In this article we introduce a new method of mitigating the problem of long wait times for low-priority customers in a two-class queuing system. To this end, we allow class 1 customers to be upgraded to class 2 after they have been in queue for some time. We assume that there are ci servers at station i, i=1, 2. The servers at station 1 are flexible in the sense that they can work at either station, whereas the servers at station 2 are dedicated. Holding costs at rate hi are accrued per customer per unit time at station i, i=1, 2. This study yields several surprising results. First, we show that stability analysis requires a condition on the order of the service rates. This is unexpected since no such condition is required when the system does not have upgrades. This condition continues to play a role when control is considered. We provide structural results that include a c-μ rule when an inequality holds and a threshold policy when the inequality is reversed. A numerical study verifies that the optimal control policy significantly reduces holding costs over the policy that assigns the flexible server to station 1. At the same time, in most cases the optimal control policy reduces waiting times of both customer classes.

We consider the steady-state simulation output analysis problem for a process that satisfies a functional central limit theorem. We construct an estimator for the time-average variance constant that is based on excursions of a process above the minimum. The resulting estimator does not require a fixed run length, and the memory requirement can be dynamically bounded. Standardized time series methods based on excursions are also described.

Server farms, consisting of a collection of hosts and a front-end router that dispatches incoming jobs to hosts, are now commonplace. It is well known that when job service requirements (job sizes) are highly variable, then the Size-Interval task assignment policy is an excellent rule for assigning jobs to hosts, since it provides isolation for short jobs by directing short jobs to one host's queue and long jobs to another host's queue. What is not understood is how to classify a “short” job versus a “long” job. For a long time it was believed that the size cutoff separating “short” jobs from “long” ones should be chosen to balance the load at the hosts in the server farm. However, recent literature has provided empirical evidence that load balancing is not always optimal for minimizing mean response time. This article provides the first analytical criteria for when it is preferable to unbalance load between two hosts using Size-Interval task assignment and in which direction the load should be unbalanced. Some very simple sufficient criteria are provided under which we prove that the short job host should be underloaded, and likewise for the long job host. These criteria are then used to prove that the direction of load imbalance depends on moment index properties related to the job size distribution. For example, under the Bounded Pareto (BP) job size distribution with parameter α and a sufficiently high upper bound (the BP is well known to be a good model of empirical computer system workloads), we show that α determines the direction of load imbalance. For α<1, the short job host should be underloaded; for α=1, load should be balanced; and for α>1, the long job host should be underloaded. Many other job size distributions are considered as well. We end by showing that load unbalancing can have a dramatic impact on performance, reducing mean response time by an order of magnitude compared to load balancing in many common cases.

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function . For such models, we study some properties of multivariate aging of that are described by means of the multivariate aging function , which is a useful tool for describing the level curves of . Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals.

In this article, we study the asymptotic equipartition property (AEP) for asymptotic circular Markov chains. First, the definition of an asymptotic circular Markov chain is introduced. Then by applying the limit property for the bivariate functions of nonhomogeneous Markov chains, the strong limit theorem on the frequencies of occurrence of states for asymptotic circular Markov chains is established. Next, the strong law of large numbers on the frequencies of occurrence of states for asymptotic circular Markov chains is obtained. Finally, we prove the AEP for asymptotic circular Markov chains.

We propose a simple model of random interval graphs generated by immigration–death processes (also known as M/G/∞ queuing processes), where the length of each interval follows a subexponential distribution, and provide a condition under which the stationary degree distribution is also subexponential. Furthermore, we consider the conditional expectation of the cluster coefficient of a vertex given the degree and show that it vanishes in the limit as the degree goes to infinity under the same condition as that for obtaining the tail asymptotics of the stationary degree distribution.

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.