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.

Age replacement policy is commonly used in order to reduce the number of in-service failures. In this paper we define a multivariate version of this policy and develop some of its desirable properties. We also obtain an optimal age replacement policy.

The number of items of data which are irretrievable without additional effort after hashing can be greatly reduced if several hash tables are used simultaneously. Here we show that, in a multiple hashing scheme, this number has a distribution very close to Poisson. Thus choosing the number and sizes of the tables to minimize the expected number of irretrievable items is the right way to dimension a scheme.

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

We show that if an input process ζ to a queue is asymptotic stationary in some sense, satisfies a condition AB and some other natural conditions, then the output processes (w, ζ) and (w, q,ζ) are asymptotic stationary in the same sense. Here, w and q are the waiting time and queue length processes, respectively.

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 this paper we describe how the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity. We also consider the stationary case, proving large deviation principles for the state of the system in equilibrium and for departures from an equilibrium system.

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ –1, and if the traffic intensity ρ = λμ –1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

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.

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

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.

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class of subexponential distributions are asymptotically of two types as x →∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞, or it starts from a level of the form x – O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 – ρ.

We take a fresh look at some transient characteristics of an M/M/∞ queue, studied previously by Guillemin and Simonian using delicate complex analysis. Along the way we obtain the Laplace transform of the joint distribution of the duration, number of arrivals and swept area associated with a busy period of an M/M/1 queue.

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.

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 necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager II's winning probability when both managers play optimally. Finally, we show that manager II's winning probability is strictly increasing in n, is always less than c, and converges to c as n →∞, where c = 0.3275624139 · ·· is a solution of the equation ln(2) + x ln(x) = x.

Explicit formulas for the time congestion and the call blocking probability are derived in a single server loss system whose total input consists of a finite superposition of independent general stationary traffic streams with exponentially distributed service times. The results are used for studying to what extent two arrival processes with coinciding customer-stationary state distributions are similar or even identical, and whether an arrival process with coinciding customer-stationary and time-stationary state distributions is of the Poisson type.