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Infinite- and finite-buffer Markov fluid queues: a unified analysis

Part of: General theory of linear operators Computer system organization Probabilistic methods, simulation and stochastic differential equations Special processes Numerical linear algebra

Published online by Cambridge University Press:  14 July 2016

Nail Akar  and
Khosrow Sohraby
Nail Akar*
Affiliation:
Bilkent University
Khosrow Sohraby*
Affiliation:
University of Missouri-Kansas City
*
∗ Postal address: Electrical and Electronic Engineering Department, Bilkent University, 06800 Bilkent, Ankara, Turkey.
∗∗ Postal address: School of Interdisciplinary Computing and Engineering, University of Missouri–Kansas City, Kansas City, MO 64110, USA. Email address: sohrabyk@umkc.edu
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Abstract

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

Keywords

Stochastic fluid modelMarkov fluid queueperformance analysiscomputer networkspectral divide-and-conquer problemgeneralized Newton iteration

MSC classification

Primary: 60K25: Queueing theory
Secondary: 65C40: Computational Markov chains 65F15: Eigenvalues, eigenvectors 68M20: Performance evaluation; queueing; scheduling 47A15: Invariant subspaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 557 - 569
Copyright
Copyright © Applied Probability Trust 2004 

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