Modelling mobility: A discrete revolution

We introduce a new approach to model and analyze mobility. It is fully based on discrete mathematics and yields a class of mobility models, called the Markov Trace model. It can be viewed as the discrete version of the Random Trip model: including all variants of the Random Way-Point model . We derive fundamental properties and explicit analytical formulas for the stationary probability distributions yielded by the Markov Trace model. Besides having a per-se interest, such results can be exploited to compute formulas and properties for concrete cases by just applying counting arguments. We apply the above general results to the discrete version of the Manhattan Random Way-Point. We get explicit formulas for the total stationary distribution and for two important conditional distributions: the agent spatial and the agent destination ones. Our method makes the analysis of complex mobile systems a feasible task. As a further evidence of this important fact, we 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 modular version of the Markov Trace model, we get explicit formulas for the stationary distributions yielded by this vehicular-mobile model as well.