Stability of multiclass spatial birth-and-death processes with wireless-type interactions

With the densification of wireless networks and the increasing standards of wireless communication protocols, the study of the stochastic stability of wireless networks becomes increasingly important. Here, we study a multiclass spatial birth-and-death process on a compact region of the Euclidean plane modeling wireless interactions as seen in a telecommunication network: users arrive at a constant rate and leave the network at a rate inversely proportional to a shot-noise created by other interfering users in the network. The novelty of this work lies in the addition of service differentiation, inspired by bandwidth partitioning introduced in the latest generation of wireless networks. In this setup, users choose a fixed number of transmission channels, and only interfere with transmissions on the channels they use. We restrict our study to symmetric networks, where users transmitting on the same  number of bands have the same stochastic properties in the network.

We define a general mathematical framework using stochastic geometry and queuing the ory tools to study this category of processes and we prove the existence of a critical arrival rate above which the system is always unstable. We then introduce a discretization of the dynamics to reduce the study of stability to which of a queuing network in a countable state space. We then find a closed form of the critical arrival rate using fluid limits arguments. In a second part, we define a heuristics to estimate the steady-state densities of users in the network. The first one relies on a Poisson approximation of the steady-states processes and allows us to define a heuristic for the critical arrival rate of the network. The second heuristic uses a cavity approximation and a second-order approximation to improve the performance.