SOME PROBLEMS IN THE THEORY OF PROVISIONING AND OF DAMS

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Abstract
The paper begins with a review of two problems in the theory of provisioning with a discrete stock considered by Pitt (1946), and a short account of Moran's work (1954) in the theory of finite dams. It is pointed out that these provide two different methods of attack, each appropriate to certain conditions, on problems in the probability theory of a general storage function S(t) defined at time t by S(t) = I(t) - D(t) - F(t), where f(t), D(t), F(t) are respectively an input, output and overflow function. This storage function is identified with the stock deficit in provisioning theory, or the dam content in dam theory, so that any problem and its solution in the one theory has an exact analogue in the other. The paper continues with the use of Pitt's results of the theory of provisioning in two analogous cases of the infinite discrete dam. This is followed by the application of Moran's methods of the theory of dams in some analogous problems of provisioning with a discrete finite stock; exact solutions are obtained for the discrete and continuous cases of a particular problem in which ordering and replacement times coincide. The paper closes with an exact solution of the general storage problem in the case of a finite continuous storage function S(t), fed by a discrete input function of Poisson type, with a continuous output which has a steady rate when S(t) ≠ 0, and is zero if S(t) = 0, and an overflow function such that S(t) never exceeds a prescribed value.

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