This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons.

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable X i . The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Y i from time i to time i − 1. Assume that (X i , Y i ), i ∈ N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

The subject of geometrical probability has recently shown signs of renewed vigour which is partly due to the attractive nature of the many unsolved problems which it contains, and partly the ever widening field of applications. These applications are so diverse and so scattered in the literatures of different branches of science that when Professor M.G. Kendall and I published a monograph on the subject in 1963 some interesting papers were ignored. Others have since appeared and the purpose of the present paper is to survey, in a very brief manner, work not described in that book. In fact the present bibliography, which contains 94 items, is disjoint from the bibliography in our joint monograph.

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Let X1, X2, ·· ·be stationary normal random variables with ρn = cov(X0, Xn). The asymptotic joint distribution of and is derived under the condition ρn log n → γ [0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.

A new measure of the ageing process called the vitality measure is introduced. It measures the ‘vitality' of a time period in terms of the increase in average lifespan which results from surviving that time period. Apart from intrinsic interest, the vitality measure clarifies the relationship between the familiar properties of increasing hazard and decreasing mean residual life. The main theorem asserts that increasing hazard is equivalent to the requirement that mean residual life decreases faster than vitality. It is also shown for general (i.e. not necessarily absolutely continuous) distributions that the properties of increasing hazard, increasing failure rate, and increasing probability of ‘sudden death' are all equivalent.

Expressions are derived for the expected waiting times for the customers of two types who arrive in batches (in a compound Poisson fashion) at a single-server queueing station with no waiting room. Those who cannot get served immediately keep returning to the system after random exponential amounts of time until they get served. The result is shown to agree with similar results for three special cases studied in the literature.

This paper is concerned with the optimality equation for the average costs in a denumerable state semi-Markov decision model. It will be shown that under each of a number of recurrency conditions on the transition probability matrices associated with the stationary policies, the optimality equation has a bounded solution. This solution indeed yields a stationary policy which is optimal for a strong version of the average cost optimality criterion. Besides the existence of a bounded solution to the optimality equation, we will show that both the value-iteration method and the policy-iteration method can be used to determine such a solution. For the latter method we will prove that the average costs and the relative cost functions of the policies generated converge to a solution of the optimality equation.

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.

The use of alternating renewal processes to model the behaviour of a repairable system is discussed, and properties of two measures of availability are reviewed. It is shown how the introduction of forward recurrence times to failure enables these measures to be generalised.

In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f0(t)k(log f0 (t)), where f0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in ℝ and derive estimators of the q th moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements on q -flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.

Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovič iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.

The asymptotic behaviour of a superadditive bisexual Galton–Watson branching process is studied. Sufficient conditions for the almost sure and L1 convergence of the suitably normed process are given. Finally, a first approach to the study of the L1 convergence for a superadditive bisexual Galton–Watson branching process under the Z log+Z condition is considered.

A conditional Poisson process (often called a double stochastic Poisson process) is characterized as a random time transformation of a Poisson process with unit intensity. This characterization is used to exhibit the jump times and sizes of these processes, and to study their limiting behavior. A conditional Poisson process, whose intensity is a function of a Markov process, is discussed. Results similar to those presented can be obtained for any process with conditional stationary independent increments.

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

In this paper a unified approach is used for proving relationships between customer-stationary and time-stationary characteristics of service systems with varying service rate and point processes. This approach is based on an intensity conservation principle for general stationary continuous-time processes with imbedded stationary marked point processes. It enables us to work under weaker independence assumptions than usual in queueing theory.