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On a generalized storage model with moment assumptions

Published online by Cambridge University Press:  14 July 2016

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.

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 kj, 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.

Type
Research Papers
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.

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