We show that the GEM process has strong ordering properties: the probability that one of the k largest elements in the GEM sequence is beyond the first ck elements (c > 1) decays superexponentially in k.

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = N i (∞), as a function of N i (0).

We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξi , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

Introducing the concept of overall station balance which extends the notion of station balance to non-Markovian queueing networks, several necessary and sufficient conditions are given for overall station balance to hold. Next the concept of decomposability is introduced and it is related to overall station balance. A particular case corresponding to a Jackson-type queueing network is considered in some more detail.

Equipping the edges of a finite rooted tree with independent resistances that are inverse Gaussian for interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constellations of the parameters, to show that the total resistance is reciprocal inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is established and, for the case of finite trees, an exact distributional and independence result is derived for the conditional model given the total resistance.

This paper establishes structural properties for the throughput of a large class of queueing networks with i.i.d. new-better-than-used service times. The main result obtained in this paper is applied to a wide range of networks, including tandems, cycles and fork-join networks with general blocking and starvation (as well as certain networks with splitting and merging of traffic streams), to deduce the concavity of their throughput as a function of system parameters, such as buffer and initial job configurations, and blocking and starvation parameters. These results have important implications for the optimal design and control of such queueing networks by providing exact solutions, reducing the search space over which optimization need be performed, or establishing the convergence of optimization algorithms. In order to obtain results for such disparate networks in a unified manner, we introduce the framework of constrained discrete event systems (CDES), which enables us to characterize any permutable and non-interruptive queueing network through its constraint set. The main result of this paper establishes comparison properties of the event occurrence processes of CDES as a function of the constraint sets, which are then translated into the above-mentioned concavity of the throughput as a function of system parameters in the context of queueing networks.

Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n < 0, at least one W ij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[supn≥0S n > x] → (−𝔼X n )−1 ∫x ∞ ℙ[X n > u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given.

First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

An infinite dam with input formed by a compound Poisson process is considered. As an output policy, we adopt the P λ M -policy. The stationary distribution and expectation of the level of water in the reservoir are obtained.

The primary objective in the present paper is to gain fundamental understanding of the performance achievable in ATM networks as a function of the various system characteristics. We derive limit theorems that characterize the achievable performance in terms of the offered traffic, the admissible region, and the revenue measure. The insights obtained allow for substantial simplifications in the design of real-time connection admission control algorithms. In particular, we describe how the boundaries of admissible regions with convex complements may be linearized - thus reducing the admissible region - so as to obtain a convenient loss network representation. The asymptotic results for the achievable performance suggest that the potential reduction in revenue is immaterial in high-capacity networks. Numerical experiments confirm that the actual reduction is typically negligible, even in networks of moderate capacity.

The traffic equations are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the traffic equations are therefore of great importance. This note shows that the new condition stating that each transition is covered by a minimal closed support T-invariant, is necessary and sufficient for the existence of a solution for the traffic equations for batch routing queueing networks and stochastic Petri nets.

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

In this paper we study an approximation of system reliability using one-step conditioning. It is shown that, without greatly increasing the computational complexity, the conditional method may be used instead of the usual minimal cut and minimal path bounds to obtain more accurate approximations and bounds. We also study the conditions under which the approximations are bounds on the reliability. Some further extensions are also presented.

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Henderson et al. (1994), as well as those of Chao et al. (1996).

The risk reserve process of an insurance company within a deteriorating Markov-modulated environment is considered. The company invests its capital with interest rate α; the premiums and claims are increasing with rates β and γ. The problem of stopping the process at a random time which maximizes the expected net gain in order to calculate new premiums is investigated. A semimartingale representation of the risk reserve process yields, under certain conditions, an explicit solution of the problem.

Let {A i : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑i=1 ∞ A i Z i , where the Z i are independent random variables with distribution μ, which are also independent of {A i }. Under first moment assumptions imposed on {A i }, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points δ0 and δ∞ is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.