The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

The response of the photographic grain to light is a non-deterministic process which is as yet not completely understood. This response, as measured by the photographic density, is usually taken to be a function of the product of incident light intensity and exposure time interval duration, but at extreme values of either of these two quantities, this is no longer true. This latter effect is called reciprocity-law failure.

This paper discusses a probabilistic model, similar to a multiserver queue, for high-intensity reciprocity failure.

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.

Joint densities concerning in particular the value and time of the maximum over a fixed time interval, or the behavior over intervals determined by some first- and last-passage times, are determined for Brownian motion, the three-dimensional Bessel process and Brownian meander. Simple change of measure formulas permit easy passage from one process to the other. Examples are given.

The asymptotic behavior of the maximum likelihood estimator of a parameter in the drift term of a stationary ergodic diffusion process is studied under conditions in which the true drift function and true noise function do not coincide with those specified by the parametric model.

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

We observe a sequence {Xk}k≧1 of i.i.d. non-negative random variables one at a time. After each observation, we select or reject the observed variable. A variable that is rejected may not be recalled. We want to select N variables as soon as possible subject to the constraint that the sum of the N selected variables does not exceed some prescribed value C > 0. In this paper, we develop a sequential selection procedure that minimizes the expected number of observed variables, and we study some of its properties. We also consider the situation where N → ∞and C/N → α > 0. Some applications are briefly discussed.

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.

Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a → 0 as n →∞.

We show that the results may be proved in a unified and simple way by using a result of Le Cam.

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

Two classes of ergodic stationary multitype spatial point processes are constructed. These processes have the property that interactions between the types exist, but cannot be detected using standard second-order methods of analysis. Simulations indicate that the interactions can, however, be detected by using ‘empty space' techniques.

Many queueing situations such as computer, communications and emergency systems have the feature that customers may require service from several servers at the same time. They may thus be delayed until the required number of servers is available and servers may be idle when customers are waiting. We consider general server-completion-time distributions and derive approximation methods for the computation of the steady-state distribution of the number of customers in queue as well as the moments of the waiting-time distribution. Extensive computational results are reported.

We study the problem of optimal customer admission to multiserver queues. These queues are assumed to live in an extraneous environment which changes in a semi-Markovian way. Arrivals, service mechanism and random reward/cost structure may all depend on these surroundings. Included as special cases are SM/M/c queues, in particular G/M/c queues, in a random environment. By a direct inductive approach we establish optimality of a generalized control-limit rule depending on the actual environment. Particular emphasis is laid on different applications that show the versatility of the proposed setup.

In this paper we discuss the total service rendered by all the servers in certain loss systems.

An analysis is given of the first-come-first-served waiting-time process in stages 1 and 2 of a two-stage service system with k and n parallel service channels in the first and second stages respectively, and m intermediate waiting places (k, n ≧ 1, m ≧ 0).

Although the method of analysis is straightforward the details are intricate and require careful study of the location of the zeros of a high-degree polynomial.

The analysis paves the way for an extensive study of the numerical effect on waiting time of blocking in commonly encountered systems of this nature. ‘Effective service time' in stage 1, defined so as to include blocked time, is considered separately.

Using a result of Bikjalis (1971) concerning the rate of convergence in the multidimensional central limit theorem we obtain informations about some limit distributions in multivariate renewal theory.

We consider degenerate limit laws for the sequence {Xn, n}n(N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.