itap 2012, wuhan china 1 addressing optimization problems in wireless networks modeled as...

ITAP 2012, Wuhan China 1

Addressing optimization problems in wireless networks modeled as

probabilistic graphs.

Louis Petingi

Computer Science Dept.College of Staten IslandCity University of New York


Network ReliabilityEdge Reliability Model (1960s)

i. Communication network modeled as a digraph G=(V,E).

ii. Distinguished set K of terminals nodes (participating nodes) and source-node

iii. Each edge e fails independently with probability qe=1-pe.

iv. Classical Reliability (Source-to-K-terminal reliability)

Rs,K(G)= Pr { there exists an operational dipath between s and u, after deletion of failed links).

.Ks

,Ku


Operating States

G=(V,E)

K = dark vertices

operating non-operating

s

Sample space All possible subgraphs of G

ss

s s

s

s

s


Operating States

Let O be the set of operating states H of G.

G=(V,E)

K = dark vertices

OH OH OH He He

iiKs

i i

qpHpHpGR

)(}{)(.

Hp(H)=(0.4)4(0.6)2

Suppose for every edge e

qe=0.6

pe=0.4

s

s


Heuristics to estimate reliability – classical reliability

Motivation of Source-to-K-terminal reliability :

Single-source broadcasting

Rs,K(G) is #P-complete - Rosenthal (Reliability and Fault Tree Analysis SIAM 1975 – for the undirected case).

Cancela and El Khadiri – (IEEE Trans. on Rel. (1995)) Monte Carlo Monte Carlo Recursive Variance ReductionRecursive Variance Reduction (RVR) for classical reliability.


Wireless Networks (Mesh) Source-to-K-Terminal reliability (digraph)

links (channels)

K = terminal nodesK = terminal nodes

sss

q(l) = prob. that link l fails.

Rs,K (G) = Pr {source s will able to send

info. to all the terminal nodes of K}


Wireless Network, link probability

communication channels (links)

digraph

yprobabilit failurelink }{)( RCprobpeq outage

Khandani et. al (2005) (capacity of wireless channel)

)1(log2||

2 SNRC nd

f

)exp(1)( 'SNRd n

eq

R bits per channel use

= E(|f|2)

12'

R

SNRSNR

f=Fading state of channelf=Fading state of channel Rayleigh r.v.Rayleigh r.v.

Sensor node Transceiver


Wireless Networks (Mesh) Nodes Redundancy (optimization)

Several applications of Monte Carlo

SNRdb = 30SNRdb = 30, , R=1 bit/channel useR=1 bit/channel use,, = E(|f|2)=1

Rs,t (G) = 0.904

Red3= 0.904 - 0.792 = 0.112

(40)(40)

(20)(20)

(25)(25)

(10)(10)11 22

33(20)(20)

(28)(28)

(15)(15)

(28)(28)

(20)(20)

tt

ss

)exp(1)( 'SNRd n

eq .33

.33.464

.8

.33

.543.543

.33

)()(),,(Re ,, xGRGRKsGd KsKsx

Red2= 0.904 – 0.693 = 0.211

Red1= 0.904 – 0.763 = 0.141


Wireless Networks (Mesh) Areas connectivity (optimization)

G =( V , E )

R egio n 1

2 2 ( ( 2 2 8 8 , , 0.543)

( ( 2 2 5 5 , , 0.464)

1 1

3 3

( ( 4 4 0 0 , , 0 0 . . 8 8 ) )

( ( 2 2 8 8 , , 0 0 . . 5 5 4 4 3 3 ) )

( ( 2 2 0 0 , , 0 0 . . 3 3 3 3 ) )

( ( 2 2 0 0 , , 0 0 . . 3 3 3 3 ) )

4 4 a

b

c

d

R egio n 2

)exp(1)( 'SNRd n

eq

OG(R1, R2) : Find in G[R1,R2 ] nodes u

and v, u V1 and v V2, such as

]),,[( ]),[( 21,21,

2

1

RRRR GRMaxGR yx

Vy

Vxvu

Mobile map 1, M1

Same transmission rate R,

Transmission power,

Noise average power (assuming additive

white Gaussian noise η).

Mobile map 2, M2

Areas differentphysicalcharacteristicsn-path loss exp,f –fading state

601.0, caR

704.0, daR

597.0, cbR

739.0, dbR

K-terminal-to-sink reliability- Motivation (sensor networks)


gateway

sensor nodestransceiver

sink-nodegateway

K-terminal-to-sink reliability

RK,s(G)= Pr { there exists an operational dipath between u and sink s, after deletion of failed links).,Ku

K terminal nodes


Operating States

G=(V,E)

K = dark vertices

s sink-node

operating non-operating

s

Sample space All possible subgraphs of G

s s

s s

s

s

s

OH OH OH He He iisK i iqp HpHpGR

)(}{)(,

K-terminal-to-sink reliability- Motivation (sensor networks)


sensor nodestransceiver

sink-nodegateway

K-terminal-to-sink reliability

RK,s(G)= Pr { there exists an operational dipath between u and sink s, after deletion of failed links).,Ku

K terminal nodes

gateway

OptimizationPut to sleep nodes

Max {RK,s(G-x): x not in K}

Binary Networks (undirected graph)


edge exists if nodes within each other range

Optimization problems in sensor nets.(Graph Theory )

Purpose:Put remaining nodes periodically to sleep to save energy as they are covered by A

2

s

1 3

4 5

Sink node s

Find minimum set of backbone nodes A (including s) such as:

1. A is a dominating-set (all remaining nodes {1 ,4 ,5 } are adjacent to at least one node of A).

A

2

s

1 3

4 5

sink node s

2. The graph induced by the vertices of A must be connected.

dominating-set NP-Complete

good transmitter good receiver poor transmitter

good receiver

no edge

node still transmitting information to closed-by nodes

Probabilistic Networks (directed graph)

14

Anti-parallel links may have different prob . of failure depending on the transmitting or receiving characteristics of nodes

.67

.8complete graph (some links may have large prob. of failure)

Contract A into S

R {1, 4, 5), S (G’) equivalent to Dominating-set

sink node S=A

1

4 5

S

G’

2

s

1 3

4

5

sink node s

R {2,3,), s (G*) equivalent to connectivity in A

2

s

1

3

4

5

GBackbone nodes A={s,2,3}

A

s

2 3

G*

A

A

R(G, A, s) = R {1, 4, 5), S * R {2,3,), s (G*)

RK,s (G) calculated using Monte Carlo

ITAP 2012, Wuhan China


Optimization Problem - choose backbone set of size 3

Assumption about the nodes

1) each node has SNR=1000.

2) transmission rate R = 1 bit per channel use.

3) fading state of each channel has expected value =1. 4) n=2 (open space).

R(G, A,s) =0.5579

R(G, A,s) =0.4413


Final remarks and future work

I. Probabilistic networks are more realistic as nodes transmitting/receiving characteristics maybe very different, and probability of the communication can be accurately evaluated.

II. Simulation problems are well-defined, given the characteristic of the nodes are known, and calculations are done in conjunction with suitable network reliability models .

III. Binary networks assumptions of why two nodes are connected are sometimes not well-defined (unless similarities between nodes are assumed).

IV. Graph Theoretical parameters (widely used to used to measure performance objectives of wireless networks) sometimes are computationally expensive (NP-Complete, or NP-hard) and equivalent reliability measures can be evaluated efficiently using Monte-Carlo techniques.

V. Specify optimization problems in communication (determine performance objectivesperformance objectives to be evaluated).

VI.VI. ImproveImprove (analyze) edge reliability models (integrate antenna gains and nodes interference metrics).

THANK YOU!



References [CE1] H. Cancela, M. El Khadiri. A recursive variance-reduction algorithm for

estimating communication-network reliability. IEEE Trans. on Reliab. 4(4), (1995), pp. 595-602.

[KMAZ] E. Khandani, E. Modiano, J. Abounadi, L. Zheng, Reliability and Route Diversity in Wireless networks, 2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16-18, 2005.

[PET1] Petingi L., Application of the Classical Reliability to Address Optimization Problems in Mesh Networks. International Journal of Communications 5(1), (2011), pp. 1-9.

[PET2] Petingi L., Introduction of a New Network Reliability Model to Evaluate the Performance of Sensor Networks. International Journal of Mathematical Models and Methods in Applied Sciences 5-(3), (2011), pp. 577-585.

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