We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.

Consider a particular bidimensional risk model, in which two insurance companies divide between them in different proportions both the premium income and the aggregate claims. In practice, it can be interpreted as an insurer–reinsurer scenario, where the reinsurer takes over a proportion of the insurer's losses. Under the assumption that the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure, an asymptotic expression for the ruin probability of this bidimensional risk model with constant interest rates is established.

This article derives the transient behavior of the stochastic process X(t) for which in earlier articles it was proven that for α<1, the process (X(t))1/(1−α) (approximately, under low drop probability or ECN marking probability) describes the behavior of the congestion windows in certain transport protocols in the so-called TCP paradigm. The transient distribution is found explicitly and is particularly transparent for moments like E[(X(t))k |X(0)]. The purpose of this article is to be part of the mathematical foundation when comparing the many protocols in the TCP paradigm.

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

A practical source coding scheme based on approximate string matching is proposed. It is an approximate fixed-length string matching data compression combined with a block-coder based on the empirical distribution. A lemma on approximate string matching, which is an extension of the Kac Lemma, is proved. It is shown, based on the lemma, that the deterministic algorithm converts the stationary and ergodic source, u, into an output process v, and under the assumption that v is a stationary process, after the scheme has run for an infinite time, the optimal compression ratio R(D) is achieved. This reduces the problem of the universal lossy coder to the proof of stationarity of the output process ν in the proposed algorithm. The main advantages of the proposed method are the asymptotic sequential behavior of the encoder and the simplicity of the decoder.

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

We study an online capacity planning problem in which arriving patients require a series of appointments at several departments, within a certain access time target.

This research is motivated by a study of rehabilitation planning practices at the Sint Maartenskliniek hospital (the Netherlands). In practice, the prescribed treatments and activities are typically booked starting in the first available week, leaving no space for urgent patients who require a series of appointments at a short notice. This leads to the rescheduling of appointments or long access times for urgent patients, which has a negative effect on the quality of care and on patient satisfaction.

We propose an approach for allocating capacity to patients at the moment of their arrival, in such a way that the total number of requests booked within their corresponding access time targets is maximized. The model considers online decision making regarding multi-priority, multi-appointment, and multi-resource capacity allocation. We formulate this problem as a Markov decision process (MDP) that takes into account the current patient schedule, and future arrivals. We develop an approximate dynamic programming (ADP) algorithm to obtain approximate optimal capacity allocation policies. We provide insights into the characteristics of the optimal policies and evaluate the performance of the resulting policies using simulation.

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

An inventory whose stock decreases linearly with time is considered. The inventory may be replenished at the instants at which a deliveryman arrives provided that the level of the inventory does not exceed a certain threshold; deliveries are made according to a Poisson process. A partial differential equation for the distribution function of the level of the inventory is solved to yield a formula for the corresponding Laplace–Stieltjes transform. The evaluation of the transform is discussed and explicit results are obtained for the stationary case.

Tasks belonging to N priority classes arrive for processing in a single or multiserver facility. If the processing does not begin by a certain time (deterministic or random), the task is lost and a cost is incurred. We determine properties of dynamic, nonidling, nonpreemptive strategies that minimize an infinite horizon expected cost.

The introduction of the class of queueing networks called G-networks by Gelenbe has been a breakthrough in the field of stochastic modeling since it has largely expanded the class of models which are analytically or numerically tractable. From a theoretical point of view, the introduction of the G-networks has lead to very important considerations: first, a product-form queueing network may have non-linear traffic equations; secondly, we can have a product-form equilibrium distribution even if the customer routing is defined in such a way that more than two queues can change their states at the same time epoch. In this work, we review some of the classes of product-forms introduced for the analysis of the G-networks with special attention to these two aspects. We propose a methodology that, coherently with the product-form result, allows for a modular analysis of the G-queues to derive the equilibrium distribution of the network.

For discrete versions of red-and-black with a goal, for timid play to be optimal, it suffices, but is not necessary, that no win probability be less than one-half. A condition that is both necessary and sufficient is provided.

We consider an infinite series of independent and identical -/GI/∞ queues fed by an arbitrary stationary and ergodic arrival process, A1. Let Ai be the arrival process to the ith node, and let νi be the law of Ai. Denote by (·) the input–output map of the ·/GI/∞ node; that is, νi+1 = (νi). It is known that the Poisson process is a fixed point for . In this paper, we are interested in the asymptotic distribution of the departure process from the nth node, νn+1 = n(ν1), as n ∞. Using couplings for random walks, this limiting distribution is shown to be either a Poisson process or a stationary ν-Poisson process, depending on the joint distribution of A1 and the service process. This generalizes a result of Vere-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321–333) and is similar in flavor to Mountford and Prabhakar (1995, Annals of Applied Probability 5(1): 121–127), where Poisson convergence is established for departures from a series of exponential server queues using coupling methods.

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

We model the virtual contention queue in an ATM nonblocking packet switch with capacity c and input queues by a Geox1, Geox2/D/c Head-of-Line priority queueing system with Random Order Selection within each class and find the joint queue length distribution and the waiting time distribution for each class.

For each n ≥ 1, let Πn = (π1, π2, …, πn) be a random permutation of the integers 1,2, …, n. The quantity Bn = Σ measures the degree of oscillation of Πn and is an important measure in the study of sorting algorithms. In this work the asymptotic normality of Bn is derived under the assumption that, for each n ≥ 1, Πn is uniformly distributed over the n! permutations of 1,2, …, n. The proof relies on coupling (π1, π2, …, πn) with a sequence of independent and identically distributed U(0,1) random variables and offers a more probabilistic and computationally simpler alternative to the method of moments proof of Chao, Bai, and Liang (1993, Probability in the Engineering and Informational Sciences 7: 227–235).

Suppose each permutation (πl,…,πn) of ( 1, …, n) has probability 1/n!. The oscillation of (πl; …, πn) is defined as Tn = | πk − πk+1|, where πn+1 = π1. It is proved that (Tn − ETn)/(var Tn)1/2 converges in distribution to N(0,1). The connection between the oscillation and the presortedness measure is also discussed.

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

We consider a monotone multistate system with conditionally independent components given the component reliabilities, and random component reliabilities. Upper and lower bounds are derived for the moments of the random reliability function, extending results for binary systems. The second moment is given special attention, as this quantity is used to calculate the standard deviation of the system reliability estimate. The motivation for the paper is to establish a basis for uncertainty analysis and Bayesian estimation in connection with multistate systems.