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

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Affiliation:
Tokyo University of Science
Hiroyuki Takada
Affiliation:
Tokyo University of Science
Corresponding

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2002 

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