Simulation studies of circuit switched networks with dynamic alternate routing reveal the existence of hysteresis phenomena, which suggest that such networks can admit more than one regime of operation for the same offered traffic. Such behavior is also suggested by a detailed analytical model due to Marbukh and a simpler model due to Gibbens, Hunt and Kelly. In these models, a limit is taken as the network size becomes large, and one finds a limiting ODE describing the proportions of network links in different states. The possibility of multiple regimes of operation shows up through the fact that the ODE has multiple equilibrium points for certain ranges of parameters. The kinds of limits considered by Marbukh and Gibbens, Hunt and Kelly do not take into account the spatial extent of the network. In an attempt to preserve the spatial characteristics, we consider a lattice model similar to that of Gibbens, Hunt and Kelly. We derive a mean field limit for this lattice model. This is an integrodifferential equation which describes how the spatial distribution of the network evolves in time. The mean field equation also admits multiple spatially homogeneous equilibrium solutions for certain ranges of the parameters, which may be loosely thought of as the different operating regimes. This equation may be particularly useful in understanding the exchange between the operating regimes, that is, questions like "for what parameter values is a hot spot of heavy loading in the system likely to take over the whole network by knock-on effects?"

The log of the price of a share is commonly modelled as a Brownian motion with drift, $\sigma B_t + ct$, where the constants $c$ and $\sigma$ are unknown. In order to use the Black-Scholes option pricing formula, one needs an estimate of $\sigma$, though not of $c$. In this paper, we propose a new estimator of $\sigma$ based on the high, low, and closing prices in a day's trading. This estimator has the merit of being unbiased whatever the drift $c$. In common with other estimators of $\sigma$, the approximation of the true high and low values of the drifting Brownian motion by the high and low values of a random walk introduces error, often quite a serious error. We shall show how a simple correction can overcome this error almost completely.

In an independent, equiprobable allocation urn model, there are various Poisson and normal limit laws for the occupancy of single urns. Applying the Chen-Stein method, we obtain Poisson, compound Poisson and multivariate Poisson limit laws, together with estimates of their rates of convergence, for the number of chunks of $\kappa$ (fixed) adjacent urns occupied by certain numbers of balls distributed in some specified patterns. Several related results on occupancy, waiting time and spacings at certain random times are also presented.

The Ewens sampling formula arises in population genetics and the study of random permutations as a probability distribution on the set of partitions (by allelic type in a sample, or according to cycle structure, respectively) of the integer $n$ for each $n$. It may be embedded naturally in the familiar linear birth process with immigration. One consequence of this is another proof of the functional central limit theorem for the Ewens sampling formula.

We consider a series of $n$ single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then $k$ customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time $D(k,n)$ required for all $k$ customers to complete service from all $n$ queues. In particular, we investigate the limiting behavior of $D(k,n)$ as $n \rightarrow \infty$ and/or $k \rightarrow \infty$. There is a duality implying that $D(k,n)$ is distributed the same as $D(n,k)$ so that results for large $n$ are equivalent to results for large $k$. A previous heavy-traffic limit theorem implies that $D(k,n)$ satisfies an invariance principle as $n \rightarrow \infty$, converging after normalization to a functional of $k$-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of $D(k_n,n)$, where $k_n \rightarrow \infty$ as $n \rightarrow \infty$. The case of $k_n = \lbrack xn \rbrack$ corresponds to a hydrodynamic limit.

Suppose that a child is likely to be weaker than its parent and a child who is too weak will not reproduce. What is the condition for a family line to survive? Let $b$ denote the mean number of children a viable parent will have; we suppose that this is independent of strength as long as strength is positive. Let $F$ denote the distribution of the change in strength from parent to child, and define $h = \sup_\theta(-\log(\int e^{\theta t} dF(t)))$. We show that the situation is black or white: 1. If $b < e^h, \text{then} P(\text{family line dies}) = 1$. 2. If $b > e^h, \text{then} P(\text{family survives}) > 0$. Define $f(x) := E(\text{number of members in the family} \mid \text{initial strength} x)$. We show that if $b < e^h$, then there exists a positive constant $C$ such that $\lim_{x \rightarrow \infty}e^{- \alpha x}f(x) = C$, where $\alpha$ is the smaller of the (at most) two positive roots of $b \int e^{st} dF(t) = 1$. We also find an explicit expression for $f(x)$ when the walk is on a lattice and is skip-free to the left. This process arose in an analysis of rollback-based simulation, and these results are the foundation of that analysis.

Let $(X_1, X_2)$ be an $\alpha$-stable random vector, not necessarily symmetric, with $0 < \alpha < 2$. This article investigates the regression $E(X_2 \mid X_1 = x)$ for all values of $\alpha$. We give conditions for the existence of the conditional moment $E(|X_2|^p|X_1 = x)$ when $p \geq \alpha$, and we obtain an explicit form of the regression $E(X_2 \mid X_1 = x)$ as a function of $x$. Although this regression is, in general, not linear, it can be linear even when the vector $(X_1, X_2)$ is skewed. We give a necessary and sufficient condition for linearity and characterize the asymptotic behavior of the regression as $x \rightarrow \pm \infty$. The behavior of the regression functions is also illustrated graphically.

This paper establishes a necessary and sufficient condition for geometrical ergodicity for the general first-order threshold autoregressive processes. This is achieved by investigating the nonlinear dynamic behavior generated by the delay parameter of a threshold model. The ergodic region turns out to be unbounded which is different from that of a linear process.