When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns out that the total number of customers at time t has the desired property, if the network is idle at time O. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.

We consider a memoryless single-server queue in which users can purchase relative priority so as to reduce their expected waiting costs, which are linear with time. Relative priority is given in proportion to a price paid by customers present in the system. For two service disciplines, (weighted) processor sharing and (weighted) random entrance, we find the unique pure and symmetric Nash equilibrium price paid by the customers.

We present results on the finite-time behavior of the discrete-time, finite-space version of the simulated annealing algorithm. The asymptotic and finite-time behavior of the annealing algorithm under slow cooling will be shown to depend on the largest eigenvalue of a certain matrix. To illustrate the utility of our results, we study the slowly cooled annealing algorithm applied to the maximum matching problem and demonstrate a polynomial randomized approximation property of the algorithm.

We compute the first two moments and give a recursive formula for the generating function of the first k-record index for a sequence of independent and identically distributed random variables that take on a finite set of possible values. When the random variables have an infinite support, we bound the distribution of the index of the first k-record and show that its mean is infinite.

It is well known that the standard simulated annealing optimization method converges in distribution to the minimum of the cost function if the probability a for accepting an increase in costs goes to 0. α is controlled by the “temperature” parameter, which in the standard setup is a fixed sequence of values converging slowly to 0. We study a more general model in which the temperature may depend on the state of the search process. This allows us to adapt the temperature to the landscape of the cost function. The temperature may temporarily rise such that the process can leave a local optimum more easily. We give weak conditions on the temperature schedules such that the process of solutions finally concentrates near the optimal solutions. We also briefly sketch computational results for the job shop scheduling problem.

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

The line reversibility property of a two-stage multiserver system with manufacturing blocking has previously been established using a duality argument between the blocked state and the idle state. In this paper, we provide counterexamples that show this argument does not apply to systems with more than two stages. A focus of this paper is to provide a new proof for the result. Our argument is based on the duality between the service initiation and the service completion events.

We study tandem ordering of constant-service stations with unlimited buffers and arbitrary arrival process where service at each station adds a certain value to the job. The goal is to order the stations such that the total expected value of the jobs in the system or, equivalently, the work-in-process inventory cost is minimized. Assuming that the added value–service length ratio increases with the service length, we provide a branch and bound method that finds an optimal order.

Recently, some authors have suggested usage models of Markov type as a technique of specifying the estimated operational use distribution of a given program. A main purpose of such models is the derivation of random test cases allowing unbiased estimates on the (un)reliability of the program in its intended environment. In this article, we show that by a shift of the transition probabilities of the Markov chain corresponding to such a model, prior information on the errorjproneness of single-program operations can be taken into account. An unbiased unreliability estimator with reduced variance is obtained. Furthermore, it is shown that minimization of the variance leads to a special stochastic optimization problem that can be demonstrated to be convex, such that efficient solution techniques apply. Some related questions are also treated in a more general, non-Markovian framework.

Extremal arrival processes, in the sense of increasing convex order of waiting time of queueing systems, are investigated. Two types of extremal processes are proposed: one in the class of processes that have identical marginal distributions and the other in the class of bounded stochastic processes that have the same mean and covariance structure. The worst performance with regard to waiting time in the sense of increasing convex order is guaranteed when these extremal processes are fed into a first in-first out single-server queue.

This note illustrates that a combination of the approach in our previous papers (Boucherie and Boxma, 1996, Probability in the Engineering and Informational Sciences10: 261–277; Jain and Sigman, 1996, Probability in the Engineering and Informational Sciences 10: 519–531) directly leads to a Pollaczek-Khintchine form for the workload in a queue with negative customers. The same technique is also applied to risk processes with lump additions.

This note studies the comparison of finite-buffer and nonexponential batch arrival systems of the form Gx/M/c/c + N with the corresponding systems, with N replaced by N', where N' can be smaller, larger, or infinite. If N' = ∞ the service times can be arbitrarily distributed. Both comparison and error bounds are obtained for performance measures such as the throughput, the idle probability, and the active server distribution. The results are of practical interest to establish computational reductions, either by infinite-space approximation or by reduced finite truncations. Two different proof techniques will be employed: the sample path approach and the Markov reward approach. The comparison of these two techniques is of interest in itself, showing the advantage and disadvantage of each.

Recently, Derman and Ross (1995, An improved estimator of a in quality control, Probability in the Engineering and Informational Sciences 9: 411–415) derived an estimator of the standard deviation in the standard quality control model and showed that it had smaller mean squared error than the usual estimator. The new estimator was also shown to be consistent even when the underlying distribution deviates from normality, unlike the usual estimator. In this note, the mean squared error is further improved via shrinkage of the Derman-Ross estimator, and a consistent minimum variance unbiased estimator is presented. Finally, by making use of additional subgroup statistics, a minimum variance unbiased estimator is derived and further improved via shrinkage.

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

In this paper, we consider the interval availability distribution of a two-state single unit. We fit the interval availability distribution directly to a beta distribution, such that the probability of nil and a 100% availability is correct as well as the expectation and the variance. The method is fast and easy to implement and gives good results.

We consider a general discrete-time stochastic recursive model that is influenced by an external Markov chain. Our aim is to investigate the effect that the transition matrix of the external process has on the system states of the model. To answer this question, we use new stochastic ordering concepts. Especially interesting are the results for infinite-stage Markov-modulated models. We illustrate our main results by three applications: an inventory model, a consumption model, and a queueing model for a time division multiplexing system.

Suppose customers pass through a network of two queues in parallel. A statedependent routing policy gives individuals their quickest journey. The Downs-Thomson effect is any increase in the long-run expected journey time caused by an increase in the service rates. This effect may occur.

The convex (concave) parameterization of a generalized renewal process is considered in this paper. It is shown that if the interrenewal times have log concave distributions or have log concave survival functions (i.e., an increasing failure rate distribution), then the renewal process is convexly (concavely) parameterized in its mean parameterization.

In many multi-echelon inventory systems, the lead times are random variables. A common and reasonable assumption in most models is that replenishment orders do not cross, which implies that successive lead times are correlated. However, the process that generates such lead times is usually not well defined, which is especially a problem for simulation modeling. In this paper, we use results from queuing theory to define a set of simple lead time processes guaranteeing that (a) orders do not cross and (b) prespecified means and variances of all lead times in the multiechelon system are attained.

We consider scheduling a single server in a multiclass queue subject to setup times and setup costs. We examine the issue of whether or not reductions in the mean and variance of the setup time distributions can lead to degraded system performance. Provided that setups are reduced according to a stochastically smaller ordering, we show that if an optimal policy is used both for the original system and for the system with reduced setup times, then an improvement in performance is guaranteed. Even in cases for which a truly optimal policy is unknown, idling can be employed to avoid degradation of performance as setup times are cut. We extend this approach to show that system performance is monotonic with respect to service time distributions, switching costs, holding costs, and uniform reductions in the arrival rates. Extensions to sequencedependent setups and job feedback are noted.