Optimal server assignment in multi-server parallel queueing systems with random connectivities and random service failures

The problem of assignment of K identical servers to a set of N symmetric parallel queues is investigated in this paper. The parallel queueing system is considered to be time slotted and the connectivity of each queue to each server is varying randomly over time and following Bernoulli distribution with a given parameter. Each server is capable of serving at most one packet per time slot (if it is connected and assigned to a queue). At any time slot, each server can serve at most one queue and each queue can be served by at most one server. We assume that the service of a scheduled packet by a connected server fails randomly with a certain probability. The packet arrival processes to the queues are assumed to be i.i.d. and follow Bernoulli distribution with a fixed parameter. For such a symmetric system, i.e., with the same arrival, connectivity and service failure parameters for all the queues, we show that Maximum Weighted Matching (MWM) server assignment policy is delay optimal. More specifically, using stochastic ordering and dynamic coupling techniques we prove that MWM minimizes, in stochastic ordering sense, a broad range of stochastic cost functions of the queue lengths including total queue occupancy (or equivalently average queueing delay).