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Large deviations and overflow probabilities for the general single-server queue, with applications

  • N. G. Duffield (a1) and Neil O'connell (a2)

Abstract

We consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (at, vt, tR+) and a rate function I such that if (Wt, tR+) denotes the workload process, then

on the continuity set of I. In the case that at = vt = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt[8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = supt≥0Wt) decays exponentially:

and the decay rate δ is directly related to the rate function I. We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queue-length distribution has an exponential tail only if limt→∞at/vt is finite and strictly positive; otherwise, provided our conditions are satisfied, the tail probabilities decay like

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Large deviations and overflow probabilities for the general single-server queue, with applications

  • N. G. Duffield (a1) and Neil O'connell (a2)

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