In this paper we study first passage times of (reflected) Ornstein-Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry, Stadje and Zacks (2004) to the (reflected) Ornstein-Uhlenbeck case.

We consider the model of Deijfen, Häggström and Bagley (2004) for competing growth of two infection types in Rd, based on the Richardson model on Zd. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates, and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen, Häggström and Bagley (2004) conjectured that the initial configuration is basically irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which, e.g. do not include the case of a deterministic radius. Here we give a proof that does not rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.

In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

Recently, there has been considerable interest in the calculation of decay rates for models that can be viewed as quasi-birth-and-death (QBD) processes with infinitely many phases. In this paper we make a contribution to this endeavour by considering some classes of models in which the transition function is not homogeneous in the phase direction. We characterize the range of decay rates that are compatible with the dynamics of the process away from the boundary. In many cases, these rates can be attained by changing the transition structure of the QBD process at level 0. Our approach, which relies on the use of orthogonal polynomials, is an extension of that in Motyer and Taylor (2006) for the case where the generator has homogeneous blocks.

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability, and the total dividend, using methods and results from queueing theory.

We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.

We establish some interesting duality results for Markov-modulated fluid flow models. Though fluid flow models are continuous-state analogues of quasi-birth-and-death processes, some duality results do differ by the inclusion of a scaling factor.

In this paper we consider a general Lévy process X reflected at a downward periodic barrier At and a constant upper barrier K, giving a process VKt=Xt +LAt−LKt. We find the expression for a loss rate defined by lK=ELK1 and identify its asymptotics as K→∞ when X has light-tailed jumps and EX1<0.

We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.

We apply the known formulae of the RESTART problem to Markov models of software (and many other) systems, and derive new equations. We show how checkpoints might be included, with their resultant performance under RESTART. The result is a complete procedure for finding the mean, variance, and tail behavior of the job completion time as a function of the failure rate. We also provide a detailed example.

The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

Nonspatial stochastic models of populations subject to catastrophic events result in the common conclusion that the survival probability of the population is nondecreasing with respect to the random number of individuals removed at each catastrophe. The purpose of this paper is to prove that such a monotonic relationship is not true for simple spatial models based on Harris' contact processes, whose dynamics are described by hypergraph structures rather than traditional graph structures. More precisely, it is proved that, for a wide range of parameters, the destruction of (infinite) hyperplanes does not affect the existence of a nontrivial invariant measure, whereas the destruction of large (finite) cubes drives the population to extinction, a result that we depict by using the biological picture: forest fires are more devastating than tornadoes. This indicates that the geometry of the subsets struck by catastrophes is somewhat more important than their area, thus the need to consider spatial rather than nonspatial models in this context.

In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒt,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e, t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.