When traffic sources are statistically multiplexed over a common link, the sum of the peak rates of the sources exceeds the throughput of the link. The excess may be stored in a buffer, but when this overflows, information is lost. When a source is bursty, the peak rate is attained only for very short periods of time, whereas between bursts the source is idle. Because the sources are independent, the chance that many bursts arrive simultaneously is small, but these rare events do occur and the mean time until overload is a key design parameter. Here we model the multiplexor as a multidimensional Markov process with a set of forbidden states that represent the exceedance of the link capacity. We use the theory of induced Dirichlet forms to estimate the Laplace transform of the hitting time of this forbidden set. We obtain an upper bound on the probability that the link capacity is exceeded during a fixed time interval along with a lower bound for the mean time until the link capacity is exceeded. This provides the network designer with a degree of assurance about the probability and frequency of overloads.

We consider the standard first-passage percolation problem on $\mathbb{Z}^d: \{t(e): e \text{an edge of} \mathbb{Z}^d\}$ is an i.i.d. family of random variables with common distribution $F, a_{0,n} := \inf\{\sum^k_1 t(e_1): (e_1, \cdots, e_k)$ a path on $\mathbb{Z}^d$ from 0 to $n \xi_1\}$, where $\xi_1$ is the first coordinate vector. We show that $\sigma^2(a_{0,n}) \leq C_1 n$ and that $P\{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}\} \leq C_2 \exp(-C_3 x)$ for $x \leq C_4 n$ and for some constants $0 < C_i < \infty$. It is known that $\mu := \lim(1/n)Ea_{0,n}$ exists. We show also that $C_5 n^{-1} \leq Ea_{0,n} - n\mu \leq C_6 n^{5/6}(\log n)^{1/3}$. There are corresponding statements for the roughness of the boundary of the set $\tilde{B}(t) = \{\nu: \nu$ a vertex of $\mathbb{Z}^d$ that can be reached from the origin by a path $(e_1, \cdots, e_k)$ with $\sum t(e_i) \leq t\}$.

We consider solid-on-solid models of two- and three-dimensional crystal growth described by Markov rate processes whose states are representations of the entire crystal surface and whose transitions are captures of single atoms by this surface or escapes of atoms. Under natural conditions on the transition probability rates, we prove ergodicity of the Markov process. This implies the existence of states of steady growth and statistically stationary surface structure. Microscopic surface instabilities do not therefore occur under such conditions. For two dimensions, our conditions are very general; for three dimensions, they are less general.

This paper studies the philosophers' process, introduced in the finite case of Zenie after Dijkstra's dining philosophers' problem and in different contexts by Suhov and Kelly. This process is presented as a nearest particle system on $\mathbb{Z}$, where a configuration may flip from 0 to 1 at one site $x$ only if it is null for the two neighbors of $x$. It flips from 1 to 0 at a constant rate. The model is proved to be ergodic and reversible, and its stationary measure is explicitly characterized. In the finite case (configurations on $\mathbb{Z}/L\mathbb{Z})$, an explicit expression for the stationary measure is given.

A class of stochastic control problems where the payoff depends on the running maximum of a diffusion process is described. The controller must make two kinds of decision: first, he must choose a work rate (this decision determines the rate of profit as well as the proximity of failure), and second, he must decide when to replace a deteriorated system with a new one. Preventive replacement is a realistic option if the cost for replacement after failure is larger than the cost of a preventive replacement. We focus on the profit and replacement cost for a single work cycle and solve the problem in two stages. First, the optimal feedback control (work rate) is determined by maximizing the payoff during a single excursion of a controlled diffusion away from the running maximum. This step involves the solution of the Hamilton-Jacobi-Bellman (HJB) partial differential equation. The second step is to determine the optimal replacement set. The assumption that failure occurs only on the set where the state is increasing implies that replacement is optimal only on this set. This leads to a simple formula for the optimal replacement level in terms of the value function.

Stochastic control problems in which the payoff depends on the running maximum of a diffusion process are considered. Such processes provide appealing models for physical processes that evolve in a continuous and increasing manner and fail at a random time. The controller must make two decisions: first, she must choose how fast to work (this decision determines the rate of profit as well as the proximity of failure), and second, she must decide when to replace a deteriorated system with a new one. Preventive replacement becomes an important option when the cost for replacement after a failure is larger than the cost of a preventive replacement. Single-cycle and long-term average criteria are used to evaluate the control and replacement decisions. We model the process via a martingale problem formulation. This enables the long-term average control problem to be rephrased as an LP over the invariant measures of the process. We identify the invariant measures corresponding to each control and replacement decision and determine the optimal solution using an iterative scheme.

When an ecological food web is described by an acyclic directed graph, the trophic level of a species of plant or animal may be described by the length of the shortest (or the longest) food chain from the species to a green plant or to a top predator. Here we analyze the number of vertices in different levels in a stochastic model of acyclic directed graphs called the cascade model. This model describes several features of real food webs. For an acyclic directed graph $D$, define the $i$th lower (upper) level as the set of all vertices $\nu$ of $D$ such that the length of the shortest (longest) maximal path starting from $\nu$ equals $i, i = 0, 1\cdots$. In this article, we compute the sizes of the levels of a random digaph $D(n, c)$ obtained from a random graph on the set $\{1, 2,\cdots,n\}$ of vertices in which each edge appears independently with probability $c/n$, by directing all edges from a larger vertex to a smaller one. The number of edges between any two levels of $D(n, c)$ is also found.

We consider a problem of controlling the production rate of a single machine, single product, stochastic manufacturing system in order to minimize the total discounted inventory/backlog costs. The demand has two components: one is deterministic with constant rate $d$ and the other is stochastic with random demand batches. Under heavy loading (or heavy traffic) conditions, that is, when the average production capacity is close to the average demand, the control problem is approximated by a singular stochastic control problem. The approximate problem can be solved explicitly. The solution is then interpreted in terms of the actual manufacturing system and a control policy for this system is derived. We prove that the resulting policy is nearly optimal under the heavy traffic condition. This policy is characterized by a single critical level $z_0$. The commodity should be produced only when inventory is less than or equal to $z_0$: The production rate is maximal if the inventory is less than $z_0$ and equal to the deterministic component $d$ of the demand rate if the inventory is equal to $z_0$.

For a spherically symmetric multivariate normal random sample, the asymptotic distribution of the largest interpoint Euclidean distance is derived. The number of interpoint distances exceeding a high level is shown to have a limiting Poisson distribution.

Autoregressive (AR) modelling is generalized by replacing the delay operator by discrete Laguerre filters. The motivation is to reduce the number of parameters needed to obtain useful approximate models of stochastic processes, without increasing the computational complexity. Asymptotic statistical properties are investigated. Several AR model estimation results are extended to Laguerre models. In particular, it is shown how the choice of Laguerre time constant affects the resulting estimates. A Levinson-type algorithm for computing the Laguerre model estimates in an efficient way is also given. The Laguerre technique is illustrated by two simple examples.

We consider prediction of stationary max-stable processes. The usual metric between max-stable variables can be defined in terms of the $L_1$ distance between spectral functions and in terms of this metric a kind of projection can be defined. It is convenient to project onto max-stable spaces; that is, spaces of extreme value distributed random variables that are closed under scalar multiplication and the taking of finite maxima. Some explicit calculations of max-stable spaces generated by processes of interest are given. The concepts of deterministic and purely nondeterministic stationary max-stable processes are defined and illustrated. Differences between linear and nonlinear prediction are highlighted and some characterizations of max-moving averages and max-permutation processes are given.

We consider the expected performance of two greedy matching algorithms on sparse random graphs and also on random trees. In all cases we establish expressions for the mean and variance of the number of edges chosen and establish asymptotic normality.

Let $\{X(t), t \geq 0\}$ be a harmonisable, symmetric, $\alpha$-stable stochastic process and let $C_u(T)$ be the number of times that $X$ crosses the level $u$ during the time interval $\lbrack 0, T\rbrack$. Our main result is the precise numerical value of $C := \lim_{u \rightarrow \infty} u^\alpha EC_u(T)$. By way of examples, including an explicit evaluation of $EC_u$ for a stationary process and a combination of analytic and Monte Carlo techniques for some others, we show that the asymptotic approximation $EC_u \sim Cu^{-\alpha}$ is remarkably accurate, even for quite low values of the level $u$. This formula therefore serves, for all practical purposes, as a "Rice formula" for harmonisable stable processes, and should be as important in the applications of harmonisable stable processes as the original Rice formula was for their Gaussian counterparts. We also have upper and lower bounds for $EC_u$ that hold for all $u$ and that, unlike previous results in the area, also hold for all $\alpha$ and are of the correct order of magnitude for large $u$.

The formal solution to a two-factor option pricing model in which a short-term rate and a bond yield are taken as instrumental variables is shown to explode. There are no real-valued solutions to the diffusion equations written down for the long and short rate by Brennan and Schwartz.

We show that the length of the minimum spanning tree through points drawn uniformly from the $d$-dimensional torus is almost surely asymptotically equivalent to the length of the minimum spanning tree through points drawn uniformly from the $d$-cube. This result implies that the analytical expression recently obtained by Avram and Bertsimas for the minimum spanning tree (MST) constant in the $d$-torus model is in fact valid for the traditional $d$-cube model. We also show that the number of vertices of degree $k$ for the MST in both models is asymptotically equivalent with probability 1. Finally we show how our results can be extended to other combinatorial problems such as the traveling salesman problem.

Percolation with edge-passage probability $p$ and first-passage percolation are studied for the $n$-cube $\mathscr{B}_n = \{0, 1\}^n$ with nearest neighbor edges. For oriented and unoriented percolation, $p = e/n$ and $p = 1/n$ are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the opposite corner of the cube) converges in probability to 1 as $n \rightarrow \infty$. This resolves a conjecture of Aldous. When the edge-passage distribution is standard exponential, the (smaller) percolation time for unoriented edges is at least 0.88. These results are applied to Richardson's model on the (unoriented) $n$-cube. Richardson's model, otherwise known as the contact process with no recoveries, models the spread of infection as a Poisson process on each edge connecting an infected node to an uninfected one. It is shown that the time to cover the entire $n$-cube is bounded between 1.41 and 14.05 in probability as $n \rightarrow \infty$.