1. IEEE.AM/MMES Tenerife 2010 1 RELIABILITY STUDY OF MESH NETWORKS MODELED AS RANDOM GRAPHS. Louis Petingi Computer Science Dept. College of Staten Island City University of New York

2. IEEE.AM/MMES Tenerife 2010 2 Network Reliability Edge 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 q e =1-p e. Classical Reliability R K (G)= Pr { All the terminal vertices remain connected after deletion of the failed edges }.

3. IEEE.AM/MMES Tenerife 2010 3 Operating States G=(V,E) K = dark vertices operatingnon-operating

4. 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 H p(H)=(0.4) 4 (0.6) 2 Suppose for every edge e q e =0.6 p e =0.4

5. IEEE.AM/MMES Tenerife 2010 5 Diameter constrained reliability Petingi, 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. R K (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}

6. 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.

7. IEEE.AM/MMES Tenerife 2010 7 Heuristics to estimate reliability Monte Carlo Monte Carlo techniques excellent to estimate the classical reliability R K (G) as well as the Diameter constrained reliability R K (G,D). (Cancela and El Khadiri – IEEE Trans. on Rel. (1995)) Monte Carlo Recursive Variance Reduction Monte Carlo Recursive Variance Reduction (RVR) for classical reliability. Monte Carlo Recently Monte Carlo successfully applied to estimate Diameter constrained reliability.

8. IEEE.AM/MMES Tenerife 2010 8 Wireless Networks (Mesh) communication channels (links) digraph Khandani et. al (2005) (capacity of wireless channel) R bits per channel use  = E(|f| 2 ) f=Fading state of channel Rayleigh r.v. Rayleigh r.v. (Map) Mesh access point -Transceiver

9. IEEE.AM/MMES Tenerife 2010 9 Wireless Networks (Mesh) Source-to-K-Terminal reliability (digraph) links (channels) K = terminal nodes K = terminal nodes ss q(l) = prob. that link l fails. R s,K (G) = Pr {source s will able to send info. to all the terminal nodes of K}

10. IEEE.AM/MMES Tenerife 2010 10 Wireless Networks (Mesh) Nodes Redundancy (optimization) Several applications of Monte Carlo SNRdb = 30, R=1 bit/channel use, SNRdb = 30, R=1 bit/channel use,  = E(|f| 2 )=1 R s,t (G) = 0.904 Red 3 = 0.904 - 0.792 = 0.112 (40) (20) (25) (10) 1 2 3 (20) (28) (15) (28) (20) t s.33.464.8.33.543.33 Red 2 = 0.904 – 0.693 = 0.211 Red 1 = 0.904 – 0.763 = 0.141

11. IEEE.AM/MMES Tenerife 2010 11 Wireless Networks (Mesh) Areas connectivity (optimization) O G ( R 1, R 2 ) : Find in G[ R 1, R 2 ] nodes u and v, u  V 1 and v  V 2, such as Mobile map 1, M 1 Same transmission rate R, Transmission power, Noise average power (assuming additive white Gaussian noise η). Mobile map 2, M 2 Areas different physical characteristics n-path loss exp, f –fading state

12. IEEE.AM/MMES Tenerife 2010 12 Future work performance objectives 1. Specify optimization problems in communication (determine performance objectives to be evaluated). 2. Improve 2. Improve (analyze) edge reliability models (integrate antenna gains and nodes interference. Monte Carlo 3. Implementation of Monte Carlo RVR techniques to evaluate reliability adapted for parallel processing environment (CSI’s high performance computers). 4. Test 4. Test correctness of results.

13. 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.