IEEE.AM/MMES Tenerife 2010 1

RELIABILITY STUDY OF MESH NETWORKS MODELED AS

RANDOM GRAPHS.

Louis Petingi

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

IEEE.AM/MMES Tenerife 2010 2

Network ReliabilityEdge Reliability Model (1960s)

Communication Network modeled as a graph G=(V,E).

Distinguished set K of terminals vertices (participating nodes).

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

Classical Reliability

  RK(G)= Pr { All the terminal vertices remain connected   after deletion of the failed edges }.

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Operating States

G=(V,E)

K = dark vertices

operating non-operating

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Operating States

Let O be the set of operating states H of G

G=(V,E)

K = dark vertices

OH OH OH He He

iiK

i i

qpHpHGR

)(}Pr{)(

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

Suppose for every edge e

qe=0.6

pe=0.4

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Diameter constrained reliabilityPetingi, and Rodriguez (2001)

Suppose that we want to know what is the probability that the terminal nodes meet a delay constrained D.T, for some upper bound D.

RK(G,D) = Prob {After random failures of the edges, for every pair of

terminal nodes u and v, there exists an operational

path of length ≤ D}

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Applications

Videoconference, we take K to be the set of the participating nodes, and the Diameter constrained reliability gives the probability that we can find short enough paths between all of them.

To avoid congestion by looping data, assign a maximum number of hops to each data packet, to control information. In this case, the diameter constrained unreliability (the complement to one of the reliability) gives the probability that there are some nodes of the network which are not reachable by using these protocols.

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Heuristics to estimate reliability

Monte CarloMonte Carlo techniques excellent to estimate the classical reliability RK(G) as well as the Diameter constrained reliability RK(G,D).

(Cancela and El Khadiri – IEEE Trans. on Rel. (1995))

Monte Carlo Recursive Variance ReductionMonte Carlo Recursive Variance Reduction (RVR) for

classical reliability.

Recently Monte CarloMonte Carlo successfully applied to estimate Diameter constrained reliability.

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Wireless Networks (Mesh)

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.

(Map) Mesh access point -Transceiver

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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}

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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

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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

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Future work

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

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

3. Implementation of Monte CarloMonte Carlo RVR techniques to evaluate reliability adapted for parallel processing environment (CSI’s high performance computers).

4.4. Test Test correctness of results.

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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.

[PR] Petingi L., Rodriguez J.: Reliability of Networks with Delay Contraints. Congressus Numerantium (152), (2001), pp. 117-123.