S. Christian Albright

A Statistical Analysis of Hitting Streaks in Baseball

Abstract This article presents a study of hitting streaks for individual batters in the game of baseball. All baseball fans know that players go through periods where they never seem to make an out and other periods where nothing will get them to base. There is no doubt that “hot streaks” and “cold streaks” do occur. The question explored here is whether these streaks occur more (or less) frequently than would be predicted by a probabilistic model of randomness. I examined the records of many “regular” Major League players through four seasons, 1987–1990 and used several statistical methods to check for streakiness. These include standard methods such as the runs test, as well as a more complex logistic regression model with several explanatory variables. Based on all of these methods, there is no doubt that a certain number of players exhibited definite streakiness in certain years. But the evidence also suggests that the behavior of all players examined, taken as a whole, does not differ significantly f...

Structural Results for Partially Observable Markov Decision Processes

This paper examines monotonicity results for a fairly general class of partially observable Markov decision processes. When there are only two actual states in the system and when the actions taken are primarily intended to improve the system, rather than to inspect it, we give reasonable conditions which ensure that the optimal reward function and the optimal action are both monotone in the current state of information. Examples of maintenance systems and advertising systems for which our results hold are given. Finally, we examine the case where there are three or more actual states and indicate the difficulties encountered when we attempt to extend the monotonicity results to this situation.

An approximation to the stationary distribution of a multiechelon repairable‐item inventory system with finite sources and repair channels

We consider a multiechelon repairable-item inventory system where several bases are supported by a central depot. Unlike METRIC-based models, there are only a finite number of repairmen at each base and the depot, and the failure rates at the bases depend on the current number of items online. The principal objective of this article is to develop a quick and accurate approximation to the steady-state distribution of this system. A secondary objective is to compare the solution of this system with a comparable METRIC solution.

Steady-state approximations for a multi-echelon multi-indentured repairable-item inventory system

Abstract In this paper we model a two-echelon multi-indentured repairable-item inventory system. Each of several ‘bases’ has a maximum number of identical online machines, and each machine consists of several module types. At random times these machines fail because of module failures. When a machine fails, the failed module is determined and is replaced by an identical spare module if one is available. Otherwise, the module is backordered. Depending on the type of failure, the failed module is repaired at a central depot or at the base where the failure occurs. We assume that separate repairmen are devoted to each module type. Thus, when a module fails, it is repaired by one of its designated repairmen. In calculating the steady-state operating characteristics of this system, the usual Markovian approach leads to a multidimensional state space that is extremely large even for relatively small problems. Solving this multidimensional system exactly is virtually impossible because of the huge number of state. Consequently, we propose an approximation method that enables us to solve large problems relatively quickly. Although our solution is only approximate, results from a variety of test problems indicate that the approximation is quite accurate.

Markovian multiechelon repairable inventory system

In this article we model a two-echelon (two levels of repair, one level of supply) repairable-item inventory system using continuous-time Markov processes. We analyze two models. In the first model we assume a system with a single base. In the second model we expand this model to include n bases. The Markov approach gives rise to multidimensional state spaces that are large even for relatively small problems. Because of this, we utilize aggregate/disaggregate techniques to develop a solution algorithm for finding the steady-state distribution. This algorithm is exact for the single-base model and is an approximation for the n-base model, in which case it is found to be very accurate and computationally very efficient.

Allocation of Research Grants to University Research Proposals.

Abstract In this paper we examine the optimal allocation of research money to incoming research proposals in a university environment. Since a university has no obvious objective to maximize when allocating research money, but rather has several, possibly conflicting, goals it would like to achieve, the method of goal programming is proposed as a solution technique, and the meaning and estimation of the priority weights in the resulting objective function in the goal programming problem, with regard to tradeoffs between goals, are discussed thoroughly. Finally, calculations based upon hypothetical sample data re presented in order to test the sensitivity of the model to different sets of priority weights.

An Approximation To The Stationary Distribution Of A Multidimensional Markov Process

Abstract In this paper we present an approximate, non-iterative method for calculating the stationary distribution of a multidimensional Markov process. Although the method has more general applications, we illustrate it for a particular example of a repairable item inventory model with returns. We analyze two versions of this model. In the first version we assume that all failed items are repairable, whereas in the second version we assume that some of the failed items are irreparable and hence are scrapped. Each version gives rise to multidimensional state spaces that are extremely large even for problems with a relatively small number of items. Because of the large state spaces, the emphasis of this paper is on developing an approximation for the stationary distribution. We show that this approximation is not only easy to calculate but is also quite accurate across a broad range of problem parameters.

A birth–death model of advertising and pricing

This paper employs the methods currently used to solve many queuing control models in order to investigate the behavior of a firms optimal advertising and pricing strategies over time. Given that a firms market position expands or deteriorates in a probabilistic way which depends upon the current position, the rate of advertising, and the price the firm charges, we present conditions which ensure that the optimal level of advertising is a monotonic function of the firms market position, and we discuss the economic meaning of these conditions. Furthermore, although the primary focus is upon a non-competitive environment, we develop the above model as a non-zero sum, two-person stochastic game and show that an equilibrium strategy exists which is simple to compute.

Markov Models of Advertising and Pricing Decisions

This paper uses Markov decision analysis to study the properties of optimal advertising and pricing decisions in a dynamic, stochastic environment. We are primarily concerned with exhibiting and interpreting properties of the Markovian transition probabilities and one-period reward functions that imply that the optimal advertising levels and optimal prices always increase or always decrease as a function of the firms market position. This is done in both non-competitive and competitive situations. In the former, total discounted reward for a single firm is the objective, whereas in the latter we use the concept of stochastic games to construct what is known as a discounted equilibrium point for two competing firms.

A Markov-Decision-Chain Approach to a Stochastic Assignment Problem

This paper discusses a problem where jobs of random quality, or importance, arrive at random times and must be done by one of a fixed set of men on hand. The men can all complete any job in an exponential amount of time, with mean l/u, but the men are of different known qualities. Once a man finishes a job he again becomes available to do future jobs. If a man of quality p is assigned to do a job whose value is observed to be x, a reward rp,x is received. Also a cost of cp is incurred per unit time that a man of quality p is kept idle. The decision maker must observe the values of the incoming jobs and assign them to available men so as to maximize expected reward per unit time. The problem is set into the framework of a G/M/n queue with a possibly limited waiting room, and the optimization is done by techniques of Markov decision analysis.