Optimal Server Assignment in Multi-Server Queueing Systems with Random Connectivities

In this paper, we investigate the problem of assignment of K identical servers to a set of N parallel queues in a time slotted queueing system. The connectivity of each queue to each server is randomly changing with time; each server can serve at most one queue and each queue can be served by at most one server per time slot. Such multi-server queueing systems were widely applied in modeling the scheduling (or resource allocation) problem in wireless networks. It has been previously proven that Maximum Weighted Matching (MWM) is a throughput optimal server assignment policy for such queueing systems [1], [2]. In this paper, first we prove that for a symmetric system with i.i.d. Bernoulli packet arrivals and connectivities, MWM minimizes, in stochastic ordering sense, a broad range of cost functions of the queue lengths including total queue occupancy (or equivalently minimizes the average queueing delay). Then, we extend the model by considering imperfect services where it is assumed that the service of a scheduled packet fails randomly with a certain probability. We prove that the same policy is still optimal for the extended model. We also show that the results are still valid for some extended connectivity and arrival processes (e.g., Poisson arrival process). Second, for the multiserver queueing system with general stationary connectivity and arrival processes, we derive a linear algebraic characterization of the network stability region polytope which is needed to solve general utility optimization problems in such queueing systems.