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Analysis of stochastic fluid queues driven by local-time processes

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Takis Konstantopoulos ,
Andreas E. Kyprianou ,
Paavo Salminen  and
Marina Sirviö
Takis Konstantopoulos*
Affiliation:
Heriot-Watt University
Andreas E. Kyprianou*
Affiliation:
University of Bath
Paavo Salminen*
Affiliation:
Åbo Akademi University
Marina Sirviö*
Affiliation:
Åbo Akademi University
*
∗ Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.
∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
∗∗∗ Postal address: Department of Mathematics, Åbo Akademi University, Turku, FIN-20500, Finland.
∗∗∗ Postal address: Department of Mathematics, Åbo Akademi University, Turku, FIN-20500, Finland.
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Abstract

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We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

Keywords

Local timefluid queueLévy processSkorokhod reflectionperformance analysisPalm calculusinspection paradox

MSC classification

Primary: 60G10: Stationary processes 60G55: Point processes 60G51: Processes with independent increments; L'evy processes 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1072 - 1103
Copyright
Copyright © Applied Probability Trust 2008 

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