We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

In this paper we study the asymptotic normality of discrete-time Markov control processes in Borel spaces, with possibly unbounded cost. Under suitable hypotheses, we show that the cost sequence is asymptotically normal. As a special case, we obtain a central limit theorem for (noncontrolled) Markov chains.

In this paper we obtain an approximation for the duration distribution of the excursion set generated by the minimum of two independent F random processes above a high threshold u. Moreover, we obtain a closed-form approximation for the mean duration of the conjunction of these two F processes. As an illustration, we conduct a simulation study to compare the performances of the approximated distribution and the exact distribution.

A semi-numerical method is derived to compute the Laplace transform of the equilibrium busy period probability density function in a fluid queue with constant output rate when the buffer is nonempty. The input process is controlled by a continuous-time semi-Markov chain (CTSMC) with n states such that in each state the input rate is constant. The holding time in states with net positive output rate - so-called emptying states - is assumed to be an exponentially distributed random variable, whereas in states with net positive input rate - so-called filling states - it may have an arbitrary probability distribution. The result is demonstrated by applying it to various systems, including fluid queues with two on-off input sources. The latter exercise in part shows consistency with prior results but also solves the problem in the case where there are two emptying states. Numerical results are presented for selected examples which expose discontinuities in the busy period distribution when the number of emptying states changes, e.g. as a result of increasing the fluid arrival rate in one or more states of the controlling CTSMC.

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability αi / 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

In this paper we study the stability of queueing systems with impatient customers and a single server operating under a FIFO (first-in-first-out) discipline. We first give a sufficient condition for the existence of a stationary workload in the case of impatience until the beginning of service. We then provide a weaker condition of existence on an enriched probability space using the theory of Anantharam et al. (1997), (1999). The case of impatience until the end of service is also investigated.

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.

We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximum utility, given by the average of a concave function of each commodity flow, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects Wv ⊆ W such that w ∈ Wv with probability p, independently for all v ∈ V and all w ∈ W. Given two vertices v1, v2 ∈ V, we set v1 ∼ v2 if and only if Wv1 ∩ Wv2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

Measures of divergence or discrepancy are used either to measure mutual information concerning two variables or to construct model selection criteria. In this paper we focus on divergence measures that are based on a class of measures known as Csiszár's divergence measures. In particular, we propose a measure of divergence between residual lives of two items that have both survived up to some time t as well as a measure of divergence between past lives, both based on Csiszár's class of measures. Furthermore, we derive properties of these measures and provide examples based on the Cox model and frailty or transformation model.

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

The Ehrenfest urn is a model for the diffusion of gases between two chambers. Classic research deals with this system as a Markovian model with a fixed number of balls, and derives the steady-state behavior as a binomial distribution (which can be approximated by a normal distribution). We study the gradual change for an urn containing n (a very large number) balls from the initial condition to the steady state. We look at the status of the urn after kn draws. We identify three phases of kn: the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the amount of gas in each chamber is normally distributed, with parameters that are influenced by the initial conditions. In the linear phase a different normal distribution applies, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. At the superlinear stage the mix is nearly perfect, with a nearly perfect symmetrical normal distribution in which the effect of the initial conditions is completely washed away. We give interpretations for how the results in different phases conjoin at the ‘seam lines’. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via martingale theory.

System signatures are useful tools in the study and comparison of coherent systems. In this paper, we define and study a similar concept, called the joint signature, for two coherent systems which share some components. Under an independent and identically distributed assumption on component lifetimes, a pseudo-mixture representation based on this joint signature is obtained for the joint distribution of the lifetimes of both systems. Sufficient conditions are given based on the respective joint signatures of two pairs of systems, each with shared components, to ensure various forms of bivariate stochastic orderings between the joint lifetimes of the two pairs of systems.