In queueing systems with heterogneeous processors and multiclass job flows, weighted queue length policies are known to achieve maximal throughput, that is , stabilize the system under the maximum possible arival rates, when the buffers are of infinite capacity. However, very little is known regrding the delay or buffer overflow performance of weighted queue length policies in such queueing systems when the buffers are offinite capacity. In this paper, we consider a time-slotted"fluid" cessors 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 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.
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