An aggregation technique of ‘near complete decomposable' Markovian systems has been proposed by Courtois [3]. It is an approximate method in many cases, except for some queuing networks, so the error between the exact and the approximate solution is an important problem. We know that the error is O(ε), where ε is defined as the maximum coupling between aggregates. Some authors developed techniques to obtain a O(ε k ) error with k > 1 error with k > 1, while others developed a technique called ‘bounded aggregation’. All these techniques use linear algebra tools and do not utilize the fact that the steady-state probability vector represents the distribution of a random variable. In this work we propose a stochastic approach and we give a method to obtain stochastic bounds on all possible Markovian approximations of the two main dynamics: short-term and long-term dynamics.

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to the k-process case.

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.

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.

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.

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.

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o= i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

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.

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

This paper presents a new proof of Sengupta's invariant relationship between virtual waiting time and attained sojourn time and its application to estimating the virtual waiting time distribution by counting the number of arrivals and departures of a G/G/1 FIFO queue. Since this relationship does not require any parametric assumptions, our method is non-parametric. This method is expected to have applications, such as call processing in communication switching systems, particularly when the arrival or service process is unknown.

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.

Let {Xn } be the Lindley random walk on [0,∞) defined by . Here, {An } is a sequence of independent and identically distributed random variables. When for some r > 1, {Xn } converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s > 1 such that for every initial state . In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is and satisfies φ ‘(r 0) = 0.

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.

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

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.

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.

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.

We prove strong convergence of the proportions Un /Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.