Spatial homogenization in a stochastic network with mobility

TL;DR

This paper analyzes a stochastic mobile network model, revealing a spatial homogenization property at the fluid level and establishing stability conditions using martingale techniques.

Contribution

It introduces a novel fluid limit approach to identify the stability region and proves spatial homogenization in both stable and unstable regimes.

Findings

Spatial homogenization occurs at the fluid level.

Stability region characterized via fluid limit and homogenization.

Martingale construction aids in estimating exit times.

Abstract

A stochastic model for a mobile network is studied. Users enter the network, and then perform independent Markovian routes between nodes where they receive service according to the Processor-Sharing policy. Once their service requirement is satisfied, they leave the system. The stability region is identified via a fluid limit approach, and strongly 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 such as long as the network is not empty. In the unstable regime, spatial homogenization almost surely holds asymptotically as time goes to infinity (on the normal scale), telling how the system fills up. One of the technical achievements of the paper is the construction of a family of martingales associated to the multidimensional process of interest, which makes it possible to get crucial estimates for certain exit times.