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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.
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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 

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Footnotes

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

References

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