Bounds on the asymptotic buffer overflow probabilities of a parallel processing system

In queueing systems with heterogeneous processors and multiclass job flows, weighted queue length policies are known to achieve maximal throughput, that is, stabilize the system under the maximum possible arrival rates, when the buffers are of infinite capacity. However, very little is known regarding the delay or buffer overflow performance of weighted queue length policies in such queueing systems when the buffers are of finite capacity. We consider a time-slotted "fluid" queueing system consisting of two heterogeneous processors in parallel and two queues with finite capacity buffers. There are two classes of job flows. We present some preliminary results that use techniques of large deviations to derive upper and lower bounds on the asymptotic buffer overflow probabilities under any stabilizing scheduling policy (including weighted queue length policies) as the capacities of the buffers tend to infinity. The queueing system has applications in a number of wired and wireless telecommunication networks, computer systems, and flexible manufacturing systems.