Perfect simulation and stationarity of a class of mobility models

Abstract

Abstract-We define "random trip", a generic mobility model for independent mobiles that contains as special cases: the random waypoint on convex or non convex domains, random walk with reflection or wrapping, city section, space graph and other models. We use Palm calculus to study the model and give a necessary and sufficient condition for a stationary regime to exist. When this condition is satisfied, we compute the stationary regime and give an algorithm to start a simulation in steady state (perfect simulation). The algorithm does not require the knowledge of geometric constants. For the special case of random waypoint, we provide for the first time a proof and a sufficient and necessary condition of the existence of a stationary regime. Further, we extend its applicability to a broad class of non convex and multi-site examples, and provide a ready-to-use algorithm for perfect simulation. For the special case of random walks with reflection or wrapping, we show that, in the stationary regime, the mobile location is uniformly distributed and is independent of the speed vector, and that there is no speed decay. Our framework provides a rich set of well understood models that can be used to simulate mobile networks with independent node movements. Our perfect sampling is implemented to use with ns-2, and it is freely available to download from http://ica1www.epfl.ch/RandomTrip.

Perfect Simulations for Random Trip Mobility Models

“…In this section we first recall the definition of random trip, a class of mobility models defined in [6], and its perfect simulation.…”

“…In this paper, we confine our formalism to restricted random waypoint with only one sub-domain. The general case bears some notational complexity; see [6] for definition, properties, and perfect sampling.…”

“…This is achieved in [6] by using Palm calculus, a set of formulae that relate the distributions of a system at an arbitrary point in time and at an arbitrary transition instant (here the trip-end times) -see for example [9] for more explanations. Note that those two distributions are different: if we sample a mobile speed in the city graph at an arbitrary trip-end time (this is called the "Palm" distribution), we obtain a uniform distribution over…”

“…Le Boudec and Vojnović [6] recently used Palm calculus to provide a generic framework for finding the steady state distribution of the mobility state for a large class of random mobility movements -they call, random trip models. In this work, we derive the steady state distributions for some random trip mobility models and implement perfect sampling in a tool, whose output is directly used by de-facto standard ns-2 network simulator.…”

“…Further, any random trip mobility model has a unique time-stationary distribution if and only if the mean trip duration averaged at trip transition instants is finite (Theorem 3 [6]). For the city section example, this means that U q E £ and the average pause time is finite [6].…”