A set of ni arms of type i, i = 1,…, L, is available. A pull of arm of type i occupies a duration Vi at the end of which a reward Ci and Ni1,…, NiL new arms are obtained, while all other arms are frozen. A Gittins priority order of types is obtained and shown to yield the maximal discounted reward from this branching process of arms.

The most probable equilibrium configuration in a circuit-switched network is determined, in the limit of very large scale, by a primal optimization problem whose dual determines the blocking probabilities. This approximation can be inadequate for moderate scale, and so the blocking probabilities are usually determined by the Erlang fixed-point principle. In this paper, it is shown that there is a “softened” form of the dual natural to the problem. This gives determinations of the blocking probabilities agreeing with those derived from the Erlang principle as far as terms of order (scale)-1. It is more attractive than the Erlang principle, however, in that it has a mathematical basis, provides an extremal principle by its nature, yields immediately the geometric distribution of the numbers of free circuits, and clearly agrees with the “hard” dual in the limit of large scale.

From an urn containing colored balls, one ball is drawn and replaced by a random number of differently colored balls, with the distribution of the added balls depending only on the color of the ball drawn. Under mild regularity conditions, the proportions of different colors will converge to deterministic limits. Two applications of this standard result are described.

An improved simulation estimator of the expected total delay of the first n customers in the systems GI/M/k and GI/G/I is obtained by replacing the (raw data) delay of customer i by its expected delay given the state of the system upon its arrival. It is improved in the sense that the sum of the first n such values has the same mean and smaller variance than the usual (raw) estimator. The only constraint on the arrival process is that it be independent of the process of service times.

The group-testing problem for a binomial set of items is considered. It is desired to classify all items as good or defective with a minimum expected number of group tests. An improvement over the information lower bound, via a weak concavity property, is made for the minimum expected number of group tests.

The state of a system is modelled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process of rate λ. If the state of the system at arrival of the repairman does not exceed a certain threshold, he/she increases it by a random amount, otherwise no action is taken. Costs are assigned to each visit of the repairman, to each repair, and to the system being in state 0. It is shown that there exists a unique arrival rate λ which minimizes the average cost per unit time over an infinite horizon.

A new partial ordering generated by the set of star-shaped functions is introduced. This ordering is equivalent to the stochastic comparison of the length-biased distributions which are frequently appropriate for certain natural sampling plans in biometry, reliability, and survival analysis studies. It is shown that the length-biased ordering fits in the framework of stochastic and variatiility orderings. It enjoys many properties similar to those of the stochastic and variability orderings.

We introduce in this paper a new measure of component importance, called redundancy importance, in coherent systems. It is a measure of importance for the situation in which an active redundancy is to be made in a coherent system. This measure of component importance is compared with both the (Birnbaum) reliability importance and the structural importance of a component in a coherent system. Various models of component redundancy are studied, with particular reference to k/out / of / n systems, parallel-series systems, and series-parallel systems.

Systems are studied that consist of interfering components which are alternatively active (busy) and passive (idle) for random periods. Such systems naturally arise from the performance evaluation of computer models. A concrete invariance condition is imposed on the interferences allowed. Under this condition, the steady-state vector of active components is shown to be of product form as well as to be robust to distributional forms of active and passive periods. The proof is simple and self-contained. A number of concrete and generic examples is provided. These include resource sharing mechanisms, hierarchical circuit switchings, parallel processing, priorities, and breakdowns.

Analysis of the initial transient problem of Monte Carlo steady-state simulation motivates the following question for Markov chains: when does there exist a deterministic Tsuch that P{X(T) = y|(0) = x} = ®(y), where ρ is the stationary distribution of X? We show that this can essentially never happen for a continuous-time Markov chain; in discrete time, such processes are i.i.d. provided the transition matrix is diagonalizable.

We give an extremely simple argument to prove that in infinite user-commurlication channels, under the Aloha protocol, the number of successful transmissions is finite with probability 1. The same result is then shown to hold for those back-off protocols whose transmission probabilities are bounded away from 0.