Special algorithms have been developed to compute an optimal (s, S) policy for an inventory model with discrete demand and under standard assumptions (stationary data, a well-behaved one-period cost function, full backlogging and the average cost criterion). We present here an iterative algorithm for continuous demand distributions which avoids any form of prior discretization. The method can be viewed as a modified form of policy iteration applied to a Markov decision process with continuous state space. For phase-type distributions, the calculations can be done in closed form.

The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (i0,0) of the process. By showing that the expected return time to (i0, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0,t) is then established by showing the asymptotic independence of these two.

A result proved by Saunders et al. (1984) is generalised. This generalisation, and its proof, may help to explain some curiosities noted in the original result.

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

Empirical Paretian distributions play an important role in urban demography, size distributions of firms and income distributions; hence the addition of Paretian random variables is of interest. First, we give the asymptotic behavior (for large values of the variable) of the density function of a sum of n independently distributed Paretian variables. We then obtain the limiting distribution of an infinite sum of (i.i.d) Paretian variables and link our results with the theory of stable distributions.

Recently many authors (e.g. Shaked (1982), Deshpande and Kochar (1983), Sathe (1984)) have established relations between the dispersive ordering and some other partial orderings of distributions. This note presents connections which the dispersive ordering has with monotone failure rate distributions.