Conditioned Asymptotics for Tail Probabilities in Large Multiplexers

Consider a buffer whose input is a superposition of L independent identical sources, and which is served at rate sL. 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] ß e \GammaLI(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 fun..