In this article, we observe that processes with the uniform order statistics property (UOSP) can be characterized by the condition that their first n epoch times have a joint [ell ]∞≤-spherical density, n ≥ 1. Some related results, and some further properties of [ell ]∞≤-spherical densities, are also given. We also extend some of the results regarding the UOSP to the more general (not necessarily uniform) order statistics property. Finally, we develop a theory of discrete-time discrete-state processes with the UOSP, where the need to consider multiple jumps, at a single time point, arises.

In this article, we study a single-server queue with FIFO service and cyclic interarrival and service times. An efficient approximative algorithm is developed for the first two moments of the waiting time. Numerical results are included to demonstrate that the algorithm yields accurate results. For the special case of exponential interarrival times, we present a simple exact analysis.

We show that winning the game New Age Solitaire only depends on the number of rising sequences in the deck used. The probability of winning for the special case of a new deck that is shuffled using the GSR shuffle (and two variants) are studied. We show that this game pinpoints the Achilles' heel of the GSR shuffle as is demonstrated using the variation distance.

In a system with one queue and several service stations, it is a natural principle to route a customer to the idle station with the distributionwise shortest service time. For the case with exponentially distributed service times, we use a coupling to give strong support to that principle. We also treat another topic. A modified version of our methods brings support to the design principle: It is better with few but quick servers.

We investigate fast simulation techniques for estimating the unreliability in large Markovian models of highly reliable systems for which analytical/numerical techniques are difficult to apply. We first show mathematically that for “small” time horizons, the relative simulation error, when using the importance sampling techniques of failure biasing and forcing, remains bounded as component failure rates tend to zero. This is in contrast to naive simulation where the relative error tends to infinity. For “large” time horizons where these techniques are not efficient, we use the approach of first bounding the unreliability in terms of regenerative-cycle-based measures and then estimating the regenerative-cycle-based measures using importance sampling; the latter can be done very efficiently. We first use bounds developed in the literature for the asymptotic distribution of the time to hitting a rare set in regenerative systems. However, these bounds are “close” to the unreliability only for a certain range of time horizons. We develop new bounds that make use of the special structure of the systems that we consider and are “close” to the unreliability for a much wider range of time horizons. These techniques extend to non-Markovian, highly reliable systems as long as the regenerative structure is preserved.

In this article, we investigate single-server finite-capacity queues where the partial acceptance rule is applied. In particular, we focus on the monotonicity of the amount of lost (processed) work in the queues with respect to the directionally convex order of work or interarrival processes. We first compare the queues that differ only in their work processes and show that if the work processes are directionally convex ordered, so is the amount of work lost (or processed) in the systems. Next, we compare the queues that differ only in their interarrival processes and show that if the interarrival processes are directionally convex ordered, so is the amount of work lost (or processed) in the systems. Using these results, we establish the formula that gives the upper bound of work-loss probability based only on the marginal distributions of work and interarrival processes. Numerical experiments using the data of actual-LAN (local area network) traffic show that the derived formula gives tight bounds sufficient for practical use.

The purpose of this article is to study several preservation properties of stochastic comparisons based on the mean inactivity time order under the reliability operations of convolution and mixture. Characterizations and relationships with the other well-known orders are given. Some examples of interest in reliability theory are also presented. Finally, testing in the increasing mean inactivity time class is discussed.

In this article, we introduce the notion of the Markov chain fields on the generalized Bethe trees or generalized Cayley trees, and some strong limit theorems on the frequencies of states and ordered couples of states, including the Shannon–McMillan theorem on Bethe tree TB,N and Cayley tree TC,N, are obtained. In the proof, a new technique in the study of the strong limit theorem in probability theory is applied.