We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

In the setting of the classical Cramér–Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = {Xt: t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, St=sup_{s≤ t}Xs, then Albrecher and Hipp studied Xt - γ St,t≥ 0, where γ∈(0,1) is the rate at which tax is paid. This model has been generalised to the setting that X is a spectrally negative Lévy process by Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = {St: t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, Ut:=Xt -∫(0,t]γ(S_u)dSu,t≥ 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫0∞ (1-γ(s))d s =∞. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.

In this paper we explore the first passage times of constant-elasticity-of-variance (CEV) processes with two-sided reflecting barriers. The explicit Laplace transforms of the first passage times are derived. Our results can include analytic formulae concerning Laplace transforms of first passage times of reflected Ornstein–Uhlenbeck processes, reflected geometric Brownian motions, and reflected square-root processes.

In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, we derive recursive equations for the steady-state distributions of the virtual waiting times in M/G/1 vacation systems with a general vacation time and two vacation rules.

The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.

An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U, V, and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y+(t), Y−(t), and Y0(t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y+(t) is derived. We also obtain the probability law of X(t) = Y+(t) - Y−(t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

Let Sn, k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn, k is connected is at least ε, then the probability that Sn, k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn⌋ is s-connected (i.e. remains connected after the deletion of any s − 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best-adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive.

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

In this paper we study the asymptotic behavior of a general class of product-form closed queueing networks as the population size grows large. We first characterize the asymptotic behavior of the normalization constant for the stationary distribution of the network in exact order. This result then enables us to establish the asymptotic behavior of the system performance metrics, which extends a number of well-known asymptotic results to exact order. We further derive new, computationally simple approximations for performance metrics that significantly improve upon existing approximations for large-scale networks. In addition to their direct use for the analysis of large networks, these new approximations are particularly useful for reformulating large-scale queueing network optimization problems into more easily solvable forms, which we demonstrate with an optimal capacity planning example.

A correspondence formula between the laws of dual Markov chains on Z with two transition jumps is established. This formula contributes to the study of random walks in stationary random environments. Counterexamples with more than two jumps are exhibited.

The notion of the signature is a basic concept and a powerful tool in the analysis of networks and reliability systems of binary type. An appropriate definition of this concept has recently been introduced for systems that have ν possible states (with ν ≥ 3). In this paper we analyze in detail several properties and the most relevant aspects of such a general definition. For simplicity's sake, we focus our attention on the case ν = 3. Our analysis will however provide a number of hints for understanding the basic aspects of the general case.

We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C1, C2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C1, C2,…); here ‘∨’ denotes the maximum operator.

In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t. The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established.

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

We propose an efficient semi-numerical approach to compute the steady-state probability distribution for the number of requests at arbitrary and at arrival time instants in PH/M/c-like systems with homogeneous servers in which the inter-arrival time distribution is represented by an acyclic set of memoryless phases. Our method is based on conditional probabilities and results in a simple computationally stable recurrence. It avoids the explicit manipulation of potentially large matrices and involves no iteration. Owing to the use of conditional probabilities, it delays the onset of numerical issues related to floating-point underflow as the number of servers and/or phases increases. For generalized Coxian distributions, the computational complexity of the proposed approach grows linearly with the number of phases in the distribution.