Spatial homogenization in a stochastic network with mobility

A stochastic model for a mobile network is studied. Users enterthe network, and then perform independent Markovian routes between nodes,where they receive service according to the Processor-Sharing policy. Oncetheir service requirement is satisfied, they leave the system. The stability re-gion is identified via a fluid limit approach, and strongly relies on a “spatialhomogenization” property: At the fluid level, customers are instantaneouslydistributed across the network according to the stationary distribution of theirMarkovian dynamics and stay distributed as such as long as the network isnot empty. In the unstable regime, spatial homogenization almost surely holdsasymptotically as time goes to infinity (on the normal scale), telling how thesystem fills up. One of the technical achievements of the paper is the construc-tion of a family of martingales associated to the multidimensional process ofinterest, which makes it possible to get crucial estimates for certain exit times.