Spatial homogenization in a stochastic network with mobility (Q2268729)

The paper investigates a Markovian model for a mobile network. The customers arrive in the network according to a Poisson process with intensity \(\lambda \), move independently in the network according to some Markovian dynamics with a common rate matrix \(Q\). Service requirements are exponentially distributed with mean 1, customers are served at each node they visit according to the Processor-Sharing discipline until their demand has been satisfied. The total capacity of the network (the sum of all individual capacities) is denoted by \(\mu \), it corresponds to the instantaneous output rate of the network when no node is empty. The system is open, it may be unstable, \(\lambda <\mu \) is the stability condition, in contrast to the Jackson networks (where each node has to satisfy some constraint) here only the global quantities \(\lambda \) and \(\mu \) matter. Surprisingly, proving that is sufficient requires very technical tools, including the use of fluid limits and martingale techniques. This martingale is multidimensional generalization of the martingale built for the \(M/M/\infty \) queue. The stability region is identified via a fluid limit approach and relies on a ``spatial homogenization'' property: at the fluid level customers are instantaneously distributed across the network according to the stationary distribution of their Markovian dynamics and stay distributed as long as the network is not empty.