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On a general storage problem and its approximating solution
Published online by Cambridge University Press: 01 July 2016
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.
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- Copyright © Applied Probability Trust 1981
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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