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A reduced-peak equivalence for queues with a mixture of light-tailed and heavy-tailed input flows

Part of: Special processes Computer system organization Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Sem Borst  and
Bert Zwart
Sem Borst*
Affiliation:
CWI, Amsterdam
Bert Zwart*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: sem@cwi.nl
∗∗ Postal address: Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

Keywords

Fluid queuesheavy-tailed on periodslarge deviationsqueue length asymptotics

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling 90B18: Communication networks 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 793 - 805
Copyright
Copyright © Applied Probability Trust 2003 

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