Conditioned asymptotics for tail probabilities in large multiplexers
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Consider a buffer whose input is a superposition of L independent identical sources, and which is served at rate s L. Recent work has shown that, under very general circumstances, the stationary tail probabilities for the queue of unfinished work Q in the buffer have the asymptotics P[Q > Lb] e-LI(b)for large L. Here the shape function, I, is obtained from a variational expression involving the transient log cumulant generating function of the arrival process. In this paper, we extend this analysis to cover time-dependent asymptotics for Markov arrival processes subject to conditioning at some instant. In applications we envisage that such conditioning would arise due to knowledge of the queue at a coarse-grained level, for example of the number of current active sources. We show how such partial knowledge can be used to predict future tail probabilities by use of a time dependent, conditioned shape function. We develop some heuristics to describe the time-dependent shape function in terms of a reduced set of quantities associated with the underlying arrivals process and show how to calculate them for renewal arrivals and a class of ON-OFF arrivals. This bypasses the full variational calculation of the shape function for such models. 1998 Elsevier Science B.V.
