.1

We note that the likelihood of the occurrence that all previous symbols transmitted by all RXs

are “0” is very small. Thus, we approximate Pr(

λ DI [ j] = 0|W j−1

TX

)≈ 0 and Pr

(λ D

I [ j] > 0|W j−1TX

)≈

1. Using these approximations in (3.17), we obtain QFC[ j] ≈ QFC

[j|λ D

I [ j] > 0]. We then note

that Q]FC[ j]|

ξ=ξad,SDFC [ j] = QFC

[j|λ D

I [ j] > 0]. Thus, QFC[ j] is accurately approximated by Q]

FC[ j]

when ξ = ξad,SDFC [ j].

122 Appendix B


We take the first derivative of (3.24) with respect to ξ . However, Q]FC[ j] is a discrete function

with respect to ξ , which makes Q]FC[ j] not differentiable in terms of ξ . To tackle this challenge,

we approximate the sum in (3.25) with an integral with respect to η , i.e.,

Λ ≈∫

ξ

η=0exp(−λ

DI [ j]−λ

D,tots,h [ j]

) (λ DI [ j]+λ

D,tots,h [ j]

)η

(η !)dη . (B.13)

Using the continuous approximation of Λ in (B.13) and ∂∫ x

t=0 f (t)dt/∂ x = f (x), we take

the first derivative of (3.24) with respect to ξ as ∂ Q]FC[ j]/∂ ξ = P1ψ1(ξ )− (1−P1)ψ2(ξ ),

where ψb(ξ ), b ∈ 0,1, is given by

ψb(ξ ) =2K

∑h=1

[Pr(WRX


)(ξ !)−1

× exp(−λ

DI [ j]− λ

D,tots,h [ j]

)(λ

DI [ j]+ λ

D,tots,h [ j]

)ξ]

. (B.14)

Comparing (B.14) with (3.13), we find that ψb(s[ j]) = LSDb [ j]. We recall that ξ

ad,SDFC [ j] is

the solution to LSD1 [ j] = LSD

0 [ j] in terms of s[ j]. Hence, ξad,SDFC [ j] is the solution to P1ψ1(ξ )−

(1− P1)ψ2(ξ ) = ∂ Q]FC[ j]/∂ ξ = 0 if P1 = 1

2 . Therefore, ξad,SDFC [ j] is the optimal ξ which

minimizes (3.24).

B.5 Proof of Proposition 3.1

The problem (3.26) has K+1 optimization variables and the evaluation of its Hessian requires

very high computational complexity. To decrease the complexity, we first consider the simplest

case with K = 2 and investigate the Hessian of Q]FC[ j] with respect to S1 for a fixed ξ . To this

end, we take the first derivative of Q]FC[ j] with respect to S1. In (3.24), Λ is a discrete function in

terms of S1, which makes the derivative cumbersome. If we approximate Λ using (B.13), there

is no closed-form for the first derivative of (B.13) with respect to S1. To overcome this chal-

lenge, we approximate Λ by another continuous approximation, i.e., the continuous regularized

incomplete Gamma function. By doing so, we have Λ ≈ Γ(dξe, λ D

I [ j]+ λD,tots,h [ j]

)/Γ (dξe),

where Γ (γ ,δ ) is the incomplete Gamma function and the Gamma function Γ (γ) is a special

case of Γ (γ ,δ ) with δ = 0. Applying this approximation to (3.24), we obtain the continuous

approximation of Q]FC[ j]. Using ∂ Γ (γ ,δ )/∂δ = −exp (−δ )δ γ−1, we take the first derivative

§B.6 Proof of Lemma 3.3 123

of Q]FC[ j] as

∂ Q]FC[ j]

∂ S1≈ 1

Γ (dξe)

(1

∑a1=0

1

∑a2=0

((1−P1)α(a1,a2) −P1β (a1,a2))exp (−Ξ(a1,a2))

× (Ξ(a1,a2))−1+dξeΩ(a1,a2)

), (B.15)

where Ξ(a1,a2) = σ1S1 +σ2(N− S1) + a1ν1S1 + a2ν2(N− S1) and Ω(a1,a2) = σ1−σ2 +

a1ν1− a2ν2, where σk and νk are given in (3.29) and (3.30), respectively. We then find the

second derivative of Q]FC[ j] with respect to S1 as

∂ 2Q]FC[ j]

∂S12 = − 1

Γ (dξe)

(1

∑a1=0

1

∑a2=0

exp (−Ξ(a1,a2)) (Ξ(a1,a2))−2+dξe (Ω(a1,a2))

2ϖ

),

(B.16)

where

ϖ =((1−P1)α(a1,a2)−P1β (a1,a2)) (1−dξe+Ξ(a1,a2)) . (B.17)

In (B.16), all terms are nonnegative except for ϖ . Thus, if ϖ > 0 holds for each summand,

(B.16) is nonnegative. However, the condition ϖ > 0 is not always valid for a1 ∈ 0,1 and

a2 ∈ 0,1. Thus, for a fixed ξ , ∂ 2Q]FC[ j]/∂ξ 2 is not always nonnegative, which means that

the Hessian of Q]FC[ j] with respect to S and ξ is not always positive semidefinite.


Using (3.28), we simplify (B.15) as

∂ Q]FC[ j]

∂ S1=

exp(−Φ1−2νS1)

Φ1Φ2Γ (dξe)(α(P1−1)+βP1)ν

(exp(2νS1)Φ

dξe1 Φ2−exp(Nν)Φdξe2 Φ1

),

(B.18)

where Φ1 = N(ν +σ)−νS1 and Φ2 = Nσ + νS1. It can be shown that S1 =N2 is the one of

the solutions of (B.18). Hence, Q]FC[ j] has a local minimum or maximum when S1 =

N2 . We

then apply (3.28) and S1 =N2 to (B.16) to obtain the second derivative of Q]

FC[ j] at S1 =N2 . By

doing so, we have

∂ 2Q]FC[ j]

∂S12 |S1=

N2=

4exp(−12 N(ν + 2σ))

N2(ν + 2σ)2Γ (dξe)ν

2(N(ν + 2σ)

2)dξeΥ(ξ ), (B.19)

124 Appendix B

where all terms are nonnegative except for Υ(ξ ). Hence if Υ(ξ ) > 0, (B.19) is nonnegative

and Q]FC[ j] achieves a local minimum at S1 =

N2 ; otherwise, it achieves a local maximum.


Based on Lemma 3.3, Q]FC[ j] achieves a local minimal value at S1 =

N2 when Υ(ξ )> 0. Based

on Lemma 3.1, the approximation of QFC[ j] by Q]FC[ j] is tight when ξ = ξ

ad,SDFC [ j]. Thus,

we can prove that QFC[ j] always achieves a local minimal value at S1 = N2 by proving that

Υ(ξ ) > 0 always holds when ξ = ξad,SDFC [ j]. That is to say, we need to prove Υ(ξ ad,SD

FC [ j]) > 0.

Based on the proof of Lemma 3.2, we also recall that ξad,SDFC [ j] is the solution to P1ψ1(ξ )−(1−

P1)ψ2(ξ ) = ∂ Q]FC[ j]/∂ ξ = 0 if P1 =

12 . Thus, ξ

ad,SDFC [ j] satisfies the condition: ψ1(ξ

ad,SDFC [ j])−

ψ2(ξad,SDFC [ j]) = 0. Applying S1 = N/2 to (B.14), we write ψb(ξ

ad,SDFC [ j]), where b ∈ 0,1,

using ν and σ as

ψb(ξad,SDFC [ j])

=(ξ

ad,SDFC [ j]!

)−1[

Pr (0,0|b) exp (−σN) (σN)ξad,SDFC [ j]

+Pr (0,1|b)exp(−σN− νN

2

)(σN +

νN2

)ξad,SDFC [ j]

+Pr (1,0|b)exp(−σN− νN

2

)(σN +

νN2

)ξad,SDFC [ j]

+Pr (1,1|b)exp (−σN−νN) (σN +νN)ξad,SDFC [ j]

], (B.20)

where Pr(WRX

j,h|b)= Pr

(WRX


). In (B.20), we have Pr (0,1|0) = Pr (1,0|0) =

α and Pr (0,1|1) = Pr (1,0|1) = β based on (3.28). We then approximate Pr (0,0|0) = Pr (1,1|1)≈1 and Pr (0,0|1) = Pr (1,1|0) ≈ 0, which is tight when the error probability of the TX−RXk

link is small. Using these approximations, α , and β , we rewrite ψ0(ξad,SDFC [ j]) and ψ1(ξ

ad,SDFC [ j])

as

ψ0(ξad,SDFC [ j])≈

(ξ

ad,SDFC [ j]!

)−1exp (−σN) (σN)ξ

ad,SDFC [ j]+ 2α

(ξ

ad,SDFC [ j]!

)−1exp(−σN− νN

2

)×(

σN +νN2

)ξad,SDFC [ j]

(B.21)

§B.7 Proof of Theorem 3.3 125

and

ψ1(ξad,SDFC [ j]) ≈

(ξ

ad,SDFC [ j]!

)−1exp (−σN−νN) (σN +νN)ξ

ad,SDFC [ j]+ 2β

(ξ

ad,SDFC [ j]!

)−1

× exp(−σN− νN

2

)(σN +

νN2

)ξad,SDFC [ j]

, (B.22)

respectively. Using ψ1(ξad,SDFC [ j])−ψ2(ξ

ad,SDFC [ j]) = 0 and some basic manipulations, we obtain

β −α =12

exp(−νN

2

)(σN +νNσN + νN

2

)ξad,SDFC [ j](

exp (νN)

(σN

σN +νN

)ξad,SDFC [ j]

−1

).

(B.23)

Applying (B.23) and P1 =12 to (3.27), we have Υ(ξ ad,SD

FC [ j]) = θ1θ24 exp

(−νN

2

)(σN+νNσN+ νN

2

)ξad,SDFC [ j]

,

where θ1 =(

exp (νN) (σN/(σN +νN))ξad,SDFC [ j]−1

)and θ2 =

(2+N(ν + 2σ)−2ξ

ad,SDFC [ j]

).

To prove Υ(ξ ad,SDFC [ j])> 0, we only need to prove θ1θ2 > 0, since all other terms in Υ are non-

negative. If θ1 > 0, we have ξad,SDFC [ j] < νN/log ((σN +νN)/νN). Applying log(x) ≤ x−1,

where x > 0, we further lower-bound ξad,SDFC [ j] by ξ

ad,SDFC [ j] < σN, which leads to θ2 > 0. If

θ1 < 0, we have ξad,SDFC [ j] > νN/log ((σN +νN)/νN). Applying log(x) ≥ 1− 1/x, where

x > 0, we further upper-bound ξad,SDFC [ j] by ξ

ad,SDFC [ j] > (σN + νN), which leads to θ2 < 0 if

νN/2 > 1. Although the validity of νN/2 > 1 depends on the value of ν and N, it is generally

valid. This is because νN/2 > 1 means that at least one signaling molecule is expected at the

FC if the decision at RXk is “1” and it is a reasonable condition to be satisfied. Since θ1 and θ2

are always both negative or positive, θ1θ2 > 0 holds, which leads to Υ(ξ ad,SDFC [ j]) > 0. There-

fore, QFC[ j] achieves a local minimum at S1 = N/2 when K = 2 in a symmetric topology.

126 Appendix B

Appendix C

Appendix C

C.1 Proof of Theorem 4.1

To evaluate Nim

(~b,τ)

for a circular passive observer S0 centered at any~b with radius R0, we

first review the channel response at the point defined by ~r at the time τ due to an impulse

emission of one molecule from the point at (0,0) at time τ = 0 into an unbounded 2D environ-

ment, C (~r,τ). Based on [135, eq. (3.4)] and the fact that the molecule degradation introduces

a decaying exponential term as in [116, eq. (10)], C (~r,τ) is given by

C (~r,τ) =1

(4πDτ)exp(− |~r|

2

4Dτ− kτ

). (C.1)

We note that Nim

(~b,τ)

for a circular passive observer S0 centered at~b can be obtained by

integrating C (~r1,τ) over S0, where ~r1 is a vector from (0,0) to a point within the RX S0. Using

this method and (C.1), we write Nim

(~b,τ)

as

Nim

(~b,τ)=∫ R0

r=0

∫ 2π

θ=0C (~r1,τ) rdθdr,

=∫ R0

r=0

∫ 2π

θ=0

1(4πDτ)

exp(−|~r1|2

4Dτ− kτ

)rdθdr,

=∫ R0

r=0

∫ 2π

θ=0

1(4πDτ)

exp

(−|~b|2 + r2 + 2|~b|r cosθ

4Dτ− kτ

)rdθdr. (C.2)

Applying [136, eq. 3.339] to (C.2), we rewrite (C.2) as

Nim

(~b,τ)=

14πD

exp(−kτ)∫ R0

r=0

rτ

exp(−|~b|2 + r2

4Dτ)2πI0(

|~b|r2Dτ

)dr, (C.3)

and In(z) is the modified nth order Bessel function of the first kind. We note that there is

no closed-form expression for (C.3). To facilitate the evaluation of (C.3), we apply I0(z) ≈∑

4i=1 αi exp(βiz) [115, eq. 7] to (C.3) to arrive the approximate expression for Nim

(~b,τ)

for a

127

128 Appendix C

circle observer in (4.3). This completes the proof.


Using |~b|= 0, we simplify (C.2) as

Nim,self (τ) =∫ R0

r=0

∫ 2π

θ=0

exp(− r2

4Dτ− kτ

)(4πDτ)

dθ dr,

=∫ R0

r=0

r(2Dτ)

exp(− r2

4Dτ− kτ

)dr. (C.4)

We then apply [136, eq. 2.33.12] given by

∫xm exp (−βxn)dx = − (γ−1)!

exp (−βxn)

n

×

(γ−1

∑k=0

xnk

k!β γ−k

), γ =

m+ 1n

, (C.5)

to (C.4) to solve (C.4). We finally arrive at (4.5), which completes the proof.


We note that integrating (4.3) over τ incurs very high complexity. Thus, we simplify (C.2) by

assuming that the concentration of molecules throughout the circular RX is uniform and equal

to that at the center of the RX, i.e.,

Nim

(~b,τ)≈ πR2

0C(~b,τ)

. (C.6)

Eq. (C.6) is accurate if |~b| is relatively large and thus it is inaccurate when |~b|= 0. Based

on (4.2) and (C.6), we evaluate Nct

(~b,∞

)as

Nct

(~b,∞

)≈ πR2

0

∫∞

τ=0qC(~b,τ)

dτ . (C.7)

We then employ [136, eq. 3.471]

∫∞

0xν−1 exp

(−β

x− γx

)dx = 2

(β

γ

) ν

2

Kν(2√

βγ), (C.8)

to solve (C.7) as (4.6). This completes the proof.

§C.4 Proof of Theorem 4.4 129


By applying (4.5),∫

∞

=0 exp(−px)dx = 1/p [136, eq. 3.310], and [136, eq. 3.324.1]

∫∞

0exp(−β

x− γx

)dx =

β

γK1(√

βγ), (C.9)

to (4.2), we evaluate Nct,self (∞) as (4.8). This completes the proof.


We first write

ENagg

(~b|λ

)= E ∑

~a∈Φ(λ )

N(~b|~a). (C.10)

Using Campbell theorem [38], we rewrite (C.10) as

E ∑~a∈Φ(λ )

N(~b|~a)=∫ R1

|~r|=0

∫ 2π

ϕ=0N(~b|~r)λ |~r|dϕ d|~r|, (C.11)

where~r is a vector from (0,0) to a point within the environment circle S1 and ϕ is the supple-

mentary angle of the angle between~r and~b. We note that N(~b|~r) is obtained by multiplying

(C.1) by the emission rate q, integrating over S0, and then integrating over all time up to infinity,

i.e.,

N(~b|~r) =∫

∞

τ=0

∫ R0

|~r0|=0

∫ 2π

θ=0qC(~d,τ)|~r0|dθ d|~r0|dτ ,

=∫

∞

τ=0

∫ R0

|~r0|=0

∫ 2π

θ=0

q(4πDτ)

exp

(− |

~d|2

4Dτ− kτ

)|~r0|dθ d|~r0|dτ ,

=∫

∞

τ=0

∫ R0

|~r0|=0

∫ 2π

θ=0

q(4πDτ)

exp

(−|

~l|2 + |~r0|2 + 2|~l||~r0|cosθ

4Dτ− kτ

)|~r0|dθ d|~r0|dτ ,

(C.12)

where~l is a vector from~r to~b, i.e., ~l =~b−~r, ~r0 is a vector from~b to a point within the RX

circle S0, ~d is a vector from ~r to ~r0, and θ is the supplementary angle of the angle between~l and ~r0. According to the law of cosines, we obtain |~l|2 = |~b|2 + |~r|2 + 2|~b||~r|cosϕ and

|~d|2 = |~l|2 + |~r0|2 + 2|~l||~r0|cosθ . We then apply |~l| =√|~b|2 + |~r|2 + 2|~b||~r|cosϕ to rewrite

130 Appendix C

(C.12) as

N(~b|~r) =∫

∞

τ=0

∫ R0

|~r0|=0

∫ 2π

θ=0

q(4πDτ)

exp

(−Υ(~b)2

4Dτ− kτ

)|~r0|dθ d|~r0|dτ , (C.13)

where Υ(~b) is given in .

Υ(~b) =

√Ω(~b)+ |~r0|2 + 2

√Ω(~b)|~r0|cosθ , (C.14)

and Ω(~b) = |~b|2 + |~r|2 + 2|~b||~r|cosϕ . Applying [136, eq. 3.471] to (C.13), we obtain

N(~b|~r) =∫ R0

|~r0|=0

∫ 2π

θ=0

q2Dπ

K0

(√kD

Υ(~b)

)|~r0|dθ d|~r0|. (C.15)

We finally substitute (C.15) into (C.11), we arrive at (4.9).


We first derive LN†agg(~xi|λ )

(s) as

LN†agg(~xi|λ )

(s)

= EΦ

exp−sN†

agg(~xi|λ )

. (C.16)

We recall that the ith bacterium observes molecules in the environment released from all

bacteria (also including the molecules released from itself). Thus, we have

N†agg (~xi|λ ) = ∑

~x j∈Φ(λ )

N (~xi|~x j)

= N (~xi|~xi)+ ∑~x j∈Φ(λ )/~xi

N (~xi|~x j) , (C.17)

where N (~xi|~xi) = Nself and Nself is given in (4.8). We then write the second term of the second

line in (C.17) as

∑~x j∈Φ(λ )/~xi

N (~xi|~x j) = ∑~a∈Φ(λ)

N (~xi|~a) , (C.18)

where λ =(λπR2

1−1)/πR2

1. We consider a new density λ to keep the average number of

bacteria the same after the approximation of (C.18). Applying (C.17) and (C.18) to (C.16), we

§C.7 Proof of Remark 4.4 131

rewrite (C.16) as

LN†agg(~xi|λ )

(s)

= EΦ

exp−s ∑

~a∈Φ(λ )

N(~xi|~a)+Nself

,

= EΦ

exp−s ∑

~a∈Φ(λ )

N(~xi|~a)+Nself

,

= EΦ

exp−s ∑

~a∈Φ(λ )

N(~xi|~a)exp−sNself

= exp(−sNself

)EΦ

∏

~a∈Φ(λ )

exp−sN(~xi|~a)

. (C.19)

Using PGFL for the PPP [38, eq. (4.8)], we rewrite (C.19) as (4.18). This completes the

proof.

C.7 Proof of Remark 4.4

Recalling K = ∑~xi∈Φ(λ ) B(~xi,Φ), we directly write EK (instead of using the MGF of K) as

EK= EK= E ∑~xi∈Φ(λ )

Pr (B(~xi,Φ) = 1), (C.20)

where K is the mean of K for a given instantaneous realization of Φ. Using Campbell-Mecke

theorem of PPPs [38, eq. (8.7)] given by

E∑x∈Φ

h(x,Φ)= λ

∫R2

E(h(x,Φ))dx, (C.21)

to (C.20), we rewrite (C.20) as

E ∑~xi∈Φ(λ )

Pr (B(~xi,Φ) = 1)=∫ R1

|~r1|=0EPr (B(~r1,Φ) = 1)λ2π|~r1|,d|~r1|. (C.22)

Applying (4.30) and (4.15) to (C.22), we arrive at (4.42), which completes the proof.

132 Appendix C

C.8 Proof of Remark 4.5

Based on (4.41), we write the second moment of K as

E(K)2 ≈ (EK)2 +EK. (C.23)

Using (C.23) and VarK= E(K)2− (EK)2, we obtain VarK ≈EK. In addition, as

discussed in Remark 4.3, the approximation used in (4.34) is more accurate when the density

λ is lower. That complete the proof.

Appendix D

Appendix D

D.1 Derivation of Performance metrics

Due to the transport delay experienced by the molecules that arrive at the RX, the RX may

receive the molecules released from the current and all previous symbol slots. Based on (5.2),

we obtain the probability that the molecule being released in the kth symbol slot arrives during

the nth symbol slot, i.e., F((n− k + 1)T )− F((n− k)T ). We denote Nobn,k as the number

of molecules that arrive during the nth slot that were released at the beginning of the kth

symbol slot. We then have Nobn = ∑

nk=1 Nob

n,k = ∑n−1k=1 Nob

n,k +Nobn,n, where ∑

n−1k=1 Nob

n,k is the ISI

and Nobn,n is from the intended molecular signal. Since the molecules released in a given slot

are transported independently and have the same probability to arrive during the nth slot, Nobn,k

follows a binomial distribution, i.e.,

Nobn,k ∼ XkB(N,F((n− k+ 1)T )−F((n− k)T )). (D.1)

We note that modeling Nobn,k with the binomial distribution makes the analysis of Nob

n cum-

bersome, since a sum of Binomial RVs is not in general a Binomial RV. Fortunately, Nobn,k can

be accurately approximated by a Poisson distribution when N is large and F((n− k+ 1)T )−F((n− k)T ) is small with NF((n− k+ 1)T )−F((n− k)T ) < 10. By doing so, we rewrite

Nobn,k as

Nobn,k ∼ XkP(N(F((n− k+ 1)T )−F((n− k)T ))). (D.2)

The sum of independent Poisson RVs is also a Poisson RV whose mean is the sum of the

means of the individual Poisson RVs. As such, we have

Nobn ∼ P (γ) . (D.3)

where γ = N ∑nk=1 Xk(F((n− k + 1)T )−F((n− k)T )). In the following, we aim to derive

Pr(Nobn < ξ ), since it lays the foundation for deriving all performance metrics in this paper.

133

134 Appendix D

Based on (D.3), the CDF of the Poisson RV Nobn is written as

Pr(Nobn < ξ |X1:n) =

ξ

∑j=1

exp(−γ)γ j

j!. (D.4)

We note that the the large number of summation terms in (D.4) makes (D.4) have very high

computational complexity when ξ is large. To facilitate the evaluation when ξ is large, we

further approximate Nobn as a Gaussian RV as follows:

Nobn ∼ N(γ ,γ), (D.5)

where γ = N ∑nk=1 Xk(F((n− k+ 1)T )−F((n− k)T )). The Gaussian approximation for Nob

n

in (D.5) is accurate when γ > 10. We define X1:n = X1,X2, . . . ,Xn as the subsequence of the

symbols transmitted by the TX. Based on (D.5), we obtain the conditional CDF of the Gaussian

RV Nobn for the given X1:n as

Pr(Nobn < ξ |X1:n) =

12

(1+ erf

(ξ −0.5− γ√

2γ

)), (D.6)

where 0.5 is a continuity correction. Using (D.4) or (D.6), we obtain the following conditional

probabilities for the given X1:n−1 as:

Pr(Yn = 0|Xn = 0,X1:n−1) = Pr(Nobn < ξ |Xn = 0,X1:n−1), (D.7)

Pr(Yn = 1|Xn = 0,X1:n−1) = 1−Pr(Nobn < ξ |Xn = 0,X1:n−1), (D.8)

Pr(Yn = 0|Xn = 1,X1:n−1) = Pr(Nobn < ξ |Xn = 1,X1:n−1), (D.9)

and

Pr(Yn = 1|Xn = 1,X1:n−1) = 1−Pr(Nobn < ξ |Xn = 1,X1:n−1). (D.10)

Using (D.7)-(D.10), we first derive the conditional mutual information between channel

input and output and the conditional symbol error probability given the subsequence of the

previous symbols transmitted by the TX, X1:n−1. To assess the overall system communication

performance when transmitting different sequences of symbols, we then evaluate the average

mutual information and the average symbol error probability over all realizations of X1:n and

all symbol slots from 1 to n.

Mutual Information: We derive the conditional mutual information between Xn and Yn for

136 Appendix D

where Ψk is a set that includes all realizations of X1:k−1.

Throughput: We derive the throughput, i.e., the maximal average mutual information, as

C = maxξ

1n

n

∑k=1

∑X1:k−1∈ΨkI(Xk;Yk|X1:k−1)

2k−1 bits/slot. (D.19)

Error Probability: We derive the symbol error probability in the nth slot for the given

X1:n−1 as

Q[n|X1:n−1] = (1−P1)Pr(Yn = 1|Xn = 0,X1:n−1)+P1Pr(Yn = 0|Xn = 1,X1:n−1). (D.20)

We derive the average symbol error probability over all realizations of X1:n−1 and all sym-

bol slots from 1 to n as

Q =1n

n

∑k=1

∑X1:k−1∈ΨkQ[k|X1:k−1]

2k−1 . (D.21)

D.2 Proof of Corollary 5.1

Since Q is the sum of Q[n|X1:n−1] based on (D.21), we need to prove that Q∗[n|X1:n−1]→ 0

when N→∞, where Q∗[n|X1:n−1] =minξ

Q[n|X1:n−1]. Assuming P1 =12 , we first rewrite (D.20)

as

Q[n|X1:n−1] =12+

14

[erf

(ξ −0.5− (N(Y1 +Y2))√

2(N(Y1 +Y2))

)−erf

(ξ −0.5−NY2√

2NY2

)], (D.22)

where Y1 = (F(T )− F(0)) and Y2 = ∑n−1k=1 Xk(F((n− k + 1)T )− F((n− k)T )). We then

obtain the optimal ξ that minimizes Q[n|X1:n−1]. To this end, we take the first derivative of

(D.22) with respect to ξ and solve the resultant equation to derive the optimal ξ that minimizes

Q[n|X1:n−1] as

ξ∗[n|X1:n−1] =

NY1

ln ((Y1 +Y2)/Y2). (D.23)

Substituting (D.23) into (D.22), we write the optimal error probability Q[n|X1:n−1] as

Q∗[n|X1:n−1] =12+

14

[erf

( √NA√

2(Y1 +Y2)

)− erf

(√NB√2Y2

)], (D.24)

where

A =

(Y1

ln ((Y1 +Y2)/Y2)−(Y1 +Y2)

)(D.25)

§D.3 Proof of Corollary 5.2 137

and

B =

(Y1

ln ((Y1 +Y2)/Y2)−Y2

). (D.26)

If we can prove A < 0 and B > 0, then we have

limN→∞

Q∗[n|X1:n−1] =12+

14[erf(−∞) −erf (∞)] = 0. (D.27)

We now prove A < 0 and B > 0. Since Y1 > 0 and Y2 > 0, it is reasonably to assume

Y1 = xY2, x > 0. Using Y1 = xY2, we simplify the conditions A < 0 and B > 0 to x/(1+ x)−ln(1+ x)< 0 and x− ln(1+ x)> 0, respectively. We find that g(x) = x/(1+ x)− ln(1+ x) is

a decreasing function and f (x) = x− ln(1+x) is an increasing function with respect to x since

g′(x) =−x/(1+ x)2 < 0 and f ′(x) = 1−1/(1+ x)> 0 if x > 0. By inspection, we also find

g(x) = 0 and f (x) = 0 at x = 0. Thus, we have g(x)< 0 and f (x)> 0 for x > 0, which means

A < 0 and B > 0. Thus, we verify that Q∗[n|X1:n−1]→ 0 when N → ∞, which completes the

proof.

D.3 Proof of Corollary 5.2

We first prove I(Xn;Yn|X1:n−1) ≤ 1bits/slot. As per the Shannon entropy of probability distri-

butions for single parties, we have I(Xn;Yn|X1:n−1)≤minH(Xn|X1:n−1),H(Yn|X1:n−1). Based

on definition of entropy, the maximal H(Xn|X1:n−1) and H(Yn|X1:n−1) is 1bits/slot when

Pr(X1 = 0) = P1 = 12 and Pr(Y1 = 0) = 1

2 . Thus, the mutual information is bounded by

I(Xn;Yn|X1:n−1) ≤ 1bits/slot.

We then prove that Q→ 0 is a sufficient condition for I(Xn;Yn|X1:n−1) = 1bits/slot. Based

on (D.20), Q[n|X1:n−1]→ 0 means Pr(Yn = 1|Xn = 0,X1:n−1)→ 0 and Pr(Yn = 0|Xn = 1,X1:n−1)→0. Applying these two expressions to (D.12) and (D.15), we obtain I(Xn;Yn|X1:n−1) = 1bits/slot,

which proves Q→ 0 is a sufficient condition. We finally prove that Q→ 0 is a necessary condi-

tion for I(Xn;Yn|X1:n−1) = 1bits/slot. Since H(Yn|X1:n−1) ≤ 1 and H(Yn|Xn,X1:n−1) ≥ 0, thus

I(Xn;Yn|X1:n−1) = 1bits/slot is achieved only when H(Yn|X1:n−1) = 1 and H(Yn|Xn,X1:n−1) =

0. H(Yn|Xn,X1:n−1) = 0 means H(Yn|Xn = 0,X1:n−1) = 0 and H(Yn|Xn = 1,X1:n−1) = 0 based

on (D.15). There are four cases leading to H(Yn|Xn = 0,X1:n−1) = 0 and H(Yn|Xn = 1,X1:n−1) =

0 including:

1. Pr(Yn = 0|Xn = 1,X1:n−1) = 0 and Pr(Yn = 1|Xn = 0,X1:n−1) = 0;



4. Pr(Yn = 0|Xn = 1,X1:n−1) = 1 and Pr(Yn = 1|Xn = 0,X1:n−1) = 1.

138 Appendix D

Since case 4) does not satisfy Pr(Yn = 0|Xn = 1,X1:n−1) + Pr(Yn = 1|Xn = 0,X1:n−1) ≤ 1,

case 4) is not valid. Moreover, cases 2) and 3) result in Pr(Yn = 0|X1:n−1) = 1 and Pr(Yn =

1|X1:n−1) = 1, respectively, which leads to H(Yn|X1:n−1) = 0. Thus, they are not valid either.

We note that only case 1) satisfies both H(Yn|X1:n−1) = 1 and H(Yn|Xn,X1:n−1) = 0 and case

1) leads to Q→ 0. Thus, Q→ 0 is a necessary condition. Therefore, we prove Q→ 0 is a

sufficient and necessary condition for I(Xn;Yn|X1:n−1) = 1bits/slot.

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