Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T19:23:23.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On a generalized storage model with moment assumptions

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri  and
Samuel W. Woolford
Prem S. Puri*
Affiliation:
Purdue University
Samuel W. Woolford*
Affiliation:
Worcester Polytechnic Institute
*
∗Postal address: Department of Statistics, Mathematical Sciences Building, Purdue University, Lafayette, IN 47906, U.S.A.
∗∗Postal address: Department of Mathematical Science, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn), n = 0, 1, ·· ·}, with 0 = T0 < T1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj(·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

Keywords

MARKOV RENEWAL PROCESSSTORAGE MODELSLIMIT DISTRIBUTIONSTOTAL DEMAND LOSTAVERAGE STATIONARY INPUTSTORAGE LEVELSUPERCRITICALCRITICAL AND SUBCRITICAL CASES
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 2 , June 1981 , pp. 473 - 481
Copyright
Copyright © Applied Probability Trust 

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

The research of this author was supported in part by U.S. National Science Foundation Grant No. MCS77–04075, at Purdue University.

References

[1] Ali Khan, M. S. and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J. Appl. Prob. 5, 72–83.Google Scholar
[2] Balagopal, K. (1979) Some limit theorems for the general semi-Markov storage model. J. Appl. Prob. 16, 607–617.Google Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
[5] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
[6] Hoel, P. G., Port, S. C. and Stone, C. J. (1972) Introduction to Stochastic Processes. Houghton Mifflin, Boston.Google Scholar
[7] Kendall, D. G. (1957) Some problems in the theory of dams. J. R. Statist. Soc. B 19, 207–212.Google Scholar
[8] 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
[9] Moran, P. A. P. (1959) The Theory of Storage. Wiley, New York.Google Scholar
[10] Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.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.Google Scholar
[12] Puri, P.S. and Senturia, J. (1972) On a mathematical theory of quantal response assays. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 4, 231–247.Google Scholar
[13] Senturia, J. and Puri, P. S. (1973) A semi-Markov storage model. Adv. Appl. Prob. 5, 362–378.Google Scholar
[14] Senturia, J. and Puri, P. S. (1974) Further aspects of a semi-Markov storage model. Sankhya A 36, 369–378.Google Scholar
[15] Woolford, S. W. (1979) On a Generalized Storage Model. . Purdue University Libraries.Google Scholar