CENS, April 20 2007 1

Modeling Wireless Sensor Networks

Bhaskar KrishnamachariMing Hsieh Department of Electrical Engineering

USC Viterbi School of Engineering

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Overview

• Mathematical modeling provides fundamental insights into:

1. Link layer behavior

2. Protocol design

3. Scaling and architecture

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1. Modeling Link Layer Behaviorin Low Power Wireless Networks

Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007.

Dongjin Son, Bhaskar Krishnamachari, John Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ’06 + Ongoing work

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• Two simplified models form the basis of >95% of the literature on wireless networks:

X

Circular radio range with perfect reception within &zero reception outside

Collision with simultaneous transmissions within range

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Link Quality Variation with Distance

From Wooet al. ‘03

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An Explanatory Model• Basic idea: compose the following two functions (a) SNR

versus distance with (b) PRR versus SNR

Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007.

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Bimodal PRR Distribution• A majority of the links are either good (above 90%) or bad (below 10%), matching empirical findings (e.g., Cerpa et al. ’05)

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Expectation and Varianceof Packet Reception Rate

Justifies the presence of “long links”

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• Models Incorporated into simulators:– TOSSIM (Berkeley)– Castalia (NICTA, Australia)

• Standalone code at http://ceng.usc.edu/~anrg/downloads.html

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Conservative protocol assumption: always a collision

Concurrent Transmissions

Reality: SINR makes the difference

Son, Krishnamachari, Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ‘06

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SINR-view of Interference

)2(8)exp5.01( 10 lfSINRPRR

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Feasibility of Concurrent Transmissions

P1g11/(P2g21+N) ≥

P2g22/(P1g12+N) ≥

S1 R1

R2S2

g11g12

g21

g22

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Linear Topology Case

Counter-intuitive“embedding” of simultaneousconversations

S1 R1 R2S2

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2. MAC Design for ScalableData Collection

Kiran Yedavalli, Bhaskar Krishnamachari, "Enhancement of the IEEE 802.15.4 MAC Protocol for Scalable Data Collection in Dense Sensor Networks", USC Computer Engineering Technical Report CENG-2006-14, November 2006.

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State of the Art: IEEE 802.15.4

• Specifies both PHY and MAC layers for low-power, low-rate embedded wireless networks.

• The MAC protocol is a slotted CSMA with binary exponential back-off

• 256 nodes allowed by standard

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p-persistent CSMA Model

…

: Idle Slot

: Collision Slot

: Successful Slot

epoch

…… … ……

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Delay and Energy Expressions

1. Average expected epoch delay

2. Average expected epoch energy consumption

ξR: Energy Consumption per node per time slot in the Receive State

ξT: Energy Consumption per node per time slot in the Transmit State

1

( 1)(1 )[ ](1 )

n

n n

L L pE Tnp p

1

2 1

( 1)(1 )[ ](1 ) (1 )

n

n R Tn n

L L p LE Ep p p

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Optimality• Delay

• Energy2

1 , 1( , )

2 ( 1)( 1), 1

( 1)( 1)

Topt

Ln

p n Ln n n L n

Ln n L

2 2 ( 1)( 1) 4 ( 1)( 1)( , ) ,

( 1)( 1) 2 ( 1)( 1)E T

optR

n n n L L n np n L

n n L L n

If ξR = ξT, the same transmission probability optimizes both delay and energy simultaneously.

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A Useful Optimality Criterion

0 10 20 30 40 50 60 70 80 90 1000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Number of nodes in an epoch (n)

E[T

Idle

,n]/E

[Tn]

L = 5

poptT (n,L)

poptT (n,L) - 0.003

poptT (n,L) + 0.003

,[ ] 2 1, , ( , ) , ,[ ] ( 1)( 2 1 1)Idle nT

optn

E T L Lp p n LE T L L

When the number of contending nodes is high, this provides sensitive feedback that can be used to adapt the access rate

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Receiver Feedback Enhancement

• Receiver performs measurement and broadcast

• Window update rule:

• All contending nodes change the window size simultaneously

,

2 1( 1)( 1)( 2 1 1)

,Current

Idle Currentnext

Current

L LWTL L

WT

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Results

50 55 60 65 70 75 80 85 90 95 1000

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40

60

80

Number of Contending Nodes

Thro

ughp

ut (K

bps)

Packet Length = 50 Bytes

IEEE 802.15.4Enhanced IEEE 802.15.4

50 55 60 65 70 75 80 85 90 95 1000

2

4

6

8

Number of Contending Nodes

Ene

rgy

(mJo

ules

) IEEE 802.15.4Enhanced IEEE 802.15.4

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3. Fair and Efficient Rate Control for Data Gathering

Avinash Sridharan and Bhaskar Krishnamachari, "Maximizing Network Utilization with Max-Min Fairness in Wireless Sensor Networks," to be presented at 5th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), April 2007.

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Problem Formulation• Allocate rates to each

source to (a) ensure fairness, and (b) efficient use of available bandwidth.

• Closely related prior work by Rangwala et al. SIGCOMM ’06 – focuses primarily on fairness and proposes a TCP-like AIMD mechanism

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• Receiver capacity interference model – source rates from node’s sub-tree and its interfering neighbors’ sub-trees must not exceed available bandwidth

Problem Formulation

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Validating Capacity Model

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• Bottleneck rate turns out to be the minimum supply/demand ratio:

• This can be calculated easily given the tree, interference graph, and receiver bandwidths

P1: Solving for Fairness

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P2: Solving for Efficiency

• Duality-based approach based on the classic work on optimization flow control by Low & Lapsley ’99

• Introduce new dual variables (shadow prices) that weigh resource constraints

• Yields distributed algorithms with market auction interpretation

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Structure of P2’s Lagrange Dual

• Each router sets a price for its bandwidth

• The rate for each source depends on sum-price of routers affected by its flow

sum-price

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• Increment the shadow prices in the direction of the negative sub-gradient (determined by source rates)

• Choose source-rates to maximize component function (determined by shadow prices)

• In general, this could be a very slow iterative process…

Subgradient Optimization

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Good News

• Numerical evidence: setting all shadow prices to 1 provides near-optimal solutions in one iteration!

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Resulting Heuristic

1. First determine and allocate min rate to all sources

2. Give rank to each source that is inversely proportional to the number of downstream receivers whose bandwidth it consumes;

3. Allocate saturating rates to flows, in rank order

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Simulation Results

CDF of difference from optimal solution

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Ongoing Work

• Test-bed Implementation

• Cross-layer extensions

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4. Fundamental Scaling Lawsfor Store and Query Sensor Networks

Joon Ahn and Bhaskar Krishnamachari, "Fundamental Scaling Laws for Energy-Efficient Storage and Querying in Wireless Sensor Networks", ACM MobiHoc, May 2006.

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• Race between increasing supply and demand:- Energy and storage- Application-specific event and query traffic

• The winner of this race determines scalability.

In a Nutshell

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• N nodes deployed in a 2D area with constant density for time T

• m atomic events and qi queries for the ith event, all uniformly distributed

• Can create ri replicas for event i to reduce search cost (at the expense of increased replication cost)

Preliminaries

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Data-Centric Querying Approaches

• Unstructured: expanding ring searches, random walks.

• Structured: Geographic Hash Table, DIFS, DIM

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Energy Cost Scaling

• Creplication = c1

r : # of copies of an event

N : # of nodes

• Csearch(unstructured) = c2 • Csearch(structured) = c3

EVENTEVENT REPLICATIONUNSTRUCTURED QUERYSTRUCTURED QUERY

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Energy Optimization Formulation

S : total storage sizem : the total number of eventsqi : the query rate for ith eventri : the number of copies of ith event

Cs(ri) : the expected minimum search cost of ith event

Cr(ri) : the expected replication cost of ith event

Cr(r) = c1 Cs(r) = c2

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Optimization Solution

Minimizer

The Optimized Total Cost

(inactive constraint)

(active constraint)

qi : # of queries for event i

N : # of nodesS : total storage

sizem : # of events

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Optimal Total Cost

Simplified, assuming : q : # of queries per event

N : # of nodesS : total storage

sizem : # of eventsif

if

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Illustration of Energy Scaling

m : # of eventsq : # of queries

per event

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I - Storage and Energy Scalability Results

Energy ConditionThe energy requirement per node is boundedif and only if mq1/2 = O(N1/4)

Energy constraint is stricter than storage constraint

m : # of eventsq : # of queries per eventN : # of nodes

Storage ConditionA network scales efficiently with bounded storage per node

if mq1/2 = o(N3/4)

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II - Fixed Energy Budget Results

S – successful operation region

N : # of nodese: per-node energy budget

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III - Network Lifetime Scaling Results

Network Lifetime as a function of Network Size

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Summary• Only certain classes of applications can be sustained in arbitrarily

large sensor networks.

• Specifically, if mq1/2 = O(N1/4) for unstructured networks, and mq2/3 = O(N1/2) for structured networks:

a. The network can operate with bounded energy and storage per node.

b. The network lifetime does not decrease with network size for a given energy budget.

• The results can be reinterpreted to understand how to tier sensor networks into zones with localized queries

• These results generalize in a straightforward manner to 1D and 3D deployments. 3D deployments are inherently more scalable.

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Final Thoughts

“In theory, theory and practice are the same; in practice, they’re different.”

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Thanks