We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

A stochastic epidemic model is defined in which infected individuals have different severities of disease (e.g. mildly and severely infected) and the severity of an infected individual depends on the severity of the individual he or she was infected by; typically, severe or mild infectives have an increased tendency to infect others severely or, respectively, mildly. Large-population properties of the model are derived, using branching process approximations for the initial stages of an outbreak and density-dependent population processes when a major outbreak occurs. The effects of vaccination are considered, using two distinct models for vaccine action. The consequences of launching a vaccination program are studied in terms of the effect it has on reducing the final size in the event of a major outbreak as a function of the vaccination coverage, and also by determining the critical vaccination coverage above which only small outbreaks can occur.

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings computing the well-known Erlang formula for blocking probabilities in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for the blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulae and simulation results even for relatively small resource capacities and relatively large values of the blocking probabilities.

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

We study the phenomenon of entrainment in processor sharing networks, whereby, while individual network resources have sufficient capacity to meet demand, the requirement for simultaneous availability of resources means that a network may nevertheless be unstable. We show that instability occurs through poor control, and that, for a variety of network topologies, only small modifications to controls are required in order to ensure stability. For controls which possess a natural monotonicity property, we give some new results for the classification of the corresponding Markov processes, which lead to conditions both for stability and for instability.

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

In a sequence of independent Bernoulli trials the probability for success in the kth trial is pk, k = 1, 2, …. The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in which pk = a/(a + b + k − 1), a > 0, b ≥ 0. Also, the limit behaviour of the longest failure string in the first n trials is considered. For b = 0, the strings correspond to cycles in random permutations.

We develop an integration by parts technique for point processes, with application to the computation of sensitivities via Monte Carlo simulations in stochastic models with jumps. The method is applied to density estimation with respect to the Lebesgue measure via a modified kernel estimator which is less sensitive to variations of the bandwidth parameter than standard kernel estimators. This applies to random variables whose densities are not analytically known and requires the knowledge of the point process jump times.

With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Zd \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Zd \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R2, is given.

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

We study the perpetual American option characteristics in the case where the underlying dynamics involve a Brownian motion and a point process with a stochastic intensity. No assumption on the distribution of the jump size is made and we work with an arbitrary positive or negative jump. After proving the existence of an optimal stopping time for the problem and characterizing it as the hitting time of an optimal boundary, we provide closed-form formulae for the option value, as well as for the Laplace transform of the optimal stopping time. These results are then applied to the analysis of a real option problem when considering the impact of a fundamental and brutal change in the investment project environment. The consequences of this impact, that can seriously modify, positively or negatively, the project's future cash flows and therefore the investment decision, are illustrated by numerical examples.

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with examples arising in queueing theory and insurance risk.