Let T be the first exit time of Brownian motion W(t) from a region ℛ in d-dimensional Euclidean space having a smooth boundary. Given points ξ0 and ξ1 in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε |W(0) = ξ0, W(ε) = ξ 1} as ε → 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

This paper presents a general theory of random tessellations (i.e. stochastic aggregates of disjoint and space-filling cells) in d-dimensional Euclidean space. Some particular models of random tessellations are discussed in detail.

Let W be a sum of Bernoulli random variables and Uλ a Poisson random variable having the same mean λ = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and Uλ are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and Pólya sampling, of capture-recapture, of spacings and of matching and ménage.

‘Divide and conquer’ is a traditional approach in various fields of applied mathematics. In reliability, only modules have been proposed to decompose complex coherent systems. However, a system may include no modules, except ‘trivial’ ones. This paper is the second step of a study concerned with module generalizations : in the first step, pseudo-modules have been retained as the most general coherent subsystems which can yield a complete extension of the fundamental results concerning modules; in addition, it has been proved that they concern any binary coherent system; in this paper, it is shown that pseudo-modular decompositions are the most general coherent decompositions which can yield a complete extension of all the ‘refined bounds’ for the interval reliability and the availability currently proposed in terms of modular decompositions.

This study also yields some fundamental results to extend all the ‘refined bounds’ currently proposed for multistate coherent systems, using an easier approach.

We consider two storage/production systems in which items are produced continuously over time with fixed rate. In the first system items have infinite lifetime, while in the second system the lifetime of the items are finite and fixed. The inventory level distributions and other important functionals associated with these storage systems are derived. This derivation is accomplished by an analogy existing between the storage systems and certain queueing systems and a finite dam model. Optimization problems connected with these systems are also considered.

A road which narrows at a bottleneck from an ∞-lane road to a one-lane road is studied with the aid of two stochastic processes. Special attention is given to headways and gaps. At the bottleneck an equilibrium headway can be viewed as the maximum of a shifted exponential random variable and a minimum headway. After the bottleneck the situation becomes far more complicated. However, limiting results are obtained for headways and gaps at a large distance from the bottleneck. The asymptotic behavior of headways and gaps is largely determined by the behavior of the desired speed distribution at the lower extreme of its support.

This paper is concerned with a bivariate Markov process {Xt, Nt; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Perturbation analysis is an efficient approach to estimating the sensitivities of the performance measures of a queueing network. A new notion, called the realization probability, provides an alternative way of calculating the sensitivity of the system throughput with respect to mean service times in closed Jackson networks with single class customers and single server nodes (Cao (1987a)). This paper extends the above results to systems with finite buffer sizes. It is proved that in an indecomposable network with finite buffer sizes a perturbation will, with probability 1, be realized or lost. For systems in which no server can directly block more than one server simultaneously, the elasticity of the expected throughput can be expressed in terms of the steady state probability and the realization probability in a simple manner. The elasticity of the throughput when each customer’s service time changes by the same amount can also be calculated. These results provide some theoretical background for perturbation analysis and clarify some important issues in this area.

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

It is shown that no condition of the form E[g(N)] =∞, where N denotes a typical family size, is sufficient for explosiveness of a Markov branching process.

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

We show that the simple characterizations given by Takahashi for the independence and the total dependence of a multivariate extreme value distribution do not hold for the larger class of maximum infinitely divisible (max i.d.) distributions. This holds also for sup self-decomposable distributions.

Relations of a Markov chain describing the repeated splitting of polygons to processes known from queueing theory are pointed out.

A system is discussed which is subject to a conditional Poisson stream of failures, whose intensity function representing the cumulative damage level is a Markov chain.

We show for the three-node Jackson network studied in [3] that a customer’s sojourn times in nodes 1 and 3 are positively correlated. We actually prove a stronger result, that the two sojourn times are associated random variables. Our proof uses a stochastic ordering argument similar to that in [4].