Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T22:08:34.519Z Has data issue: false hasContentIssue false

On a general storage problem and its approximating solution

Published online by Cambridge University Press:  01 July 2016

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.

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1981 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Abramowitz, M. and Stegun, I. (1964) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Barndorff-Nielsen, O. and Cox, D. R. (1979) Edgeworth and saddlepoint approximations with statistical applications (with discussion). J. R. Statist. Soc. B 41, 279312.Google Scholar
Bhattacharya, R. N. and Ghosh, J. K. (1978) On the validity of the formal Edgeworth expansion. Ann. Statist. 6, 434451.Google Scholar
Brockwell, P. and Chung, K. L. (1975) Emptiness times of a dam with stable input and general release functions. J. Appl. Prob. 12, 212217.Google Scholar
Çinlar, E. and Pinsky, M. (1972) On dams with additive inputs and a general release rule. J. Appl. Prob. 9, 422429.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Cramér, H. (1926) On some classes of series used in mathematical statistics. Proc. 6th Scand. Math. Congr., Copenhagen, 329425.Google Scholar
Cramér, H. (1928) On the composition of elementary errors. Skand. Aktuartidskr. 11, 13–74; 141–180.Google Scholar
Davis, P. J. and Rabinowitz, P. (1975) Methods of Numerical Integration. Academic Press, New York.Google Scholar
Gaver, D. P. and Miller, R. G. (1962) Limiting distributions for some storage problems. In Studies in Probability and Management Sciences, ed. Arrow, K. J. et al., Stanford University Press.Google Scholar
Harrison, J. M. and Resnick, S. I. (1976) The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347358.Google Scholar
Karlin, S. (1966) An Introduction to Stochastic Processes. Academic Press, New York.Google Scholar
Keilson, J. and Mermin, N. D. (1959) The second-order distribution of shot noise. IRE Trans. Inf. Theory IT-5, 7577.Google Scholar
Kendall, M. G. and Stuart, A. (1977) The Advanced Theory of Statistics, Vol. 1. Griffin, London.Google Scholar
Lindley, D. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277289.CrossRefGoogle Scholar
Mikkhlin, S. G. (1957) Integral Equations and their Applications. Pergamon Press, Oxford.Google Scholar
Moran, P. A. P. (1968) An Introduction to Probability Theory. Clarendon Press, Oxford.Google Scholar
Moran, P. A. P. (1969) A theory of dams with continuous input and a general release rule. J. Appl. Prob. 6, 8893.Google Scholar
Pogorzelski, W. (1966) Integral Equations and their Applications. Pergamon Press, Oxford.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Rubinovitch, M. (1974) Queues with random service output (abstract). Adv. Appl. Prob. 6, 207208.Google Scholar
Shao, T. S., Chen, T. C. and Frank, R. M. (1964) Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials. Technical Report 00.1100. I.B.M., New York.Google Scholar
Smith, N. M. H. and Yeo, G. F. (1981) Multidimensional Gauss quadrature in storage theory.Google Scholar
Wallace, D. L. (1958) Asymptotic approximations to distributions. Ann. Math. Statist. 29, 635654.Google Scholar
Yeo, G. F. (1974) A finite dam with exponential release. J. Appl. Prob. 11, 122133.Google Scholar
Yeo, G. F. (1976) A dam with general release rule. J. Austral. Math. Soc. 19B, 469477.Google Scholar