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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

Information

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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