On the Singular Behavior of a Queueing System with Random Connectivity

In this paper, we study the stationary dynamics of a processing system comprised of several parallel queues and a single server of constant rate. The connectivity of the server to each queue is randomly modulated, taking values 1 (connected) or 0 (severed). At any given time, only the currently connected queues may receive service. A key issue is how to schedule the server on the connected queues in order to maximize the system throughput. We investigate the behavior of two dynamic schedules, when the loading of the system exceeds its capacity. It is shown that unlike many other queueing systems that exhibit a binary behavior-global stability or global instability-the system under consideration exhibits a much richer behavior, with several partial stability modes. These modes are fully determined by the underlying traffic loading. The results are obtained under very general stationary ergodic traffic flows and connectivity modulation.