This paper gives a review of some of the recent work on stochastic epidemic models and their deterministic counterparts. In particular it focusses on models for the spread of HIV infection and AIDS. The variability between realisations of an epidemic is discussed in some detail, and methods of assessing this variation are described. Numerical examples are given to illustrate various aspects of the models considered.

In queueing theory one seeks to predict in quantitative terms the congestion delays that occur when jobs or customers complete for processing resources. At present no satisfactory methods exist for the analysis of systems that allow simultaneous performance of tasks associated with a single job or customer. We present a heavy traffic analysis for the class of homogeneous fork-join networks in which jobs are routed in a feedforward deterministic fashion. We show that under certain regularity conditions the vector of total job count processes converges weakly to a multidimensional reflected Brownian motion (RBM) whose state space is a polyhedral cone in the nonnegative orthant. Furthermore, the weak limits of workload levels and throughput times are shown to be simple transformations of the RBM. As will be explained, the "steady-state throughput time" (a random variable) is expressed in terms of workload levels via the "longest path functional" of classical PERT/CPM analysis.

Consider first-passage percolation on $\mathbb{Z}^d$. A classical result says, roughly speaking, that the shortest travel time from $(0, 0,\ldots, 0)$ to $(n, 0, \ldots, 0)$ is asymptotically equal to $n \mu$, for some constant $\mu$, which is called the time constant, and which depends on the distribution of the time coordinates. Except for very special cases, the value of $\mu$ is not known. We show that certain changes of the time coordinate distribution lead to a decrease of $\mu$; usually $\mu$ will strictly decrease. Two examples of our results are: (i) If $F$ and $G$ are distribution functions with $F \leq G, F \not\equiv G$, then, under mild conditions, the time constant for $G$ is strictly smaller than that for $F$. (ii) For $0 < \varepsilon_1 < \varepsilon_2 \leq a < b$, the time constant for the uniform distribution on $\lbrack a - \varepsilon_2, b + \varepsilon_1 \rbrack$ is strictly smaller than for the uniform distribution on $\lbrack a, b\rbrack$. We assume throughout that all our distributions have finite first moments.

A variation is given of the van den Berg-Kesten inequality on the probability of disjoint occurrence of events enabling it to apply to random variables, rather than just to events, associated with various subsets of an index set. This is used to establish superadditivity of a certain family of generating functions associated with first-passage percolation. This leads to improved estimates for the rates of convergence of the expected values of certain passage times.

The Stein-Chen approach is used to obtain bounds on the Poisson approximation of a random field, in both a random variable and a stochastic process sense. The hypotheses are Dobrushin's condition or, alternatively, positive dependence combined with a bound on decay of correlations. Rates of convergence are derived which supplement the limit theorems of Berman. The results have application to certain Gibbs states at both high and low temperature.

We introduce a new method (to be referred to as the variational method, VM) for estimating the parameters of Gibbs distributions with random variables ("spins") taking values in a Euclidean space $\mathbb{R}^n, n \geq 1$, from complete or degraded data. The method applies also to the case of iid random variables governed by exponential families, and appears to be new even in this case. For complete data, the VM is computationally more efficient than, and as reliable as, the maximum pseudo-likelihood method. For incomplete data, the VM leads to an estimation procedure reminiscent of, but simpler than, the EM algorithm. In the former case, we show that under natural assumptions a certain form of the variational estimators is strongly consistent and asymptotically normal. We also present numerical experiments that demonstrate the computational efficiency and accuracy of the variational estimators.

In this paper we consider the Gibbs sampler dynamics with simulated annealing and partially parallel updating scheme, as proposed by Trouve. It is known that in some cases the support of the limiting measure does not coincide with the set of global maxima of the underlying energy function. We provide some new simple examples of this undesirable behavior. However, we also prove that for one-dimensional binary models with nearest neighbor interaction the algorithm works "generically." We prove also that for two-dimensional models with nearest neighbor ferromagnetic constant interactions the algorithm works.

We view a $\bullet/M/K$ node having $K$ exponential servers of service rate $\mu$ as a map on the space of stationary ergodic arrival processes of rate $\lambda, \lambda < K\mu$. It is well known that the Poisson process of rate $\lambda$ is a fixed point of this map. We prove there is no other fixed point.

Consider a (space-time) realization $\omega$ of a critical or subcritical one-dimensional branching Brownian motion. Let $Z_x(\omega)$ for $x \geq 0$ be the number of particles which are located for the first time on the vertical line through $(x, 0)$ and which do not have an ancestor on this line. In this note we study the process $Z = \{Z_x; x \geq 0\}$. We show that $Z$ is a continuous-time Galton-Watson process and compute its creation rate and offspring distribution. Here we use ideas of Neveu, who considered a similar problem in a supercritical case. Moreover, in the critical case we characterize the continuous state branching processes obtained as weak limits of the processes $Z$ under rescaling.

Kac has considered the integral of the absolute value of a Brownian motion process and determined the Laplace-Stieltjes transform of the distribution function of this integral. In this paper explicit expressions are given for the distribution and the moments of this integral.

We consider a probabilistic, discrete-time predator-prey model of the following kind: There is a population of predators and a second one of prey. The predator population evolves according to an ordinary supercritical Galton-Watson process. Each prey is either killed by a predator in which case it cannot reproduce, or it survives and reproduces independently of all other population members and according to the same offspring distribution with mean greater than 1. The resulting process $(X_n, Y_n)_{n \geq 0}$, where $X_n$ and $Y_n$, respectively, denote the number of predators and prey of the $n$th generation, is called a Galton-Watson predator-prey process. The two questions of almost certain extinction of the prey process $(Y_n)_{n \geq 0}$ given $X_n \rightarrow \infty$, and of the right normalizing constants $d_n, n \geq 1$ such that $Y_n/d_n$ has a positive limit on the set of nonextinction, are completely answered. Proofs are based on a reformulation of the model as a certain two-district migration model.

We present a simple nonstandard construction of a global Euler flow and some classes of measures invariant with respect to the flow, including examples of non-Gaussian ones. We also obtain existence of statistical solutions of the Euler equation for a wide class of initial measures.

This study concerns travel times in a stochastic network in which units move among the nodes that process the units. The network may be closed or open, there may be several types of units and the processing at each node may depend on the numbers of units at the other nodes. This is a Jackson network when the nodes operate independently. The travel time on a "route" in this network is the time it takes for an arbitrary unit to traverse one of a series of nodes that constitute the route, when the network is in equilibrium. An example is the time for a unit to move from one set of nodes to another. We present an expression for the expectation of a general travel time. We also characterize the distribution of the travel time, and the sojourn times at the nodes, on an overtake-free path. This includes the known results on the product-form distribution of sojourn times at the nodes on overtake-free paths in Jackson networks.

We prove the following results on Bernoulli bond percolation on the sites of the $d$-dimensional lattice, $d \geq 2$, with parameters $M$ (the maximum distance over which an open bond is allowed to form) and $\lambda$ (the expected number of open bonds with one end at the origin), when the range $M$ becomes large. If $\lambda_c(M)$ denotes the critical value of $\lambda$ (for given $M$), then $\lambda_c(M) \rightarrow 1$ as $M \rightarrow \infty$. Also, if we make $M \rightarrow \infty$ with $\lambda$ held fixed, the percolation probability approaches the survival probability for a Galton-Watson process with Poisson $(\lambda)$ offspring distribution. There are analogous results for other "spread-out" percolation models, including Bernoulli bond percolation on a homogeneous Poisson process on $d$-dimensional Euclidean space.