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We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.
We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express pĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in pĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.
We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.
We consider a two-stage tandem queue with two parallel servers and two queues. We assume that initially all jobs are present and that no further arrivals take place at any time. The two servers are identical and can serve both types of job. The processing times are exponentially distributed. After being served, a job of queue 1 joins queue 2, whereas a job of queue 2 leaves the system. Holding costs per job and per unit time are incurred if there are jobs holding in the system. Our goal is to find the optimal strategy that minimizes the expected total holding costs until the system is cleared. We give a complete solution for the optimal control of all possible parameters (costs and service times), especially for those parameter regions in which the optimal control depends on how many jobs are present in the two queues.
In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.
Let {νε, ε>0} be a family of probabilities for which the decay is governed by a large deviation principle, and consider the simulation of νε0(A) for some fixed measurable set A and some ε0>0. We investigate the circumstances under which an exponentially twisted importance sampling distribution yields an asymptotically efficient estimator. Varadhan's lemma yields necessary and sufficient conditions, and these are shown to improve on certain conditions of Sadowsky. This is illustrated by an example to which Sadowsky's conditions do not apply, yet for which an efficient twist exists.
We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.
Diffusion models in economics, finance, queueing, mathematical biology, and electrical engineering often involve reflecting barriers. In this paper, we study the analytical representation of transition densities for reflected one-dimensional diffusions in terms of their associated Sturm-Liouville spectral expansions. In particular, we provide explicit analytical expressions for transition densities of Brownian motion with drift, the Ornstein-Uhlenbeck process, and affine (square-root) diffusion with one or two reflecting barriers. The results are easily implementable on a personal computer and should prove useful in applications.
We consider a finite dam under the policy, where the input of water is formed by a Wiener process subject to random jumps arriving according to a Poisson process. The long-run average cost per unit time is obtained after assigning costs to the changes of release rate, a reward to each unit of output, and a penalty that is a function of the level of water in the reservoir.
We use a sample-path technique to derive asymptotics of generalized Jackson queueing networks in the fluid scale; that is, when space and time are scaled by the same factor n. The analysis only presupposes the existence of long-run averages and is based on some monotonicity and concavity arguments for the fluid processes. The results provide a functional strong law of large numbers for stochastic Jackson queueing networks, since they apply to their sample paths with probability 1. The fluid processes are shown to be piecewise linear and an explicit formulation of the different drifts is computed. A few applications of this fluid limit are given. In particular, a new computation of the constant that appears in the stability condition for such networks is given. In a certain context of a rare event, the fluid limit of the network is also derived explicitly.
In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.
We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.
In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.
We derive several algorithms for the busy period distribution of the canonical Markovian fluid flow model. One of them is similar to the Latouche-Ramaswami algorithm for quasi-birth-death models and is shown to be quadratically convergent. These algorithms significantly increase the efficiency of the matrix-geometric procedures developed earlier by the authors for the transient and steady-state analyses of fluid flow models.
Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.
In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.
The comparison of lifetimes has been treated extensively during the last decade. A wide variety of mathematical objects have been defined, which, in reliability theory, are used to quantify ageing properties. In this work, using the equilibrium variable, we give a new viewpoint on ageing properties. Moreover, we give new bounds on the moments of series systems.
As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.
Let (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.