1. ITAP 2012, Wuhan China 1 Addressing optimization problems in wireless networks modeled as probabilistic graphs. Louis Petingi Computer Science Dept. College of Staten Island City University of New York

2. ITAP 2012, Wuhan China 2 Network Reliability Edge 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 q e =1-p e. iv. Classical Reliability (Source-to-K-terminal reliability) R s,K (G)= Pr { there exists an operational dipath between s and u, after deletion of failed links).

3. ITAP 2012, Wuhan China 3 Operating States G=(V,E) K = dark vertices operatingnon-operating s Sample space All possible subgraphs of G s s s s s s s

4. ITAP 2012, Wuhan China 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 s s

5. ITAP 2012, Wuhan China 5 Heuristics to estimate reliability – classical reliability Motivation of Source-to-K-terminal reliability : Single-source broadcasting R s,K (G) is #P-complete - Rosenthal (Reliability and Fault Tree Analysis SIAM 1975 – for the undirected case). Monte Carlo Recursive Variance Reduction Cancela and El Khadiri – (IEEE Trans. on Rel. (1995)) Monte Carlo Recursive Variance Reduction (RVR) for classical reliability.

6. ITAP 2012, Wuhan China 6 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}

7. ITAP 2012, Wuhan China 7 Wireless Network, link probability 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. Sensor node Transceiver

8. ITAP 2012, Wuhan China 8 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

9. ITAP 2012, Wuhan China 9 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

10. K-terminal-to-sink reliability- Motivation (sensor networks) ITAP 2012, Wuhan China 10 sensor nodes transceiver sink-node gateway K-terminal-to-sink reliability R K,s (G)= Pr { there exists an operational dipath between u and sink s, after deletion of failed links). K terminal nodes

11. ITAP 2012, Wuhan China 11 Operating States G=(V,E) K = dark vertices s sink-node operatingnon-operating s Sample space All possible subgraphs of G s s ss s s s

12. K-terminal-to-sink reliability- Motivation (sensor networks) ITAP 2012, Wuhan China 12 sensor nodes transceiver sink-node gateway K-terminal-to-sink reliability R K,s (G)= Pr { there exists an operational dipath between u and sink s, after deletion of failed links). K terminal nodes gateway Optimization Put to sleep nodes Max {R K,s (G-x): x not in K}

13. 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

14. Probabilistic Networks (directed graph) 14 Probabilistic Networks (directed graph) Anti-parallel links may have different prob. of failure depending on the transmitting or receiving characteristics of nodes.67.8 complete 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 G Backbone 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*) R K,s (G) calculated using Monte Carlo ITAP 2012, Wuhan China

15. 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

16. 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. performance objectives V. Specify optimization problems in communication (determine performance objectives to be evaluated). VI. Improve VI. Improve (analyze) edge reliability models (integrate antenna gains and nodes interference metrics).

17. THANK YOU! ITAP 2012, Wuhan China 17

18. 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.