itap 2012, wuhan china 1 addressing optimization problems in wireless networks modeled as...
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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
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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
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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
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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
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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.
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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}
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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
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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
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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
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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
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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
)(}{)(,
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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}
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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
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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
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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
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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).
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THANK YOU!
ITAP 2012, Wuhan China 17
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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.
[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.