Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T06:14:45.006Z Has data issue: false hasContentIssue false
  • English
  • Français

On a generalized finite-capacity storage model

Published online by Cambridge University Press:  14 July 2016

Samuel W. Woolford
Samuel W. Woolford*
Affiliation:
Worcester Polytechnic Institute
*
∗ Postal address: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn (k)}, for k ≠ j. Here, the random variables Vn(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn) and (Zn, Qn, Ln), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

Keywords

LEVEL OF STORAGEOVERFLOWDEMAND LOSTLIMIT DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 663 - 674
Copyright
Copyright © Applied Probability Trust 1983 

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. (1970) Finite dams with inputs forming a Markov chain. J. Appl. Prob. 7, 291–303.CrossRefGoogle Scholar
[2] Arjas, E. and Speed, T. P. (1973) Symmetric Wiener-Hopf factorizations in Markov additive processes. Z. Wahrscheinlichkeitsth. 26, 105–118.Google Scholar
[3] Brockwell, P. J. (1972) A storage model in which the net growth rate is a Markov chain. J. Appl. Prob. 9, 129–139.Google Scholar
[4] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
[5] Hoel, P. G., Port, S. C. and Stone, C. J. (1972) Introduction to Stochastic Processes. Houghton-Mifflin, Boston.Google Scholar
[6] Lloyd, E. H. (1963) Reservoirs with serially correlated inflows. Technometrics 4, 85–93.Google Scholar
[7] Moran, P. A. P. (1954) A probability theory of dams and storage systems. Austral J. Appl. Sci. 5, 116–124.Google Scholar
[8] Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand and Reinhold, New York.Google Scholar
[9] Phatarfod, R. M., Speed, T. P. and Walker, A. M. (1971) A note on random walks. J. Appl. Prob. 8, 198–201.CrossRefGoogle Scholar
[10] Prabhu, N. U. (1958) Some exact results for the finite dam. Ann. Math. Statist. 29, 1234–1243.Google Scholar
[11] Puri, P. S. (1977) On the asymptotic distribution of the maximum of sums of a random number of i.i.d. random variables. Ann. Inst. Statist. Math. 29, 77–87.CrossRefGoogle Scholar
[12] Puri, P. S. and Woolford, S. W. (1981) On a generalized storage model with moment assumptions. J. Appl. Prob. 18, 473–481.Google Scholar
[13] Roes, P. B. M. (1970) The finite dam. J. Appl. Prob. 7, 316–326.Google Scholar
[14] Woolford, S. W. (1980) Limit distributions for normalized sums of random variables without moments with applications to a generalized storage model. Mimeograph Series, Department of Mathematical Sciences, Worcester Polytechnic Institute.Google Scholar