In this paper, we study a repair replacement model for a stochastically deteriorating system. For the expected discounted reward case, we show that the optimal replacement policy is of the form ‘replace at the time of the Nth failure'.

The problem of optimal control of a finite dam in the class of policies has been considered by Lam Yeh [6], [7]. In this paper, by using the first Dynkin formula, the same problems of specifying an optimal policy in the class of the policies to minimize the expected total discounted cost as well as the long-run average cost are considered. Both the expected total discounted cost and long-run average cost are determined explicitly, and then the optimal policy can be found numerically, Also, we obtain the transition density function and the resolvent operator of a reflecting Wiener process.

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.

In this paper we characterize the optimal class of output policies in a control model of a dam having a finite capacity. The input of water into the dam is determined by a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with a cost of K, (K ≧ 0) or decreased from M to 0 with zero cost, any such changes taking effect instantaneously. There is a reward of A monetary units for each unit of output, (A ≧ 0). The problem is to formulate an optimal output policy which maximizes the long-run average net reward per unit time.

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria:

1. (a) Minimum total long-run average cost per unit time.

2. (b) Minimum expected total discounted cost.

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

This note corrects a short and transparent proof of Harris's condition for recurrence in random walk processes.