A new method to solve steady state queueing equations

There are a great number of queueing systems, including the M^X/M^Y/c, the Gl^X/M/c and the discrete Gl/G/1 queue in which the state probabilities are determined by repeated queue equations. This paper gives a simple, efficient and numerically stable algorithm to caiculate the state probabilities and measure of performance for such systems. The method avoids both complex arithmetric and matrix manipulations.     

A note on splitting the bump in an elimination factorization

This exposition presents a method for incorporating a technique known as “splitting the bump” within an elimination form reinversion algorithm. This procedure is designed to reduce fill-in during reinversion and should improve the efficiency of linear programming systems which already use the superior elimination form of the inverse.     

Optimal control of price through restricted production

Consider a regulated monopolist whose current profits would be maximized if they could charge a price p?, where p? exceeds the current market price. By reducing production below current consumer demand the monopolist can create an illusion of a shortage and induce the regulator to allow a price increase. Conditions are given for which the production rate that maximizes the monopolist's expected discounted profits over an infinite horizon will have the property that the amount of unsatisfied consumer demand will be a non-increasing function of current market price.     

Computing the discounted return in markov and semi-markov chains

This paper addresses the problem of computing the expected discounted return in finite Markov and semi-Markov chains. The objective is to reveal insights into two questions. First, which iterative methods hold the most promise? Second, when are interative methods preferred to Gaussian elimination? A set of twenty-seven randomly generated problems is used to compare the performance of the methods considered. The observations that apply to the problems generated here are as follows: Gauss-Seidel is not preferred to Pre-Jacobi in general. However, if the matrix is reordered in a certain way and the author's row sum extrapolation is used, then Gauss-Seidel is preferred. Transforming a semi-Markov problem into a Markov one using a transformation that comes from Schweitzer does not yield improved performance. A method analogous to symmetric successive overrelaxation (SSOR) in numerical analysis yields improved performance, especially when the row-sum extrapolation is used only sparingly. This method is then compared to Gaussian elimination and is found to be superior for most of the problems generated.     

Compound distributions with efficient computation in inventory model applications

In an inventory model, the distribution of total units demanded can be considered as a compound distribution arising from the distributions of demand occurrence and individual demand size. Three such compound distributions are considered, where the number of demands is Poisson distributed. The demand size distribution will depend on the observed or desired variance-to-mean ratio. An approximation using the gamma distribution is given in terms of the cumulants of the compound distribution for both fixed and stochastic lead times.     

Multi-class inventory models with demand a function of inventory level

This paper considers the problem of maintaining an inventory of an item which can deteriorate and become useless. A periodic review procedure is used and new items ordered may experience a time lag in delivery. Items are considered to deteriorate through one or two states before becoming useless. Thus the deterioration process in each period plays the role of the usual demand process and is a function of the inventory level at the beginning of each period. For the case of no time lag in delivery, one stage deterioration, and either binomial or uniform deterioration, optimal ordering policies are obtained for the n-period dynamic model with the standard cost structure. (For the shortage probability criterion see the other paper by Iglehart and Jaquette, in this issue.) These policies are of the single critical number type. For more complicated models suboptimal policies of this same type are found.     

Optimal policies under the shortage probability criterion for an inventory model with unknown dependent demands

The principal innovation in this paper is the consideration of a new objective function for inventory models which we call the shortage probability criterion. Under this criterion we seek to minimize the total expected discounted cost of ordering subject to the probability that the stock level at the end of the period being less than some fixed quantity not exceed some prescribed number. For three different models we show that the minimum order policy is optimal. This result is then applied to a particular inventory model in which the demand distribution is not completely known. A Bayesian procedure is discussed for obtaining optimal policies.     

Optimal capacity expansion

We consider the problem of temporal expansion of the capacity of, say, a plant or road given estimates of its desired usage (demand). The basic problem is: given a sequence of predicted demands for N time periods, determine the optimal investment decision in each period to minimize a linear investment cost and a strictly convex cost of capacity. The relationship between capacity and the investment decisions is assumed to be linear, but time varying. Constraints on both the individual decisions and on the sum of the decisions are considered. An algorithm for solving this problem is derived.