On multiclass spatial birth-and-death processes with wireless-type interactions

In this paper, we study a multiclass spatial birth-and-death (SBD) processes on a compact region of the Euclidean plane modeling wireless interactions. In such a setup, users arrive at a constant rate and leave at a rate inversely proportional to a shot noise created by interfering users in the network. The novelty of this work lies in the addition of service differentiation, inspired by bandwidth partitioning present in 5G networks: users are allocated a fixed number of frequency bands and only interfere with transmissions on these bands. The first result of our work lies in the determination of the critical user arrival rate below which the system is always stochastically stable, and above which it is unstable. The analysis requires symmetry assumptions which are defined in the paper. The proof for this result uses stochastic monotonicity and fluid limit models to bound the dynamics from above and below by two adequate discrete-state Markov jump processes, for which we obtain stability and instability results. This leads to a closed form formula for the critical arrival rate. In a second part, we propose two heuristics to estimate the steady-state densities of users in the network: the first one relies on a Poisson approximation of the steady-state processes. The second one uses a cavity approximation leveraging second-order moment measures, which leads to more accurate estimates of the steady-state user densities of all classes of users. The Poisson heuristic also gives a good estimate for the critical arrival rate.