Uniqueness of Stationary Distributions in Random Access Poisson Networks
Plínio S. Dester, Pedro H. J. Nardelli, Paulo Cardieri · IEEE Communications Letters · 2021
This letter presents sufficient conditions for the existence and uniqueness of the limiting stationary distribution in random access Poisson networks with packet queueing and under Rayleigh fading and a general path loss model. This system model is traditionally used in the literature assuming uniqueness of the stationary distribution. We demonstrate here through a counterexample that this assumption might not always hold. From the sufficient conditions, an interesting and perhaps counterintuitive result emerged, that is, the arrival rate of packets per node greater than 1/e guarantees uniqueness of the limiting distribution. When that is the case, then setting the medium access probability to 1 minimizes the proportion of unstable nodes.