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This report is the second and final half of a project to exhaustively review the literature of one-on-one stochastic duels. This part contains a comprehensive compendium of all results contained in papers which were previously listed in the annotated bibliography of the preceding report. The results are given in a common notation and are systematically organized., See also Parts 1 and 2, AD-A068 008.

This report is approximately one half of the output of a project to exhaustively review the literature of one-on-one stochastic duels. It contains two main parts. The first is an exposition, identifying and tying together for the first time, by examples, the two principal analytical techniques used, namely; mixture theory and renewal theory. Secondly, it contains an exhaustive, comprehensive, and fully annotated bibliography of all research results in this theory which are known to the author. Finally, it also lists many theorems involving characteristic functions which are particularly useful for this work. (Author), Presented at the Annual ORSA Meeting (4th), Washington, DC 11 May 66.

The effect of random driver reaction time on throughput at a saturated signalized intersection is studied. The model assumes zero acceleration time and identical terminal velocities for all vehicles. Analytic expressions are derived for the average number of vehicles that pass through the intersection during a go-period. Some calculations and curves are given for the specific case of a general gamma distribution function for the reaction time variable. It is shown that a mean value model is quite accurate for mean value purposes, but that variability in reaction time does cause significant variance in throughput. (Author)

Terminating and non-terminating stochastic processes are studied. If digital simulation is used, either a time-slicing or an event-sequencing technique may be used to obtain an estimate of some measure (mean value in our case) of system performance. We consider the variance of the estimate to be the basic measure for determining the goodness of the estimate. Three points are examined: (1) unbiasedness of the estimate, (2) efficiency of the estimate, and (3) variance reduction techniques. For the nonterminating case a large class of steady state processes having a negative exponential covariance function is studied and curves and specific examples are derived. Replication is shown to be generally superior to increasing the length of a run. For the terminating case, a general expression for efficiency is derived and again replication is shown to be the superior variance reduction technique. All results are equally valid for discrete or continuous processes. (Author)

This paper continues the development of the theory of Stochastic Duels to include the distribution of the number of rounds fired. Most generally, the duel between two contestants who fire at each other with constant kill probabilities per round is considered. The time between rounds fired may be either a continuous random variable or it may be constant. These two cases are treated separately. The number of rounds available to each contestant at the beginning of the duel may be limited and is a discrete random variable. Besides the distribution of rounds fired, its first two moments and right tail are obtained. In addition to general results, special cases and specific examples are worked out. (Author)

By limiting the time between rounds to a constant, certain duels in which each side has a fixed kill probability in which strong interactions occur are investigated. The models investigated here are (1) the fundamental (one versus one) duel in which firing times are fixed and their ratio is a rational number; (2) the duel with displacements in which two contestants fire simultaneously at fixed intervals and a near miss causes a contestant to displace (and lose a firing turn); (3) multiple duels in which two versus one or two versus two fire simultaneously at fixed intervals and (4) the cluster duel in which there are several contestants on each side and firing is simultaneous. In the first three, the probability of a given side winning is obtained. In the last, the probability of m contestants on one side killing v of the n contestants on the other side at a single firing is derived. (Author)

The Theory of Stochastic Duels is extended by considering the distribution of time-to-completion of the fundamental duel. The model has fixed kill probabilities and either random or fixed time between rounds fired. Timelimitation is included. Special cases and examples are worked out. Clearly, the time-duration of combat has both tactical and logistic implications for the decision-maker. (Author)