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Storage processes with general release rule and additive inputs

Published online by Cambridge University Press:  01 July 2016

P. J. Brockwell ,
S. I. Resnick  and
R. L. Tweedie
P. J. Brockwell*
Affiliation:
Colorado State University
S. I. Resnick*
Affiliation:
Colorado State University
R. L. Tweedie
Affiliation:
CSIRO Division of Mathematics and Statistics, Melbourne
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.
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Abstract

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation

The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.

Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.

Keywords

DAM THEORYSTORAGE EQUATIONSTATIONARY DISTRIBUTIONLÉVY PROCESSGENERAL RELEASE RULEMARKOV PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 392 - 433
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Research supported by NSF Grant MCS 78-00915-01.

a

Present address: SIROMATH Pty Ltd, 1 York St., Sydney, NSW 2000, Australia.

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