The overflow process of the multiserver queue with phase-type service times and finite waiting room is a Markov renewal process. The solution for this process is obtained. If the service times are exponential the overflow process reduces to a renewal process. For the latter case explicit expressions and numerical results are given.

Known results for the two-sex birth-death population process are extended to the case where immigration of females is allowed. It is shown that the methods used follow those for the standard birth, death and immigration process.

It is well known that the law of a Brownian motion started from x > 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.

For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.

Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sups ϵ S(G)E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.

We propose a simple and efficient method of obtaining sufficient conditions for the existence of a stationary regime for multichannel fully available queueing systems with repeated calls.

A group of n independent identical machines with exponential lifetimes is considered. Repair is made simultaneously for all failed machines. It is possible to observe the ‘state parameter', i.e. the number of operating machines. It is proved that for two types of critieria (minimal cost per time unit and maximal return per time unit) the optimal repair policy is of the following type: repair when the number of failed machines reaches some prescribed number .

We consider a system of three queues in which arriving customers are assigned to other queues if rejected from entry to one queue; and thus the work load of each queue is shared. The objective function considered is a combination of holding costs, routing costs, and customer service rewards. We first establish the characteristics of optimal control policies via a Markov decision process formulation. Next, the decomposed problems with partial information for each server are considered and the results compared with the original problem are shown. Appropriate combinations of optimal solutions for the decomposed problems are then used either in approximating the centralized optimal policy or in determining a good starting policy for successive approximation of the multidimensional Markov decision process. Numerical results of specific models are also presented.

We consider a family of single-server queueing systems with two priority classes. The system operates under a dynamic priority queue discipline in which the relative priorities of customers increase with their waiting times, and which can be characterized by the urgency number. We investigate the transient as well as the steady-state behavior of the virtual waiting times of the two classes of customer as functions of the urgency number. Stochastic orderings, the joint distribution, and surprising limit results for these processes are obtained for the first time.

Recently, Massey [1] proved that the vector of queue lengths of some queueing networks is stochastically dominated at any given time by that of a corresponding system of parallel M/M/l queues. This result is interesting, even though the bounds are generally quite conservative, in that the transient behavior of independent parallel M/M/l queues is considerably easier to analyze than that of a network.

This note provides an alternative proof of a generalized form of that result.

In a recent letter, Deshpande and Kochar (1983) gave the following theorem.

Shaked (1982) gives a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports. However there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true. The difficulty arises because zero terms are discarded in the definition of the sign change operator S. Therefore the equality of and x′ does not imply that

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

We consider a family of single-server queueing systems with two priority classes. The system operates under a dynamic priority queue discipline in which the relative priorities of customers increase with their waiting times, and which can be characterized by the urgency number. We investigate the transient as well as the steady-state behavior of the virtual waiting times of the two classes of customer as functions of the urgency number. Stochastic orderings, the joint distribution, and surprising limit results for these processes are obtained for the first time.

We consider the flow of service starting times in the M/G/1/∞ queue and study some of its equilibrium properties.

We analyze and compare the performance, under overload, of a class of queueing and service disciplines for an M/M/l queue. The disciplines we study are (i) FIFO with finite buffer (M/M/1/N), (ii) FIFO with push out (M/M/1/N-PO), (iii) LIFO with push out, (iv) FIFO with time out (M/M1-TO) and (v) LIFO with time out. For each of these disciplines we obtain the performance measures such as the throughput, the moments and the distribution of the waiting time of the customers who get served and numerically compare the disciplines with respect to these measures.

This communication provides an explicit expression for the joint probability of exactly i arrivals and j service completions in a duration of length t for the simple queue. The impossibility of a similar expression for the queue-length distribution is explained.

In the early part of this century, a mechanical ‘ciphering machine’ was invented and marketed by a Berlin engineer, Alexander von Kryha. The Kryha machine performed a polyalphabetic substitution on text controlled by one of 1050 possible keys. In this lecture we describe the general principles of cryptanalysis and its interplay with probability theory and show how the Kryha machine may be cryptanalyzed.