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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
ITAP 2012, Wuhan China 2
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
ITAP 2012, Wuhan China 3
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
ITAP 2012, Wuhan China 4
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
ITAP 2012, Wuhan China 5
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.
ITAP 2012, Wuhan China 6
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}
ITAP 2012, Wuhan China 7
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
ITAP 2012, Wuhan China 8
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
ITAP 2012, Wuhan China 9
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)
ITAP 2012, Wuhan China 10
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
ITAP 2012, Wuhan China 11
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)
ITAP 2012, Wuhan China 12
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)
ITAP 2012, Wuhan China 13
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
ITAP 2012, Wuhan China 15
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
ITAP 2012, Wuhan China 16
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!
ITAP 2012, Wuhan China 17
ITAP 2012, Wuhan China 18
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.