A Note on the First Emptiness Problem of a Finite Dam with Poisson Type Inputs

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Abstract
This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.

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