We consider a slotted system with N queues, and independent and identically distributed (i.i.d.) Bernoulli arrivals at each queue during each slot. Each queue is associated with a channel that changes between "on" and "off" states according to i.i.d. Bernoulli processes. We assume that the system has K identical transmitters ("servers"). Each server, during each slot, can transmit up to C packets from each queue associated with an "on" channel. We show that a policy that assigns the servers to the longest queues whose channel is "on" minimizes the total queue size, as well as a broad class of other performance criteria. We provide several extensions, as well as some qualitative results for the limiting case where N is very large. Finally, we consider a "fluid" model under which fractional packets can be served, and subject to a constraint that at most C packets can be served in total from all of the N, queues. We show that when K = N, there is an optimal policy which serves the queues so that the resulting vector of queue lengths is "Most Balanced" (MB).
