Optimal Multi-Server Allocation to Parallel Queues with Random Connectivity and Retransmissions

We investigate an optimal scheduling problem for a discrete-time system of two parallel queues with infinite capacity, sharing two symmetrical servers. This model can be used to study a variety of scheduling problems in wireless networks. At any time slot, a queue can be served by one or two connected servers; the queue-server connectivity is assumed to be random and modeled by a two-state Markov chain. The arrivals to each queue are assumed to be independent and identically distributed. A scheduled packet completes service successfully with a given probability. Otherwise, it has to be retransmitted in a later time slot. The optimal scheduling policy is defined as the server allocation policy that minimizes, in a stochastic ordering sense, the total number of packets in the system. We prove, using a dynamic coupling method, that a "Most Balancing" policy, a policy that attempts to balance the lengths of the two queues, is optimal. We also compare the performance of the optimal policy to that of a few other policies via simulations.