Limit properties of stochastic multiple access networks

Abstract

Stochastic multiple-access networks are systems in which services are randomly requested by multiple users, and often simultaneously. We analyse such a system, where we assume that each node of the network receives a random number of new packets within each time slot and transmits them according to the centralised slotted Aloha protocol. We construct a new model for the workload in the system, while taking into account the one-way spatial interactions that occur at nodes that are simultaneously in activity at a relatively small distance apart. Our new model is a time-homogeneous and irreducible Markov chain on a countable state space. We investigate the stochastic stability, the instability, and the partial stability properties of the network and obtain appropriate conditions on the mean number of new packets. For the latter two properties, we consider single-node and two-node networks and make some conjecture for a higher number of nodes. Our proofs are mainly based on the drift analysis approach. We also study the large deviations asymptotics for the stationary workload in a single-node network and obtain new results. Besides, we obtain the stationary distribution for single-node and two-node networks in particular cases.