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Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Peter W. Glynn  and
Ward Whitt
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Abstract

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x–1 log P(W> x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n–1 log E exp (θSn) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

Keywords

WAITING TIMEWORKLOADTAIL PROBABILITIESASYMPTOTICSLARGE DEVIATIONSCUMULANT GENERATING FUNCTIONCOUNTING PROCESSEFFECTIVE BANDWIDTHSADMISSION CONTROLHIGH-SPEED NETWORKSASYNCHRONOUS TRANSFER MODE

MSC classification

Primary: 60K25: Queueing theory
Type
Part 3 Queueing Theory
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 131 - 156
Copyright
Copyright © Applied Probability Trust 1994 

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References

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