1. Stable mobility models for MANETS Kim Blackmore Roy Timo (DE/NICTA) Leif Hanlen (NICTA)

2. What is a MANET?

3. What’s in a data network? What if it’s Mobile Ad Hoc? "Application" protocols "Transport" protocols Routing

4. MANET Routing For example, DSR – dynamic source routing Each source maintains a cache of known paths (to previous destinations) Route discovery –if you want a path to somewhere not in the cache Route maintenance – if you try to use a cached path but it fails Reactive routing (or could try proactive)

5. Simulations to test routing protocols Because there are so many layers of complexity below the routing Control the lower layers to isolate effect of routing protocol Synthetic mobility model to describe node movement –Random Waypoint is most popular

6. Random Waypoint Mobility Model Start point: –Uniform at random Waypoint 1: –Uniform at random Travel to waypoint 1 –At random speed Choose waypoint 2 Repeat

7. The problem with simulations Are the results meaningful? J. Yoon, M. Liu, and B. Nobel, ``Random Waypoint Considered Harmful," in Proc. IEEE INFOCOM, 2003. J. Yoon, M. Liu, and B. Nobel, ``Sound Mobility Models," in Proc. IEEE MobiHoc, 2003. William Navidi and Tracy Camp. Stationary distributions for the random waypoint model. IEEE Transactions on Mobile Computing, 3(1), 2004. J. Boudec, M. Vojnovic, ``The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation," IEEE/ACM Trans. Networking, vol. 14 no. 6, Dec. 2006.

8. If node speed is randomly selected from [0,M], then time averaged node speed goes to zero.

9. Will simulation time-averages converge? Yes We prove a strong law of large numbers for the RWPMM Previous attempts used the “wrong” definition of stability

10. Notation S 1 S 2 S 3 S 4 S 5 S 6 S = f s 1 ; s 2 ; s 3 ; s 4 ; s 5 ; s 6 g X = Q V v = 1 S v x 1 = ( s 2 ; s 2 ; s 5 ; s 5 ) x 0 = ( s 2 ; s 3 ; s 6 ; s 6 ) V=4 nodes 6 possible positions

11. Estimating probability from simulations Generate a long sample path Count the number of times event A appears over N time steps x = x 0 ; x 1 ; x 2 ;::: P N ¡ 1 n = 0 1 A ¡ x n ; x n + 1 ;::: ¢ 1 A ¡ x n ; x n + 1 ;::: ¢ = ½ 1 ; i f x n ; x n + 1 ;::: 2 A 0 ; o t h erw i se

12. Estimate the probability is But how do we know Converges to ProbA? Converges at all? Estimating probability from simulations P ro b A ¼ 1 N P N ¡ 1 n = 0 1 A ¡ x n ; x n + 1 ;::: ¢ l i m N ! 1 1 N P N ¡ 1 n = 0 1 A ¡ x n ; x n + 1 ;::: ¢

13. A mobility model is Stable if: For each bounded function the limit exists (with probability one). f : X 1 ! R ­ f ® ( x ) = l i m N ! 1 1 N N ¡ 1 X i = 0 f ¡ x n ; x n + 1 ; x n + 2 ;::: ¢

14. A mobility model is Ergodic iff For each event A, the limit converges to the stationary mean probability of A. P ro b [ A ] = ­ 1 A ® ( x ) = l i m N ! 1 1 N N ¡ 1 X i = 0 1 A ¡ x n ; x n + 1 ; x n + 2 ;::: ¢

15. Result - The RWMM is stable and ergodic

16. Contributions Definitions of stable and ergodic MANET mobility models RWPMM is stable and ergodic

17. Where to next? Stable routing protocols DSR is stable.. THE END