Analysis of Random Mobility Models with Partial Differential Equations

In this paper, we revisit two classes of mobility models which are widely used to represent users' mobility in wireless networks: random waypoint (RWP) and random direction (RD). For both models, we obtain systems of partial differential equations which describe the evolution of the users' distribution. For the RD model, we show how the equations can be solved analytically both in the stationary and transient regime, adopting standard mathematical techniques. Our main contributions are 1) simple expressions which relate the transient duration to the model parameters and 2) the definition of a generalized random direction model whose stationary distribution of mobiles in the physical space corresponds to an assigned distribution.

1 Introduction

Mobility models play a fundamental role in the analysis and design of wireless systems [1], [2]. In the past several years, researchers have proposed a number of mobility models for the purpose of simulating the movement of users in a wireless network. Two widely used models are the Random Waypoint model (RWP) [3] and the Random Direction model (RD) [4]. In both models, users independently follow a sequence of linear segments and traverse each segment at a constant speed. The two models differ in how a user chooses the next segment to traverse: Under the RWP model, a user selects a random destination point within the space; instead, under the RD model, a user chooses a direction to travel in and a duration for the travel. In both cases, the speed on a segment is taken from some given distribution. Moreover, before starting to travel on the new segment, users can stop for a random time, thus alternating phases in which they move with phases in which they keep still.