Balking (refusing to join the queue) and reneging (leaving the queue after entering) are considered. The model assumes (1) Customers arrive from a single infinite source in a Poisson stream (2) Arriving customers balk with probability n/N where n is the number in system and N is the maximum number allowed in the system (3) Joining customers renege if service does not begin by a certain time, which is a random variable with negative exponential distribution (4) A single-service facility operates on a first-come, first-served basis with negative exponential service time distribution. For the steady state, the following are obtained the state probabilities, mean number in queue and system, the probability of R or more in system, the probabilities of balking, waiting, reneging, and acquiring service, the customer loss rate, the distribution and mean value of time in queue for customers who acquire service, and the corresponding results for those who renege. All of these results are also obtained for a pure balking system (no reneging) by setting the reneging parameter equal to zero. This case may be interpreted as the simple machine interference problem for which some of our results appear to be new.

A definition of the problem of the initial transient with respect to the steady-state mean value has been formulated. A set of criteria has been set forth by which the efficacy of any proposed rule may be assessed. Within this framework, five heuristic rules for predicting the approximate end of transiency, four of which have been quoted extensively in the simulation literature, have been evaluated in the M/M/l situation. All performed poorly and are not suitable for their intended use.

The relative importance of kill probabilities, rates of fire, and other parameters in a duel is considered by examining their effect on the most natural measure of effectiveness, the probability that a given side will win. In the “fundamental” duel, two contestants, A and B, fire at each other until one is killed. The time between rounds fired is a random variable of known but different density function for each, and each has a different known but fixed kill probability. The duel starts with both duelists having unloaded weapons and unlimited ammunition supplies. The general solution is obtained in quadrature form and specific solutions are derived for a particular firing time distribution and for certain variations in initial conditions. Finally, the general solution and a specific example for the duel with random initial surprise are obtained.

The tension and torsion helical springs of round cross section have been analyzed by a “thin-slice” method [1]. The effects of both curvature and pitch are included. The results of this analysis are given and discussed in this paper. Stresses, deflections, curvature changes, diametral contractions, and coupling effects are included.

The one-on-one stochastic duel is extended to the general two-on-one duel for the first time. The state equations, win probabilities, mean value, and variance functions are derived. The case where one side has Erlang (2) firing times and the other is negative exponential is compared with the corresponding "Stochastic Lanchester" and Lanchester models to demonstrate their nonequivalence.