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Storage processes with general release rule and additive inputs
Published online by Cambridge University Press: 01 July 2016
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.
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- Copyright © Applied Probability Trust 1982
Footnotes
Research supported by NSF Grant MCS 78-00915-01.
Present address: SIROMATH Pty Ltd, 1 York St., Sydney, NSW 2000, Australia.
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