For Markov chains of M/G/1 type that are not skip-free to the left, the corresponding G matrix is shown to have special structure and be determined by its first block row. An algorithm that takes advantage of this structure is developed for computing G. For non-skip-free M/G/1 type Markov chains, the algorithm significantly reduces the computational complexity of calculating the G matrix, when compared with reblocking to a system that is skip-free to the left and then applying usual iteration schemes to find G. A similar algorithm to calculate the R matrix for G/M/1 type Markov chains that are not skip-free to the right is also described.

We study the problem of preemptive scheduling of jobs in a two-machine open shop. Jobs require processing on both machines, but the order does not matter. We define the D-LERPT (double longest expected remaining processing time) policy as the policy that first processes jobs that have not yet been processed by either machine (double jobs), in decreasing order of expected remaining processing times, and then processes jobs that require processing on only one machine in any order. We show that D-LERPT stochastically minimizes the makespan when preemption is not permitted and jobs (but not machines) are stochastically identical, and that D-LERPT minimizes the makespan in the increasing convex sense when preemption is permitted and the machines are stochastically identical and processing times are exponential or geometric with a job dependent rate.

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

In this paper, asymptotic estimates for the blocking probability of a call pertaining to a given route in a large multi-rate circuit-switched network are given. Concentrating on low load and critical load conditions, these estimates are essentially derived by using probability change techniques applied to the distribution of the number of occupied links. Such estimates for blocking probabilities are also given a uniform expression applicable to both load regimes. This uniform expression is numerically validated via simple examples.

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

A sequence of irreducible closed queueing networks is considered in this paper. We obtain that the queue length processes can be approximated by reflected Brownian motions. Using these approximations, we get rates of convergence of the distributions of queue lengths.

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

Design engineers are well aware that a system where active spare allocation is made at the component level has a lifetime stochastically larger than the corresponding system where active spare allocation is made at the system level. In view of the importance of hazard rate ordering in reliability and survival analysis, Boland and El-Neweihi (1995) recently investigated this principle in hazard rate ordering and demonstrated that it does not hold in general. They showed that for a 2-out-of-n system with independent and identical components and spares, active spare allocation at the component level is superior to active spare allocation at the system level. They conjectured that such a principle holds in general for a k-out-of-n system when components and spares are independent and identical. We prove that for a k-out-of-n system where components and spares have independent and identical life distributions active spare allocation at the component level is superior to active spare allocation at the system level in likelihood ratio ordering. This is stronger than hazard rate ordering, thus establishing the conjecture of Boland and El-Neweihi (1995).

In this article, we assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables. A formula for evaluating the rate of occurrence of failures for the system is derived. As an application of the theory, a maintenance model for a two-component system is also studied.

A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.

We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential.

We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them.

We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.

This paper studies the expected average cost control problem for discrete-time Markov decision processes with denumerably infinite state spaces. A sequence of finite state space truncations is defined such that the average costs and average optimal policies in the sequence converge to the optimal average cost and an optimal policy in the original process. The theory is illustrated with several examples from the control of discrete-time queueing systems. Numerical results are discussed.

We consider the queueing system denoted by M/MN /1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

In a system modelled by a time-discrete deterministic model, predictions of the distribution of the members over the different classes do not result automatically in an integer valued vector. In this paper, for a constant size system, we discuss how to associate with the calculated vector an integer valued vector. Furthermore we examine whether the evolution of the calculated vectors on the one hand, and the evolution of the associated integer valued vectors on the other hand, have the same properties.

A bookmaker makes a book on a horse race: he offers odds against the various horses winning the race, and gamblers accept bets at those odds when they find the odds attractive. The book at a particular time consists of the bookmaker's winnings according to the different outcomes of the race if the race were run at that time. We consider strategies the bookmaker might adopt when deciding how to alter his quoted odds as bets accumulate. The bookmaker is assumed to behave conservatively in the sense that he tries to minimise his expected maximum loss over all possible outcomes of the race.

This paper studies optimal routing and jockeying policies in a two-station parallel queueing system. It is assumed that jobs arrive to the system in a Poisson stream with rate λand are routed to one of two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, μ1 and μ2. Jockeying between stations is permitted. The jockeying cost is cij when a job in station i jockeys to station j, i ≠ j. There is no cost when a new job joins either station. The holding cost in station j is hj, h 1 ≦ h 2, per job per unit time. We characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both discounted and long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically non-decreasing functions. We study the properties of these control functions, their relationships, and their asymptotic behavior. We show that some well-known queueing control models, such as optimal routing to symmetric and asymmetric queues, preemptive or non-preemptive scheduling on homogeneous or heterogeneous servers, are special cases of our system.