We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Let K, L ⊂ En be non-empty, closed, convex sets, K bounded, and suppose boundary sets αof K and ß of L are painted. If K undergoes a random motion such that K and L touch, the probability for a paint-to-paint contact is expressed by curvature measures of K and L. This generalizes and simplifies previous work of Molter (1986) on infinite cylinders L touching a convex body K.

In the multi-armed bandit problem, the decision-maker must choose each period a single project to work on. From the chosen project she receives an immediate reward that depends on the current state of the project. Next period the chosen project makes a stochastic transition to a new state, but projects that are not chosen remain in the same state. What happens in a two-armed bandit context if projects not chosen do not remain in the same state? We derive two sufficient conditions for the optimal policy to be myopic: either the transition function for chosen projects has in a certain sense uniformly stronger stochastic dominance than the transition function for unchosen projects, or both transition processes are normal martingales, the variance of which is independent of the history of process choices.

Certain limit theorems due to Berman involve the total time spent by Brownian motion with positive drift below an independent exponentially distributed level. Imhof has calculated the density function and shown that this random variable has two interesting probabilistic properties. We give sample path arguments which explain these two facts.

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

We present a new approach to the processor-sharing queue that allows us to study the accumulated work process of a job that requires an amount of processing time x. Our approach simplifies the proofs of some earlier results on expected conditional response times and extends them to the M/G/φ (·) class. The approach illuminates some of the paradoxical features of these systems.

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.