Consider a feedforward network of single-server stations populated by multiple job types. Each job requires the completion of a number of tasks whose order of execution is determined by a set of deterministic precedence constraints. The precedence requirements allow some tasks to be done in parallel (in which case tasks would "fork") and require that others be processed sequentially (where tasks may "join"). Jobs of a given type share the same precedence constraints, interarrival time distributions and service time distributions, but these characteristics may vary across different job types. We show that the heavy traffic limit of certain processes associated with heterogeneous fork-join networks can be expressed as a semimartingale reflected Brownian motion with polyhedral state space. The polyhedral region typically has many more faces than its dimension, and the description of the state space becomes quite complicated in this setting. One can interpret the proliferation of additional faces in heterogeneous fork-join networks as (i) articulations of the fork and join constraints and (ii) consequences of the disordering effects that occur when jobs fork and join in their sojourns through the network.

The subject of this paper is the heavy traffic behavior of a general class of queueing networks with first-in-first-out (FIFO) service discipline. For special cases that require various assumptions on the network structure, several authors have proved heavy traffic limit theorems to justify the approximation of queueing networks by reflecting Brownian motions. Based on these theorems, some have conjectured that the Brownian approximation may in fact be valid for a more general class of queueing networks. In this paper, we prove that the Brownian approximation does not hold for such a general class of networks. Our findings suggest that it may be fruitful to consider a more general class of approximating processes.

We consider open and closed multiclass queueing networks, with Poisson arrivals (for open networks), exponentially distributed class dependent service times and class dependent deterministic or probabilistic routing. The performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies. By optimizing over these sets, we obtain lower bounds on achievable performance. In the special case of single station networks (multiclass queues and Klimov's model) and homogeneous multiclass networks, the polyhedron derived is exactly equal to the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. We apply the same approach to closed networks to obtain upper bounds on the optimal throughput. We check the tightness of our bounds by simulating heuristic policies and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with a number of variables and constraints that is polynomial (quadratic) in the number of classes in the network. The $i$th order approximation leads to a convex programming problem in dimension $O(R^{i+1})$, where $R$ is the number of classes in the network, and can be solved efficiently using techniques from semidefinite programming.

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables and define $M_n = \max\big\{\sum_{\nu \in \pi}X_\nu: \pi \text{a self-avoiding path of length} n \text{starting at the origin}\big\}$, $N_n = \max\big\{\sum_{\nu \in \xi}X_\nu:\xi \text{a lattice animal of size} n \text{containing the origin}\big\}$. In a preceding paper it was shown that if $E\{X^d_0(\log^+ X_0)^{d+a}\} < \infty$ for some $a > 0$, then there exists some constant $C$ such that w.p.1, $0 \leq M_n \leq N_n \leq Cn$ for all large $n$. In this part we improve this result by showing that, in fact, there exist constants $M,N < \infty$ such that w.p.1, $M_n/n \rightarrow M$ and $N_n/n \rightarrow N$.

We study asymptotics of the number of linear extensions of the random $G_{n,p}$ partial order, where $p$ is fixed and $n \rightarrow \infty$. In particular, it is shown that the distribution is asymptotically $\log$-normal.

In this paper we study open generalized Jackson networks with general arrival streams and general service time distributions. Assuming that the arrival rate does not exceed the network capacity and that the service times possess conditionally bounded second moments, we deduce stability of the network by bounding the expected waiting time for a customer entering the network. For Markovian networks we obtain convergence of the total work in the system, as well as the mean queue size and mean customer delay, to a unique finite steady state value.

The standard Foster-Lyapunov approach to establishing recurrence and ergodicity of Markov chains requires that the one-step mean drift of the chain be negative outside some appropriately finite set. Malyshev and Men'sikov developed a refinement of this approach for countable state space chains, allowing the drift to be negative after a number of steps depending on the starting state. We show that these countable space results are special cases of those in the wider context of $\varphi$-irreducible chains, and we give sample-path proofs natural for such chains which are rather more transparent than the original proofs of Malyshev and Men'sikov. We also develop an associated random-step approach giving similar conclusions. We further find state-dependent drift conditions sufficient to show that the chain is actually geometrically ergodic; that is, it has $n$-step transition probabilities which converge to their limits geometrically quickly. We apply these methods to a model of antibody activity and to a nonlinear threshold autoregressive model; they are also applicable to the analysis of complex queueing models.

Let $Q$ and $M$ be random variables with given joint distribution. Under some conditions on this joint distribution, there will be exactly one distribution for another random variable $R$, independent of $(Q,M)$, with the property that $Q + MR$ has the same distribution as $R$. When $M$ is nonnegative and satisfies some moment conditions, we give an improved proof that if the upper tail of the distribution of $Q$ is regularly varying, then the upper tail of the distribution of $R$ behaves similarly; this proof also yields a converse. We also give an application to random environment branching processes, and consider extensions to cases where $Q + MR$ is replaced by $\Psi(R)$ for random but nonlinear $\Psi$ and where $M$ may be negative.

A conditioning and martingale method is used to relate the asymptotics of uniformly integrable general branching processes in varying environment to the behaviour of their expectations.

We provide a short and elementary proof of the Gittins index theorem for the multi-armed bandit problem, for the case where each bandit is modeled as a finite-state semi-Markov process. We also indicate how this proof can be extended to the branching bandits and Klimov problems.

We consider a sequence matching problem involving the optimal alignment score for contiguous subsequences, rewarding matches and penalizing for deletions and mismatches. This score is used by biologists comparing pairs of DNA or protein sequences. We prove that for two sequences of length $n$, as $n \rightarrow \infty$, there is a phase transition between linear growth in $n$, when the penalty parameters are small, and logarithmic growth in $n$, when the penalties are large. The results are valid for independent sequences with iid or Markov letters. The crucial step in proving this is to derive a large deviation result for matching with deletions. The longest common subsequence problem of Chvatal and Sankoff is a special case of our setup. The proof of the large deviation result exploits the Azuma-Hoeffding lemma. The phase transition is also established for more general scoring schemes allowing general letter-to-letter alignment penalties and block deletion penalties. We give a general method for applying the bounded increments martingale method to Lipschitz functionals of Markov processes. The phase transition holds for matching Markov chains and for nonoverlapping repeats in a single sequence.

In this paper a simple problem of combined singular stochastic control and optimal stopping is formulated and solved. We find that the optimal strategies can take qualitatively different forms, depending on parameter values. We also study a variant on the problem in which the value function is inherently nonconvex. The proofs employ the generalised Ito formula applicable for differences of convex functions.

For Gaussian processes there is a simple and well-known relationship between the fractal dimension of sample paths and the fractal index of the covariance function. This property is of considerable practical interest, since it forms the basis of several estimators of fractal dimension. Motivated by statistical applications involving non-Gaussian processes, we discuss the relationship in a wider context. We show that the relationship fails in some circumstances, but nevertheless does hold in a variety of cases.