We consider the following circuit-switching problem which is one of the simplest possible extensions of the classical M/M/K/K model: there are p source centers connected to a hub by channels of various line-capacities and the hub is connected to a destination center by a common channel with its own line-capacity. A circuit requires a line from its source to the hub and another line from the hub to the destination. The holding times of circuits are independent, arbitrarily distributed random variables with means which depend on their source, and requests for circuits arrive in Poisson streams. Blocked calls are cleared. The problem is to calculate the blocking probabilities at each of the sources. The formal solution is well known but its calculation is exponentially hard in p.

We have developed a technique which extends recent results on integral representations and their asymptotic expansions to obtain the full expansion for the blocking probabilities in inverse powers of a large parameter N. The power of the method derives from two sources: first, an efficient recursive formula for calculating the general term of the expansion; second, tight upper and lower bounds which accompany the estimates. The computational complexity is polynomial in p. We report on computations.

The crossing intensity of a level by a shot noise process with a monotone response is studied, and it is shown that the intensity can be naturally expressed in terms of a marginal probability.

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

Standard models in stochastic resource allocation concern the economic processing of all jobs in some set J. We consider a set up in which tasks in various subsets of J are deemed to be alternative to one another, in that only one member of such a subset of alternative tasks will be completed during the evolution of the process. Existing stochastic scheduling methodology for single-machine problems is developed and extended to this novel class of models. A major area of application is in research planning.

A probabilistic proof is given of the fact that the departure process from two initially empty. ·/M/1 queues in tandem is unaffected when the service rates are interchanged. As a consequence of this, we show that when the sum of the service rates at the two queues is held constant the departure process stochastically increases as the service rates become equal. The proofs are based on coupling of reflected random walks.

A consecutive-k-out-of-n system consists of n identical and linearly ordered components. The system will fail if and only if at least k consecutive components fail. Let Tn be the system&s lifetime. Then, under very general conditions we prove that there is a positive constant a, so that the distribution of the random variable n(1/ka)Tn converges to a Weibull distribution, as n→∞

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying the GI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for the GI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.

The algorithm has been coded and examples of numerical results are included.

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

The Ehrenfest model is modified by drawing r balls at a time. The stationary distribution is the same as for r = 1.

This paper studies the single-server queueing system with two independent input streams: a GI/G and an M/G stream. A new proof is given of an old result which shows how this system can be transformed into an equivalent ‘single input stream’ GI/G/1 queue, and methods to study that equivalent system numerically are given. As part of the numerical analysis, algorithms are given to compute the moments and the distribution function of busy periods in the M/G/1 queue, and of other related busy periods. Special attention is given to the single-server queue with independent D/G and M/G input streams.

This work is to be used in the modeling of real-time computer systems, which can often be described as a single-server queueing system with independent D/G and M/G input streams, see for example Ott (1984b).

Conditions implied by the optimality of static priority policies in a special class of stochastic scheduling problems will be derived.

We show that Niu&s (1980) conjecture is true for Poisson arrivals and light traffic. We also show that a weak version of the conjecture is true for a special case in heavy traffic.

A simple example is given of a point process with dependent inter-event intervals for which the distributions of the counting random variables Nt are given by the same formulas as for a renewal process. This example shows the care needed in establishing that a given point process has the renewal property, but also raises interesting questions about the class of point processes with given counting distributions.

A time-dependent solution for the number in a single-server queueing system with Poisson arrivals and exponential service times is derived in a direct way.

Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].