We consider a model of Internet congestion control that represents the randomly varying number of flows present in a network where bandwidth is shared fairly between document transfers. We study critical fluid models obtained as formal limits under law of large numbers scalings when the average load on at least one resource is equal to its capacity. We establish convergence to equilibria for fluid models and identify the invariant manifold. The form of the invariant manifold gives insight into the phenomenon of entrainment whereby congestion at some resources may prevent other resources from working at their full capacity.

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: $n\approx \mathbf{r}+\beta \sqrt{\mathbf{r}}$ for some scalar β. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)

We first consider a formal weak limit, through which our queueing scheduling problem gives rise to a diffusion control problem. We show that the latter has an optimal Markov control policy, and that the corresponding Hamilton–Jacobi–Bellman (HJB) equation has a unique classical solution. The Markov control policy and the HJB equation are then used to define scheduling control policies which we prove are asymptotically optimal for our original queueing system. The analysis yields both qualitative and quantitative insights, in particular on staffing levels, the roles of non-preemption and work conservation, and the trade-off between service quality and servers’ efficiency.

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Z_n:n≥1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n↗∞, and use this to study conditional large deviations of {Y_Zn:n≥1}, where Y_n satisfies an LDP, particularly of {Z_n⁻¹Z_n+1:n≥1} conditioned on Z_n≥v_n.

This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633–664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager’s information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager’s information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azéma martingale, and we use excursion theory of Brownian motions to price risky debt.

We prove asymptotic normality of the so-called maximum likelihood estimator of the extreme value index.

Let {X_n,n≥0} be a Markov chain on a general state space ${\mathcal{X}}$ with transition probability P and stationary probability π. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n≥0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n≥0}. To be more precise, for given b≥0, define the stopping time τ=τ(b)=inf {n:S_n>b}. When a drift μ of the random walk S_n is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities P_π{τ<m} and P_π{τ<m,S_m<c}, where m≤∞, c≤b and P_π denotes the probability under the initial distribution π. When μ≠0, Brownian approximations for the first passage probabilities with correction terms are derived. Applications to sequential estimation and truncated tests in random coefficient models and first passage times in products of random matrices are also given.

Consider the d-dimensional lattice ℤ^d where each vertex is “open” or “closed” with probability p or 1−p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d)=v(d)−1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d=2 and 3 and it is an infinite collection of distinct trees for d≥4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

We consider the problem of hedging a European interest rate contingent claim with a portfolio of zero-coupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite-factor models. Indeed, under natural conditions on the model, we find that there exists a unique hedging strategy, and that this strategy has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim.

Let X={X_n:n=0,1,2,…} be an irreducible, positive recurrent Markov chain with invariant probability measure π. We show that if X satisfies a one-step minorization condition, then π can be represented as an infinite mixture. The distributions in the mixture are associated with the hitting times on an accessible atom introduced via the splitting construction of Athreya and Ney [Trans. Amer. Math. Soc. 245 (1978) 493–501] and Nummelin [Z. Wahrsch. Verw. Gebiete 43 (1978) 309–318]. When the small set in the minorization condition is the entire state space, our mixture representation of π reduces to a simple formula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3 (2001) 161–177] from which samples can be easily drawn. Despite the fact that the derivation of this formula involves no coupling or backward simulation arguments, the formula can be used to reconstruct perfect sampling algorithms based on coupling from the past (CFTP) such as Murdoch and Green’s [Scand. J. Statist. 25 (1998) 483–502] Multigamma Coupler and Wilson’s [Random Structures Algorithms 16 (2000) 85–113] Read-Once CFTP algorithm. In the general case where the state space is not necessarily 1-small, under the assumption that X satisfies a geometric drift condition, our mixture representation can be used to construct an arbitrarily accurate approximation to π from which it is straightforward to sample. One potential application of this approximation is as a starting distribution for a Markov chain Monte Carlo algorithm based on X.

This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process. A two-station example is worked out in detail, and simulation results are presented to illustrate the predictive value of the theory. This work is a generalization of Doytchinov, Lehoczky and Shreve [Ann. Appl. Probab. 11 (2001) 332–379], which developed these results for the single queue case.

We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a ψ-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224–247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings–Metropolis algorithms.

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Lévy process with zero expectation. We derive a Pollaczek–Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman–Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solutions exhibit a critical phenomenon: If the capital constraint is below some critical value, then the solution will coincide with a classical solution; above this critical value, the solution is a superposition of a classical solution and a less risky or even risk-free investment. For general risk measures and utility functionals, it is shown that there exists a solution that can be written as a deterministic increasing function of the price density.

We investigate the asymptotic behavior of one version of the so-called two-armed bandit algorithm. It is an example of stochastic approximation procedure whose associated ODE has both a repulsive and an attractive equilibrium, at which the procedure is noiseless. We show that if the gain parameter is constant or goes to 0 not too fast, the algorithm does fall in the noiseless repulsive equilibrium with positive probability, whereas it always converges to its natural attractive target when the gain parameter goes to zero at some appropriate rates depending on the parameters of the model. We also elucidate the behavior of the constant step algorithm when the step goes to 0. Finally, we highlight the connection between the algorithm and the Polya urn. An application to asset allocation is briefly described.

Generalized Pólya urn models can describe the dynamics of finite populations of interacting genotypes. Three basic questions these models can address are: Under what conditions does a population exhibit growth? On the event of growth, at what rate does the population increase? What is the long-term behavior of the distribution of genotypes? To address these questions, we associate a mean limit ordinary differential equation (ODE) with the urn model. Previously, it has been shown that on the event of population growth, the limiting distribution of genotypes is a connected internally chain recurrent set for the mean limit ODE. To determine when growth and convergence occurs with positive probability, we prove two results. First, if the mean limit ODE has an “attainable” attractor at which growth is expected, then growth and convergence toward this attractor occurs with positive probability. Second, the population distribution almost surely does not converge to sets where growth is not expected and almost surely does not converge to “nondegenerate” unstable equilibria or periodic orbits of the mean limit ODE. Applications to stochastic analogs of the replicator equations and fertility-selection equations of population genetics are given.

We address the problem of simulating efficiently from the posterior distribution over the parameters of a particular class of nonlinear regression models using a Langevin–Metropolis sampler. It is shown that as the number N of parameters increases, the proposal variance must scale as N^−1/3 in order to converge to a diffusion. This generalizes previous results of Roberts and Rosenthal [J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (1998) 255–268] for the i.i.d. case, showing the robustness of their analysis.

In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Lé vy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d≤3, then this solution can be represented as a classical random field in L²(μ), where μ is the probability law of the L évy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform.

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τ is the first exit time of iterated Brownian motion from the solid, then P(τ>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(τ>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator $\frac{1}{8}\Delta^{2}-\frac{\partial}{\partial t}$.

Let $A_n=(a_{ij})_{i,j=1}^n$ be an $n×n$ positive matrix with entries in $[a,b], 0<a≤b$. Let $X_{n}=(\sqrt{a_{ij}}x_{ij})_{i,j=1}^{n}$ be a random matrix, where $\{x_{ij}\}$ are i.i.d. $N(0,1)$ random variables. We show that for large $n$, $\det (X_{n}^{T}X_{n})$ concentrates sharply at the permanent of $A_n$, in the sense that $n^{-1}\log (\det(X_{n}^{T}X_{n})/\operatorname {per}A_{n})\to_{n\to\infty}0$ in probability.

We study the large space and time scale behavior of a totally asymmetric, nearest-neighbor exclusion process in one dimension with random jump rates attached to the particles. When slow particles are sufficiently rare, the system has a phase transition. At low densities there are no equilibrium distributions, and on the hydrodynamic scale the initial profile is transported rigidly. We elaborate this situation further by finding the correct order of the correction from the hydrodynamic limit, together with distributional bounds averaged over the disorder. We consider two settings, a macroscopically constant low density profile and the outflow from a large jam.