Understanding the Simulation of Mobility Models with Palm Calculus

The simulation of mobility models such as the random waypoint often cause subtle problems, for example the decay of average speed as the simulation progresses, a difference between the long term distribution of nodes and the initial one, and sometimes the instability of the model. All of this has to do with time averages versus event averages. This is a well understood, but little known topic, called Palm calculus. In this paper we first give a very short primer on Palm calculus. Then we apply it to the random waypoint model and variants (with pause time, random walk). We show how to simply obtain the stationary distribution of nodes and speeds, on a connected (possibly non convex) area. We derive a closed form for the density of node location on a square or a disk. We also show how to perform a perfect (i.e. transient free) simulation without computing complicated integrals. Last, we analyze decay and explain it as either convergence to steady state or lack of convergence.