Let X be a random vector and A a matrix. Let M be the maximal coordinate of the vector AX. For given marginal distributions of the coordinates of X, we present sharp bounds on the expectations of convex increasing functions of M. We derive joint distributions of X that achieve some of these bounds, and under these “worst case” distributions we study the joint distribution of M and the index of the largest coordinate of AX. Some possible applications are PERT network analysis and design of experiments.

μ-Geometric ergodicity of two-dimensional versions of the ALOHA and coupled processors models is verified by checking μ-geometric recurrence. Ergodicity and convergence of the Laplace-Stieltjes transforms in a neighborhood of 0 are necessary and sufficient conditions for the first model. The second model is exponential, for which ergodicity suffices to establish the required results.

The loss probabilities of customers in the Mx/GI/1/k, GI/Mx/l/k and their related queues such as server vacation models are compared with respect to the convex order of several characteristics, for example, batch size, of the arrival or service process. In the proof, we give a characterization of a truncation expression for a stationary distribution of a finite Markov chain, which is interesting in itself.

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

Separation is one measure of distance from stationarity for Markov chains. Strong stationary times provide bounds on separation and so aid in the analysis of mixing rates. The precise connection between separation and strong stationary times was drawn by Aldous and Diaconis (1987) (Advances in Applied Mathematics 8: 69−97) for discrete time chains. We develop the corresponding foundational theory for continuous time chains; several new and interesting mathematical issues arise.

A set of stochastic jobs is to be processed on a single machine that is subject to breakdown. All jobs make progress as they are processed in the absence of machine breakdowns. However, breakdowns cause setbacks to (possibly) all jobs in the system, except those that have already been completed. With machines subject only to fairly mild restrictions on this “damage” process, we demonstrate the existence of a nonpreemptive policy that is optimal in the class of all preemptive ones.

Suppose we have a single asset that we would like to sell. As time goes by, independent and identically distributed offers with a common known distribution F are given to us. At any given moment, we may either accept the current offer or reject it, thereby losing it forever. The rate at which offers arrive follows a nonhomogeneous Poisson process whose instantaneous intensity is under our control, using advertizing in a manner to be described. Our objective is, roughly, that of maximizing the total discounted expected reward composed of the offer we decide to accept, minus the total advertizing costs.

An approximation scheme for evaluating Wiener integrals by simulating Brownian motion is studied. The rate of convergence and numerical results are given, including an application to the heat equation by using the Feynman-Kac formula.

A simple method of calculating Stieltjes integrals is proposed. The method is essentially the two-point (trapezoidal) rule from numerical analysis. Two theorems yielding error bounds are given. When error requirements are modest (two or three significant decimal places) the method is fast and inexpensive. Examples are given solving the Renewal equation from reliability theory. A program for an HP-41C hand calculator is given, which solves the renewal equation.

Tijms H.C. and Van Ommeren J.C.W. (1989), Asymptotic Analysis for Buffer Behavior in Communication Systems, Probability in the Engineering and Informational Sciences, 3, 1–12.