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A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa  and
Hiroyuki Takada
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
Hiroyuki Takada*
Affiliation:
Tokyo University of Science
*
∗ Postal address: Department of Information Sciences, Tokyo University of Science, Noda, Chiba 278-8510, Japan.
∗ Postal address: Department of Information Sciences, Tokyo University of Science, Noda, Chiba 278-8510, Japan.
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Abstract

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

Keywords

Matrix exponentialfluid queueMarkov modulatedbatch fluidGI/M/1-type queuestationary distributionhitting probability

MSC classification

Primary: 60K25: Queueing theory 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J75: Jump processes 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 604 - 618
Copyright
Copyright © Applied Probability Trust 2002 

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