Consider a large number of interacting queues with symmetric interactions. In the asymptotic limit, the interactions between any fixed finite subcollection become negligible, and the overall effect of interactions can be replaced by an empirical rate. The evolution of each queue is given by a time inhomogeneous Markov process. This may be considered a technique for writing dynamic Erlang fixed-point equations. We explore this as a tool to approximate sojourn time distributions.

Known results in random graph theory lead easily to a quantitative result on the number of multiplications needed in a matrix factorization algorithm, under the assumption that non-zero entries are randomly distributed.

In this paper we analyze an asymmetric 2 × 2 buffered switch, fed by two independent Bernoulli input streams. We derive the joint equilibrium distribution of the numbers of messages waiting in the two output buffers. This joint distribution is presented explicitly, without the use of generating functions, in the form of a sum of two alternating series of product-form geometric distributions. The method used is the so-called compensation approach, developed by Adan, Wessels, and Zijm.

In this paper we describe a class of discrete time processes that can be used to model packet arrival streams in packetized communication. Mathematically, (K(t)) can be seen as a discrete time self-exciting point process, as a multitype branching process, or as an epidemic with immigration of infected people. The purpose of this paper is to show that this class of models simultaneously is quite useful and analytically more tractable than is obvious at first glance. It is shown that certain probabilities can reliably be computed using generating function methods, and expressions are given for the second order properties and for the asymptotic index of dispersion.

In a previous paper, we defined and studied a branching process input stream that can be used to model the organization of packets (in packetized communication) in messages. In this paper we derive a central result on the effect that such streams have on a packet switch (or queue) and further study one particular model (the case of so-called short-range dependence), where the distribution of the number of packets in a buffer can explicitly be found (albeit in the form of a transform). The solution to this particular model suggests an approximation for the average buffer content in the general case.

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

The two-dimensional correlated Wiener process (or Brownian motion) with drift is considered. The Fokker-Planck (or Kolmogorov forward) equation for the Wiener process (X1(t), X2(t)) is solved under absorbing boundary conditions on the lines x1= h1 and x2 = h2 and a fixed starting point (x0,1, x0,2). The first passage (or first exit) time when the process leaves the domain G = ( −∞, h1) × ( −∞, h2) is derived.

This paper proposes some new statistics based on k-NN distances for assessing the uniformity of a set of points belonging to a d-dimensional bounded region. Size and power of the tests have been estimated and compared to those of the modified Hopkins' statistic.

We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.