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

Published online by Cambridge University Press:  24 October 2008

N. G. Duffield  and
Neil O'connell
N. G. Duffield
Affiliation:
School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland. e-mail: duffieldn@dcu.ie
Neil O'connell
Affiliation:
Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland. e-mail: oconnell@stp.dias.ie
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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, t∈R+) and a rate function I such that if (Wt, t∈R+) 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

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 363 - 374
Copyright
Copyright © Cambridge Philosophical Society 1995

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