IEEE.AM/MMES Tenerife 2010 1 RELIABILITY STUDY OF MESH NETWORKS MODELED AS RANDOM GRAPHS. Louis...
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Transcript of IEEE.AM/MMES Tenerife 2010 1 RELIABILITY STUDY OF MESH NETWORKS MODELED AS RANDOM GRAPHS. Louis...
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 }.
IEEE.AM/MMES Tenerife 2010 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
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
IEEE.AM/MMES Tenerife 2010 5
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}
IEEE.AM/MMES Tenerife 2010 6
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.
IEEE.AM/MMES Tenerife 2010 7
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.
IEEE.AM/MMES Tenerife 2010 8
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
IEEE.AM/MMES Tenerife 2010 9
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}
IEEE.AM/MMES Tenerife 2010 10
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
IEEE.AM/MMES Tenerife 2010 11
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
IEEE.AM/MMES Tenerife 2010 12
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.
IEEE.AM/MMES Tenerife 2010 13
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.