# Stability of spatial wireless systems with random admissible-set scheduling

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We examine the stability of wireless networks whose users are distributed over a compact space. Users arrive at spatially uniform locations with intensity \lambda and each user has a random number of packets to transmit with mean \beta. In each time slot, an admissible subset of users is selected uniformly at random to transmit one packet. A subset of users is called admissible when their simultaneous activity obeys the prevailing interference constraints. We consider a wide class of interference constraints, including the SINR model and the protocol model. Denote by \mu the maximum number of users in an admissible subset for the model under consideration. We will show that the necessary condition $\lamba \beta <\mu$ is also sufficient for random admissible-set scheduling to achieve stability. Thus random admissible-set scheduling achieves stability, if feasible to do so at all, for a broad class of interference scenarios. The proof relies on a description of the system as a measure-valued process and the identi??cation of a Lyapunov function. Keywords: Wireless networks, stability, Foster-Lyapunov, Harris recurrent, measure-valued process, interference constraints, SINR requirements, protocol model