1. Message Delay in Mobile Ad Hoc Networks by Robin Groenevelt In collaboration with Philippe Nain and Ger Koole

2. Example of MANET

3. Meeting times are exponential Area = 4x4 km 2 V min = 4km/hour V max = 10km/hour

4. Meeting times are exponential (2) Area = 4x4 km 2 V min = 4km/hour V max = 10km/hour

5. Meeting times are exponential (3) Area = 4x4 km 2 V min = 4km/hour V max = 10km/hour

6. MANET as a Markov Chain State i (i=1,…,N) represents number of copies in network State N+1 is when destination receives the message

7. MANET as a Markov Chain (2) P(j → j+1) = j/N P(j → N+1) = 1-j/N If C = number of copies when destination reached, then

8. T = message delay under two-hop relay protocol T*( Θ ) = E[e -ΘT ] = Laplace transform of T where S j = Sojourn time in state i MANET as a Markov Chain (3)

9. MANET as a Markov Chain (4) S j = Sojourn time in state i S j,1 = Time to enter state i+1 = exp((N-j)λ) S j,2 = Time to enter state N+1 = exp(jλ) S j = min(S j,1,S j,2 ), moreover Therefore

10. Message Delay two-hop relay This gives the Laplace transform of the message delay: From this the mean delay is easily obtained: It can be shown that

11. Unrestricted relaying If C U = number of copies when destination reached, then

12. Message Delay unrestricted relaying LST of message delay under restricted relaying: Expected delay under unrestricted relaying: where γ ≈ 0.57721 is Euler’s constant Number of copies (C U ) is uniformly distributed

13. Message Delay two-hop relaying

16. Message Delay unrestricted relaying

19. Under certain independence assumptions: For the Random Direction (RD) and the Random Waypoint (RW) mobility models: and if V 1 =v=V 2 where ω ≈ 1.3683 is the Waypoint constant. The parameter λ