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On a general storage problem and its approximating solution

Published online by Cambridge University Press:  01 July 2016

N. M. H. Smith  and
G. F. Yeo
N. M. H. Smith*
Affiliation:
University of Melbourne
G. F. Yeo*
Affiliation:
Odense University
*
∗Postal address: 10 Manica St., West Brunswick, Vic. 3055, Australia.
∗∗Postal address: Department of Mathematics, Odense University, Campusvej 55, DK-5230 Odense, Denmark.
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Abstract

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn+1(y) = ∫ l(y, w)gn(w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

Keywords

STOREQUEUEDAMRANDOM WALKLIMIT DISTRIBUTIONSHERMITE POLYNOMIALSGRAM–CHARLIER REPRESENTATIONSGAUSS QUADRATURESSIMULATION
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 567 - 602
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

Work partially carried out at the Universities of Rochester and Odense in 1978.

Partially supported by a grant from the Danish Natural Science Foundation.

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