Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T19:36:09.968Z Has data issue: false hasContentIssue false
  • English
  • Français

A semi-Markov storage model

Published online by Cambridge University Press:  01 July 2016

Jerome Senturia  and
Prem S. Puri
Jerome Senturia
Affiliation:
University of Wisconsin, Madison
Prem S. Puri
Affiliation:
Purdue University
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper a storage model is described in which fluctuations in the content are governed by a sequence of independent identically distributed (i.i.d.) random inputs and i.i.d. random releases. This sequence proceeds according to an underlying semi-Markov process. Laplace transforms of the exact distribution of the content are given for the case of negative exponential distributions for both inputs and releases. Exact expressions for limiting (in time) content distributions are found. In the general case, the asymptotic behavior of the content is described for critical and supercritical limiting conditions.

Keywords

SEMI-MARKOV PROCESSRANDOM INPUTSRANDOM RELEASESLIMIT BEHAVIORASYMPTOTICS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 362 - 378
Copyright
Copyright © Applied Probability Trust 1973 

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.)

References

[1] Ali Khan, M. S. and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J. Appl. Prob. 5, 72–83.CrossRefGoogle Scholar
[2] Arrow, K. J., Karlin, S. and Scarf, H. (1958) Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, California.Google Scholar
[3] Bliss, C. I. (1952) The statistics of bioassay, with special reference to the vitamins. Reprinted, with additions, from Vitamin Methods, Volume II. Academic Press, New York. 445–628.Google Scholar
[4] Chung, K. L. (1948) Asymptotic distribution of the maximum cumulative sum of independent random variables. Bull. Amer. Math. Soc. 54, 1162–1170.CrossRefGoogle Scholar
[5] Erdös, P. and Kac, M. (1946) On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.CrossRefGoogle Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar
[7] Finney, D. J. (1947) Probit Analysis. A Statistical Treatment of the Sigmoid Response Curve. Cambridge University Press, Cambridge.Google Scholar
[8] Finney, D. J. (1952) Statistical Method in Biological Assay. Charles Griffin, London.Google Scholar
[9] Gani, J. (1955) Some problems in the theory of provisioning and of dams. Biometrika 42, 179–200.CrossRefGoogle Scholar
[10] Gani, J. (1957) Problems in the probability theory of storage systems. J. Roy. Statist. Soc. B 19, 181–205.Google Scholar
[11] Gani, J. and Pyke, R. (1960) The content of a dam as the supremum of an infinitely divisible process. J. Math. Mech. 9, 639–652.Google Scholar
[12] Hasofer, A. M. (1966) The almost full dam with Poisson input. J. Roy. Statist. Soc. 28, 329–335.Google Scholar
[13] Hasofer, A. M. (1966) The almost full dam with Poisson input: further results. J. Roy. Statist. Soc. 28, 448–455.Google Scholar
[14] Karlin, S. and Fabens, A. (1962) Generalized renewal functions and stationary inventory models. J. Math. Anal. Appl. 5, 461–487.CrossRefGoogle Scholar
[15] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
[16] Lloyd, E. H. (1963) Reservoirs with serially correlated inflows. Technometrics 5, 85–93.Google Scholar
[17] Lloyd, E. H. and Odoom, S. (1965) A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Prob. 2, 215–222.Google Scholar
[18] Moran, P. A. P. (1954) A probability theory of dams and storage systems. Austral. J. Appl. Sci. 5, 116–124.Google Scholar
[19] Moran, P. A. P. (1959) The Theory of Storage. Methuen, London.Google Scholar
[20] Prabhu, N. U. (1965) Queues and Inventories. John Wiley, New York.Google Scholar
[21] Prabhu, N. U. (1968) Some new results in storage theory. J. Appl. Prob. 5, 452–460.CrossRefGoogle Scholar
[22] Puri, P. S. (1971) On the asymptotic distribution of the maximum of sums of a random number of I. I. D. random variables. Department of Statistics, Purdue University Mimeograph Series No. 265.Google Scholar
[23] Puri, P. S. and Senturia, J. (1972) On a mathematical theory of quantal response assays. Proc. Sixth Berkeley Symp. on Math. Statist, and Prob. (Biology-Health Section).Google Scholar
[24] Puri, P. S. and Senturia, J. (1972) An infinite depth dam with Poisson input and Poisson release. Submitted for publication.Google Scholar
[25] Pyke, R. (1961) Markov renewal processes. Ann. Math. Statist. 32, 1231–1259.CrossRefGoogle Scholar
[26] Rényi, A. (1957) On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 8, 193–199.CrossRefGoogle Scholar
[27] Senturia, J. (1972) On a mathematical theory of quantal response assays and a new model in dam theory. . Purdue University.Google Scholar
[28] Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A, 232, 6–31.Google Scholar
[29] Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.CrossRefGoogle Scholar
[30] Takács, L. (1970) On the distribution of the maximum of sums of mutually independent and identically distributed random variables. Adv. Appl. Prob. 2, 344–354.CrossRefGoogle Scholar
[31] Widder, D. V. (1941) The Laplace Transform. Princeton University Press, Princeton.Google Scholar