Properties of Random Direction Models
A number of mobility models have been proposed for the purpose of either
analyzing or simulating the movement of users in a mobile wireless network.
Two of the more popular are the random waypoint and the random direction
models. The random waypoint model is physically appealing but difficult to
understand. Although the random direction model is less appealing physically,
it is much easier to understand. User speeds are easily calculated, unlike for
the waypoint model, and, as we will observe, user positions and directions are
uniformly distributed. The contribution of this paper is to establish this
last property for a rich class of random direction models that allow future
movements to depend on past movements. To this end, we consider finite one-
and two-dimensional spaces. We consider two variations, the random direction
model with wrap around and with reflection. We establish a simple relationship
between these two models and, for both, show that positions and directions are
uniformly distributed in steady-state for a class of Markov movement models
regardless of initial position. In addition, we establish a sample path
property for both models, namely that any piecewise linear movement applied to
a user preserves the uniform distribution of position and direction provided
that users were initially uniformly throughout the space with equal likelihood
of being pointed in any direction.
Philippe Nain
Last modified: Tue Mar 15 15:07:36 MET 2005