Asymptotic buffer overflow probabilities in multiclass multiplexers : part I : the GPS policy

In this paper and its companion [BPT96] we consider a multiclass multiplexer, with segregated buffers for each type of traffic, and under specific scheduling policies for sharing bandwidth we seek the asymptotic (as the buffer size goes to infinity) tail of the buffer overflow probability for each buffer. We assume correlated arrival and service processes that are usually used in modeling bursty traffic. Here we consider the generalized longest queue first policy (GLQF) and in [BPT96] the generalized processor sharing policy (GPS). In the standard large deviations methodology we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. 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 overflow. We find that the GLQF policy outperforms the GPS policy with respect to loss probabilities characteristics. Our results have important implications in traffic management of high-speed networks and can be used as a basis for an admission control mechanism which guarantees different loss probability for each type of traffic.