We consider a multiclass closed queueing network model analogous to the open network models of Rybko and Stolyar and of Lu and Kumar. The closed network has two single-server stations and a fixed customer population of size n. Customers are routed in cyclic fashion through four distinct classes, two of which are served at each station, and each server uses a preemptive-resume priority discipline. The service time distribution for each customer class is exponential, and attention is focused on the critical case where all four classes have the same mean service time. Letting n approach infinity, we prove a heavy traffic limit theorem that is unconventional in three regards. First, in our heavy traffic scaling of both queue-length processes and cumulative idleness processes, time is compressed by a factor of n rather than the factor of $n^2$ occurring in conventional theory. Second, the spatial scaling applied to some components of the queue-length and idleness processes is that associated with the central limit theorem, but the scaling applied to other components is that associated with the law of large numbers. Thus, in the language of queueing theory, our heavy traffic limit theorem involves a mixture of Brownian scaling and fluid scaling. Finally, the limit process that we obtain is not an ordinary reflected Brownian motion, as in conventional heavy traffic theorems, although it is related to or derived from Brownian motion.

A set of nodes and a set of consumers are given, and to each consumer there corresponds a subset of the nodes. Each consumer has a demand, which is a load to be distributed among the nodes corresponding to the consumer. The load at a node is the sum of the loads placed on the node by all consumers. The load is balanced if no single consumer can shift some load from one node to another to reduce the absolute difference between the total loads at the two nodes. The model provides a setting to study the performance of load balancing as an allocation strategy in large systems.

The set of possible balanced load vectors is examined for infinite networks with deterministic or random demands. The balanced load vector is shown to be unique for rectangular lattice networks, and a method for computing the load distribution is explored for tree networks. An FKG-type inequality is proved. The concept of load percolation is introduced and is shown to be associated with infinite sets of nodes with identical load.

We investigate a storage model where the input and the demand are additive functionals on a Markov chain J. The storage policy is to meet the largest possible portion of the demand. We first derive results for the net input process embedded at the epochs of transitions of J, which is a Markov random walk. Our analysis is based on a Wiener-Hopf factorization for this random walk; this also gives results for the busy period of the storage process. The properties of the storage level and the unsatisfied demand are then derived.

We consider finite closed Jackson networks with N first come, first serve nodes and M customers. In the limit $M \to \infty, N \to \infty, M/N \to \lambda > 0$, we get conditions when mean queue lengths are uniformly bounded and when there exists a node where the mean queue length tends to $\infty$ under the above limit (condensation phenomena, traffic jams), in terms of the limit distribution of the relative utilizations of the nodes. In the same terms, we also derive asymptotics of the partition function and of correlation functions.

We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system and upon arrival at a station the server serves all waiting customers until the queue is empty, where the service time distribution depends on the station. The choice of the station to be visited next as well as the corresponding walking time may depend on the whole current state. Examples are systems with a greedy-type routing mechanism. Under appropriate independence assumptions it is proved that the system is stable if and only if the workload is less than 1.

We give a Taylor series expansion for the mean value of the canonical stationary state variables of open (max +)-linear stochastic systems with Poisson input process. Probabilistic expressions are derived for coefficients of all orders, under certain integrability conditions. The coefficients in the series expansion are the expectations of polynomials, known in explicit form, of certain random variables defined from the data of the (max +)-linear system.

These polynomials are of independent combinatorial interest: their monomials belong to a subset of those obtained in the multinomial expansion; they are also invariant under cyclic permutation and under translations along the main diagonal.

The method for proving these results is based on two ingredients: (1) the (max +)-linear representation of the stationary state variables as functionals of the input point process; (2) the series expansion representation of functionals of marked point processes and, in particular, of Poisson point processes.

Several applications of these results are proposed in queueing theory and within the framework of stochastic Petri nets. It is well known that (max +)-linear systems allow one to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking (manufacturing and communication), synchronized queueing networks and so on. It also contains some basic manufacturing models such as Kanban networks, Job-Shop systems and so forth. The applicability of this expansion method is discussed for several systems of this type. In the M/D case (i.e., all service times are deterministic), the approach is quite practical, as all coefficients of the expansion are obtained in closed form. In the M/GI case, the computation of the coefficient of order k can be seen as that of joint distributions in a stochastic PERT graph of an order which is linear in k .

We consider a stochastic fluid network with inputs which are independent subordinators. We show that under some natural conditions the distribution of the fluid content process converges in total variation to a proper limit and that the limiting distribution has a positive mass at the origin. As a consequence of the methodology used, we obtain upper and lower bounds for the expected values of this limiting distribution. For the two-dimensional case, under certain conditions, explicit formulas for the means, variances and covariance of the steady-state fluid content are given. Further, for the two-dimensional case, it is shown that, other than for trivial setups, the limiting distribution cannot have product form.

We study an asymptotically 4-stable process. The main result is an Itô type formula.

Let ${\Phi_t, t \geq 0}$ be a Markov process on the state space $[0, \infty)$ that is stochastically ordered in its initial state. Examples of such processes include server workloads in queues, birth-and-death processes, storage and insurance risk processes and reflected diffusions. We consider the existence of a limiting probability measure $\pi$ and an exponential "convergence rate" $\alpha > 0$ such that $$\lim_{t \to \infty} e^{\alpha t} \sup_A |P_x[\Phi_t \epsilon A] - \pi (A)| = 0$$ for every initial state $\Phi_0 \equiv x$.

The goal of this paper is to identify the largest exponential convergence rate $\alpha$, or at least to find computationally reasonable bounds for such a "best" $\alpha$. Coupling techniques are used to derive such results in terms of (i) the moment-generating function of the first passage time into state ${0}$ and (ii) solutions to drift inequalities involving the generator of the process. The results give explicit bounds for total variation convergence of the process; convergence rates for $E_x [f(\Phi_t)]$ to $\int f(y) \pi (dy)$ for an unbounded function f are also found. We prove that frequently the bounds obtained are the best possible. Applications are given to dam models and queues where first passage time distributions are tractable, and to one-dimensional reflected diffusions where the generator is the more appropriate tool. An extension of the results to a multivariate setting and an analysis of a tandem queue are also included.

A general multitype branching process with sibling dependencies is considered. The dependencies within a group of siblings are described by a joint probability measure, determined by the structure of that particular group. The process is analyzed by means of the embedded macro process, consisting of sibling groups. It is shown that the regular asymptotic behavior of the sibling-dependent process is guaranteed by conditions on the individual reproductions, and that these conditions are exactly the same as those normally required for an ordinary independent process that has the same individual marginals. Convergence results for the expected population size as well as the actual population size are given, and the stable population is described. The sibling-dependent process and the ordinary independent process with the same marginals are compared; some simple examples illustrate the differences and similarities. The results are extended to more general dependencies that are local in the family tree.

prove a large deviation principle for the density field of independent particle systems in an infinite volume. We then deduce from the one-dimensional case of this result the large deviations for the occupation times of various sets (from microscopic to macroscopic scales) and we recover the theorem established by Cox and Griffeath. An expression of the rate function is given using the Brownian motion local time as in Deuschel and Wang.

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius' version of Cramér's classical method based on analytic moment generating functions. We thus generalize Cramér's and Kubilius's theorems on large deviations.