Modelling Mobility: A Discrete Revolution

TL;DR

This paper introduces the Markov Trace Model, a discrete mathematical framework for analyzing mobility patterns, providing explicit formulas for stationary distributions, and demonstrating its application to various complex mobile systems including vehicular networks.

Contribution

The paper presents the Markov Trace Model as a novel discrete approach for mobility analysis, with explicit formulas and applications to complex systems like vehicular mobility.

Findings

Derived explicit stationary distribution formulas for the Markov Trace Model.

Applied the model to the Manhattan Random Way-Point and vehicular systems.

Enabled analytical study of complex mobile systems using counting arguments.

Abstract

We introduce a new approach to model and analyze \emph{Mobility}. It is fully based on discrete mathematics and yields a class of mobility models, called the \emph{Markov Trace} Model. This model can be seen as the discrete version of the \emph{Random Trip} Model including all variants of the \emph{Random Way-Point} Model \cite{L06}. We derive fundamental properties and \emph{explicit} analytical formulas for the \emph{stationary distributions} yielded by the Markov Trace Model. Such results can be exploited to compute formulas and properties for concrete cases of the Markov Trace Model by just applying counting arguments. We apply the above general results to the discrete version of the \emph{Manhattan Random Way-Point} over a square of bounded size. We get formulas for the total stationary distribution and for two important \emph{conditional} ones: the agent spatial and destination distributions. Our method makes the analysis of complex mobile systems a feasible task. As a further evidence of this important fact, we first model a complex vehicular-mobile system over a set of crossing streets. Several concrete issues are implemented such as parking zones, traffic lights, and variable vehicle speeds. By using a \emph{modular} version of the Markov Trace Model, we get explicit formulas for the stationary distributions yielded by this vehicular-mobile model as well.