Asymptotic Buuer Overrow Probabilities in Multiclass Multiplexers: an Optimal Control Approach 1

We consider a multiclass multiplexer with support for multiple service classes, and dedicated bu ers for each service class. Under speci c scheduling policies for sharing bandwidth among these classes, we seek the asymptotic (as the bu er size goes to in nity) tail of the bu er over ow probability for each dedicated bu er. We assume dependent arrival and service processes as is usually the case in models of bursty tra c. In the standard large deviations methodology, we provide a lower and a matching (up to rst degree in the exponent) upper bound on the bu er over ow probabilities. We introduce a novel optimal control approach to address these problems. In particular, we relate the lower bound derivation to a deterministic optimal control problem, which we explicitly solve. Optimal state trajectories of the control problem correspond to typical congestion scenarios. We explicitly and in detail characterize the most likely modes of over ow. We specialize our results to the generalized processor sharing policy (GPS) and the generalized longest queue rst policy (GLQF). The performance of strict priority policies is obtained as a corollary. We compare the GPS and GLQF policies and conclude that GLQF achieves smaller over ow probabilities than GPS for all arrival and service processes for which our analysis holds. Our results have important implications for tra c management of high-speed networks, and can be used as a basis for an admission control mechanism which guarantees a di erent loss probability for each class.