Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues

Abstract

Our primary purpose in this paper is to contribute to the design of admission control schemes for multi-class service systems. We are motivated by emerging highspeed networks exploiting asynchronous transfer mode (ATM) technology, but there may be other applications. We develop a simple criterion for feasibility of a set of sources in terms of "effective bandwidths". These effective bandwidths are based on asymptotic decay rates of steady-state distributions in queueing models. We show how to compute asymptotic decay rates of steady-state queue length and workload tail probabilities in general infinite-capacity multi-channel queues. The model hasm independent heterogeneous servers that are independent of an arrival process which is a superposition ofn independent general arrival processes. The contribution of each component arrival process to the overall asymptotic decay rates can be determined from the asymptotic decay rates produced by this arrival process alone in a G/D/1 queue (as a function of the arrival rate). Similarly, the contribution of each service process to the overall asymptotic decay rates can be determined from the asymptotic decay rates produced by this service process alone in a D/G/1 queue. These contributions are characterized in terms of single-channel asymptotic decay-rate functions, which can be estimated from data or determined analytically from models. The asymptotic decay-rate functions map potential decay rates of the queue length into associated decay rates of the workload. Combining these relationships for the arrival and service channels determines the asymptotic decay rates themselves. The asymptotic decay-rate functions are the time-average limits of logarithmic moment generating functions. We give analytical formulas for the asymptotic decay-rate functions of a large class of stochastic point processes, including batch Markovian arrival processes. The Markov modulated Poisson process is a special case. Finally, we try to put our work in perspective with the related literature.