Lanchester attrition of interpenetrating forces

To approximate the solutions of detailed simulations of interpenetrating forces (or possibly to assist in curtailing Monte Carlo calculations), this article provides solutions to a simple problem assuming that the speed of advance is constant; the only interactions are local; Lanchester's linear or square law applies; force distributions are continuous if not initially uniform in depth. The resultant partial differential equations are solvable (1) in closed form if attrition is minimal or (2) with pain when attrition is sufficient to annihilate the leading edge of a force. This is exemplified only for the square law, where one must solve an integrodifferential equation for an ancillary function. A general solution is given for either law, and for the latter case a more complete one, assuming that initial force distributions are uniform. Useful properties of an unusual class of Bessel functions needed for this analysis are given in an appendix. Copies of computer programs are available.     

Evaluation of costs of ordering policies in large machine repair problems

This article examines a version of the machine repair problem where failures may be irreparable. Since the number of machines in the system keeps decreasing, we impose a fixed state-dependent ordering policy of the type often encountered in inventory models. Although the system is Markovian, the number of states becomes very large. The emphasis of the article, therefore, is on deriving computationally tractable formulas for the steady-state probabilities, the long-run average cost per unit time, and the vector of expected discounted costs. When the state space is so large that exact computations may be infeasible, we propose approximations which are relatively quick and simple to compute and which yield very accurate results for the test problems examined.     

Hu's precedence tree scheduling algorithm: A simple proof

This article presents a simple proof of Hu's algorithm for scheduling in minimum time a set of tasks constrained by precedence tree constraints, each task requiring a unit time to complete, and where m processors are available.     

A simple Markov model for the analysis of multiple air combat

It is proposed to describe multiple air-to-air combat having a moderate number of participants with the aid of a stochastic process based on end-game duels. A simple model describing the dominant features of air combat leads to a continuous time discrete-state Markov process. Solution of the forward Kolmogorov equations enables one to investigate the influence of initial force levels and performance parameters on the outcome probabilities of the multiple engagement. As is illustrated, such results may be useful in the decision-making process for aircraft and weapon system development planning. Some comparisons are made with Lanchester models as well as with a semi-Markov model.     

On a sequential test procedure with delayed observations

This paper considers sequential test procedures to decision problems where there exists time delays in obtaining observations.