This second meeting In the ORSA/TIMS Applied Probability series was attended by 90 participants, more than a quarter of them from Australia, Belgium, Britain, Canada, Denmark, Finland, Hungary, Italy, Israel, Japan, The Netherlands and New Zealand. The remaining participants were from the USA, among them several graduate students.

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed, further, viz., spatial nearest-neighbour lattice models, and doubly stochastic models.

There are two modes by which gross deformation can occur in a solid. Slips may occur along crystal planes, or the atomic planes may be torn apart and create microcracks. Both are highly stochastic phenomena requiring attention from probabilists. The present work attempts to understand the creation and growth of microcracks and their effect on the macro-level deformations.

Professor Neuts opened the discussion by suggesting that Applied Probability appeared to be in a precarious situation in the USA. Applied probabilists were widely spread in mathematics, statistics, electrical engineering, biomathematics and other departments; the fact that they defied classification was a drawback. They were perhaps not pure enough to be accepted by the mathematicians, but yet not applied enough for the practitioners. ‘Where’ he asked ‘do we fit in?’.

Consider a general single-server queue where the customers arrive according to a Poisson process whose rate is modulated according to an independent Markov process. The authors have previously reported on limits for the average delay in light and heavy traffic. In this paper we review these results, extend them to multiserver queues, and describe some approximations obtained from them for general delays.

Queueing networks continue to be of both theoretical interest and applied interest. One of the operations performed on a point process, representing customer flows in the network, thins a given process into two (or more) subprocesses each of which may thereafter be the arrival process to subsequent servers. A fair amount is known about the properties of each subprocess.

Using operator methods, we prove a general decomposition theorem for Jackson networks. For its transient joint queue-length distribution, we can stochastically bound it above by a network that decouples into smaller independent Jackson networks.

Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric. The graph of the asymptotic rate η of these densities as a function of the traffic intensity ρ is the caudal characteristic curve. This is an informative graph from which a number of qualitative inferences about the behavior of the queue may be drawn.

We consider a k-component system where the failure rates of the components interact, but where interaction depends only on the current ages of the components. The stationary behaviour of such systems has been studied by, for example, Franken and Streller (1980). Our main aim is to study the transient behaviour, and in particular to show that under easily verifiable conditions the transient behaviour tends to a stationary situation.

A state-space model is presented for a queueing system where two classes of customer compete in discrete-time for the service attention of a single server with infinite buffer capacity. The arrivals are modelled by an independent identically distributed random sequence of a general type while the service completions are generated by independent Bernoulli streams; the allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc -rule, is shown to be optimal when the expected long-run average criterion and the expected discounted criterion, over both finite and infinite horizons, are used. This static prioritization of the two classes of customers is done solely on the basis of service and cost parameters. The analysis is based on the dynamic programming methodology for Markov decision processes and takes advantage of the sample-path properties of the adopted state-space model.

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.

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.

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 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.

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.

One considers semi-Markov processes using phase-type distributions, such that the marginal distribution of the interevent times are identically distributed but not independent. By suitably choosing the phase-type distributions, one obtains an exponential marginal distribution.

A numerical method is proposed for computing the stationary probabilities of a Markovian system consisting of two subsystems with slight interrelation. It exploits the independent features of the subsystems for evaluating the effects of their interrelation. In a test with a simple example, the numerical method proposed required less computational burden than other common numerical methods.

A general method is developed for giving simulation estimates of the probability ψ(u, T) of ruin before time T. When the probability law P governing the given risk reserve process is imbedded in an exponential family (Pθ), one can write ψ(u, T) = EθRθ for certain random variables Rθ given by the fundamental identity of sequential analysis. Using this to simulate from Pθ rather than P, it is possible not only to overcome the difficulties connected with the case T = ∞, but also to obtain a considerable variance reduction. It is shown that the solution of the Lundberg equation determines the asymptotically optimal value of θ in heavy traffic when T = ∞, and some results guidelining the choice of θ when T < ∞ are also given. The potential of the method in complex models is illustrated by two examples.

Since the late 1960s Wiener's theory on the non-linear functionals of white noise has been widely applied to the construction of mathematical models of non-linear systems, especially in the field of biology. For such applications the main part is the measurement of Wiener's kernels, for which two methods have been proposed: one by Wiener himself and the other by Lee and Schetzen. The aim of this paper is to show that there is another method based on Hida's differential operator.