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Stochastic theory of a fluid model of producers and consumers coupled by a buffer

Published online by Cambridge University Press:  01 July 2016

Debasis Mitra
Debasis Mitra*
Affiliation:
AT & T Bell Laboratories
*
∗Postal address: AT & T Bell Laboratories, Room 2C125, 600 Mountain Avenue, Murray Hill, NJ 07974-2070, USA.
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Abstract

This paper analyzes, derives efficient computational procedures and numerically investigates the following fluid model which is of interest in manufacturing and communications: m producing machines supply a buffer, n consuming machines feed off it. Each machine independently alternates between exponentially distributed random periods in the ‘in service' and ‘failed' states. Producers/consumers have their own failure/repair rates and working capacities. When the buffer is either full or empty some of the machines in service are not utilized to capacity; otherwise they are fully utilized. Our main result is for the state distribution of the Markovian system in equilibrium which is the solution of a system of differential equations. The spectral expansion for its solution is obtained. Two important decompositions are obtained: the eigenvectors have the Kronecker-product form in lower-dimensional vectors; the characteristic polynomial is factored with each factor an explicitly given polynomial of degree at most 4. All eigenvalues are real. For each of various cases of the model, a system of linear equations is derived from the boundary conditions; their solution complete the spectral expansion. The count in operations of the entire procedure is O(m3n3): independence from buffer size exemplifies an important attraction of fluid models. Computations have revealed several interesting features, such as the benefit of small machines and the inelasticity of production rate to inventory. We also give results on the eigenvalues of a more general fluid model, reversible Markov drift processes.

Keywords

REVERSIBLE MARKOV DRIFT PROCESSESKRONECKER-PRODUCT FORM
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 646 - 676
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

†

However, the resulting deterministic models, often described as fluid, are to be distinguished from stochastic fluid models such as the one in this paper.

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