Maximizing Path Durations in Mobile Ad-Hoc Networks

1. Maximizing Path Durations in Mobile Ad-Hoc Networks Yijie Han and Richard J. La Department of ECE & ISR University of Maryland, College Park CISS, Princeton University March 22nd, 2006

2. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • Maximizing expected path durations • NS-2 simulation results • Parameter update • Conclusion & Future Directions

3. Background • Ad hoc network routing protocols • Table-driven routing protocols (proactive) • Attempt to maintain consistent, up-to-date routing information from each node to every other node in the network. • Each node maintains one or more tables to store routing information. • Example: DSDV(Destination-Sequenced Distance-Vector), WRP (Wireless Routing Protocol), etc • On-demand routing protocol (reactive) • Attempt to minimize the number of required broadcasts by providing a path only when requested • Require path/route discovery phase/mechanism • Examples: AODV( Ad-hoc On-demand Distance Vector), DSR (Dynamic Source Routing)

4. Motivation • On-demand routing protocols in ad-hoc networks • Path recovery procedure initiated when an existing path is broken • Disruption in network service to applications • Performance and overhead shaped by the distribution of link and path durations • Suggests that (expected) path duration should be taken into account when selecting a path • Reduce overhead • Provide more reliable network service to applications • Requires understanding of statistical properties of path duration

5. Existing protocols • Ad-hoc On-demand Distance Vector (AODV) • Selects the first discovered route • Dynamic Source Routing (DSR) • Selects the min-hop route • Associativity Based Routing (ABR) • Each node maintains “associativity” for each neighbor from beacons • Higher beacon counts = more stable links • Destination selects the path with the highest average associativity

6. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

7. Connectivity between nodes • {0, 1}-valued reachability process between two nodes • xij(t) = 1 – if the link (i,j) is up • xij(t) = 0 – if the link (i,j) is down • xij(t) = xji(t) – symmetric links Basic Model (for studying statistical properties of path duration) • V = {1, …, I} - set of mobile nodes moving across a domain D of R2 or R3 • - location/trajectory of node i

8. Basic Model Basic Model • Link durations • {Uij(k), k = 1, 2, ,…} and {Dij(k), k = 1, 2, …} • Uij(k) (resp. Dij(k)) – duration ofk-thup (resp. down) time • Time-varying graph (V, E(t)) t

9. Basic Model • Path discovery phase • Path available between s and d if a set of links provides connectivity • May not be unique • Routing algorithm selects one • Denote the set of links along the selected path byLsd(t) n4 n1 n2 s d n3

10. Excess Life and Path Duration • For each link • - time to live or excess life after time t • Time to live or duration of a path • Path available till one of the links goes down • Path duration = amount of time that elapses till one of the links in breaks down

11. Question:What does the distribution of look like? • In particular, when the hop counter is large • In a large scale MANET, the number of hops is expected to be large

12. Outline • Background • Basic Model • Setup – Parametric Scenario and Difficulties • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

13. Parametric Scenario • Scaling: For each fixed n = 1, 2, …, • -- set of mobile nodes • -- domain across which nodes move • Stationarity:Reachability processes jointly stationary • constitutes a stationary sequence with generic marginals • - CDF of • A pair of source and destination nodes selected at time t = 0 for each n

14. Parametric Scenario (cont’d) • Define • Excess or residual life of a link • Distribution of forward recurrence time • Follows from elementary renewal theory

15. Parametric Scenario (cont’d) • Path duration - • Explore the distributional properties of the rvs as

16. Sources of Difficulty • - random set that depends on • Assume is a deterministic sequence with for convenience Example: • Fix the domain, and randomly select the locations of the source and destination • Randomly place n2 – 2 other nodes in the domain • Transmission range decreases as 1/n • Number of hops along the shortest path increases with n

17. Sources of Difficulty (cont’d) • Dependence of reachability processes • Introduces dependence in link excess lives • Asymptotic independence – dependence in link excess lives goes away asymptotically as hop distance increases • Mixing conditions

18. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

19. Assumptions • Assumption 1: (scaling) There exists such that where • Scaling introduced for defining limit distribution parameter • Assumption 2: For every and any given there exists an integer such that • Interpretation:probability that a link duration is strictly • positive is one

20. Definitions • Array of -valued rvs • for notational convenience

21. Definitions • Let be a sequence of real numbers • Usually increases with n

22. Definitions • Sufficient condition:

23. Assumptions • Define • A sufficient condition is that there exists an arbitrarily small constant e > 0 such that for all and

24. Assumptions

25. Interpretation of Assumption 4

26. Distributional convergence • Implications: For sufficiently large hop count, the expected path duration can be approximated by

27. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

28. Proposed algorithm • Link durations seen by a node likely to depend on its own type and the types of neighbors • Different nodes with different speeds and capabilities • Each node maintains average link durations • Can maintain a separate average for each type of neighbors • Average link duration used as estimate of expected link durations (during path discovery)

29. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results - AODV • Parameter update • Conclusion & Future Directions

30. NS-2 simulation - Setup • Modified AODV routing protocol • 200 nodes in 2 km x 2 km rectangular region • Transmission range = 250 m • Two classes of nodes • Nodes with different speed (e.g., soldiers vs. jeeps or tanks) • Class 1 node speed ~ [1, 5] m/s • Class 2 node speed ~ [10, 30] m/s • Varying mixture • Class1:Class2 = 140:60, 160:40, and 180:20

31. NS-2 simulation

32. NS-2 simulation

33. NS-2 simulation

34. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

35. Estimation of expected path duration • Recall: For sufficiently large hop count, the expected path duration can be approximated by • Question: For finite hop counts, how good is this approximation? • For back-up paths • Local recovery after a link failure

36. Threshold update – local recovery • Select a back-up path only if the estimated probability of being available exceeds a certain threshold • Probability of being available estimated to be • Not accurate due to discrepancy in exp. parameter and collected IPD value (sum of inverses of expected link durations) • Target probability • Update the threshold as follows where is the threshold after n back-up path tries and is the indicator function of a back-up path being available Amount of time since last update

37. Threshold update • Define to be the indicator function of the event that a selected backup path is available when the threshold value is • and - unknown distribution of and its mean, respectively • Assume (i) is strictly increasing in , and (ii) there exists such that

38. Outline • Background • Basic Model • Setup • Distributional convergence • Proposed algorithm • NS-2 simulation results • Parameter update • Conclusion & Future Directions

39. Conclusions & Future Directions • Studied the statistical properties of path durations in MANETS • Showed distributional convergence with increasing hop count • Relationship between link durations and path duration • Proposed an algorithm for maximizing expected durations of selected paths • Stochastic approximation based algorithm for handling the discrepancy between IPD values and exponential parameters • Plan to implement with other on-demand routing protocols • Validation of assumptions • Convergence speed

40. Proposed algorithm in AODV • Each node maintains a route entry from each known dest node • Up to k paths (instead of a single path in AODV) • (i) dest seq. number, (ii) next hop, (iii) hop count, and (iv) Inverse Path Duration (IPD) • IPD = sum of the inverses of average link durations reported in a path reply message • Paths ranked based on (i) seq. number, (ii) IPD value, (iii) hop count • Request message • (i) src ID, seq. number, (ii) broadcast ID, (iii) dest ID and seq. number, and (iv) hop count to the src • Reply message • (i) dest ID, (ii) dest seq. number, (iii) IPD value, and (iv) hop count • Either an intermediate node or dest generates a reply message • Intermediate node – copy information from its entry • Dest node – initialize IPD and hop count to zero