What’s in a data network?
What if it’s Mobile Ad Hoc?"Application" protocols
"Transport"protocols
Routing
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)
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
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
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.
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
Notation
S1
S2
S3
S4
S5
S6
S = fs1;s2;s3;s4;s5;s6g
X =Q V
v=1 Sv
x1 = (s2;s2;s5;s5)
x0 = (s2;s3;s6;s6)
V=4 nodes
6 possible positions
Estimating probability from simulations
• Generate a long sample path
• Count the number of times event A appears over N time steps
x = x0;x1;x2; : : :
P N ¡ 1n=0 1A
¡xn;xn+1; : : :
¢
1A
¡xn;xn+1; : : :
¢=
½1; if xn ;xn+1; : : : 2 A0; otherwise
• Estimate the probability is
• But how do we know
• Converges to ProbA?• Converges at all?
Estimating probability from simulations
ProbA ¼ 1N
P N ¡ 1n=0 1A
¡xn ;xn+1; : : :
¢
limN ! 11N
P N ¡ 1n=0 1A
¡xn ;xn+1; : : :
¢
A mobility model is Stable if:
For each bounded function the limit
exists (with probability one).
f : X 1 ! R
f®(x) = lim
N ! 1
1N
N ¡ 1X
i=0
f¡xn;xn+1;xn+2; : : :
¢
A mobility model is Ergodic iff
For each event A, the limit
converges to the stationary mean probability of A.
Prob[A]=1A
®(x) = lim
N ! 1
1N
N ¡ 1X
i=0
1A¡xn;xn+1;xn+2; : : :
¢