Launer [6] introduced the class of life distributions having decreasing (increasing) variance residual life, DVRL (IVRL). It is shown that the DVRL (IVRL) distributions are intimately connected to the behavior of the mean residual life function of the equilibrium distribution. Some counter examples are presented to demonstrate the lack of relationship between DVRL (IVRL) and NBUE (new better than used in expectation) (NWUE; new worse than used in expectation) distributions. Finally, we obtain bounds on moments and survival functions of DVRL (IVRL) distributions. These bounds turn out to be improvements on the previously known bounds for decreasing (increasing) mean residual life (DMRL (IMRL)) distributions.

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate

,

where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

In this paper, we are concerned with the simulation of Gaussian random fields by means of iterative stochastic algorithms, which are compared in terms of rate of convergence. A parametrized class of algorithms, which includes stochastic relaxation (Gibbs sampler), is proposed and its convergence properties are established. A suitable choice for the parameter improves the rate of convergence with respect to stochastic relaxation for special classes of covariance matrices. Some examples and numerical experiments are given.

In September 2000, the Brazilian system dispatch and spot prices were calculated twice, using different inflow forecasts for that month, as in the last 5 days of August the inflows to the reservoirs in the South and Southeast regions changed 200%. The first run used a smaller forecasted energy inflow and the second used a higher energy inflow. Contrary to expectations, the spot price in the second run, with the higher energy inflow, was higher than the one found in the first run. This paper describes the problem, presents the special features of the PAR(p) model that allow the described behavior, and shows the solution taken to avoid the problem.

The relation between extropy and variational distance is studied in this paper. We determine the distribution which attains the minimum or maximum extropy among these distributions within a given variation distance from any given probability distribution, obtain the tightest upper bound on the difference of extropies of any two probability distributions subject to the variational distance constraint, and establish an analytic formula for the confidence interval of an extropy. Such a study parallels to that of Ho and Yeung [3] concerning entropy. However, the proofs of the main results in this paper are different from those in Ho and Yeung [3]. In fact, our arguments can simplify several proofs in Ho and Yeung [3].

In this article, we consider a discrete-time insurance risk model. An autoregressive model is used to model both the claim process and the premium process. The probability of ruin is examined in a model with a constant interest rate. Both exponential and nonexponential upper bounds are obtained for the ruin probability of an infinite time horizon.

In this paper, TP2 orderings of distributions and of survival functions are considered. It is shown that the first passage time of a Markov process with TP2- ordered transition distributions has an increasing failure rate (IFR). Conditions are also given for which mixtures of IFR distributions are IFR. A formula is obtained for the failure rate when the strength distribution is a function of both load and strength. This formula, in conjunction with the TP2-ordering on the strength survival functions and log-concavity of the strength and load variables, leads to an increasing failure rate for the strength distribution. Finally, two models based on this theory are presented which explain the IFR character of the strength distribution of fibrous composite materials.

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

In this paper, we provide a new technique for analyzing the nonstationary Erlang loss queueing model with abandonment. Our method uniquely combines the use of the functional Kolmogorov forward equations with the well-known Gram-Charlier series expansion from the statistics literature. Using the Gram-Charlier series expansion, we show that we can estimate salient performance measures of the loss queue such as the mean, variance, skewness, kurtosis, and blocking probability. Lastly, we provide numerical examples to illustrate the effectiveness of our approximations.

We establish strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems. Our results substantially extend previously available results on steady-state derivative estimation via IPA.

Our basic assumption is that the process under study is regenerative, but our analysis uses regenerative structure in an indirect way: IPA estimators are typically biased over regenerative cycles, so straightforward differentiation of the regenerative ratio formula does not necessarily yield a valid estimator of the derivative of a steady-state mean. Instead, we use regeneration to pass from unbiasedness over fixed, finite time horizons to convergence as the time horizon grows. This provides a systematic way of extending results on unbiasedness to strong consistency.

Given that the underlying process regenerates, we provide conditions under which a certain augmented process is also regenerative. The augmented process includes additional information needed to evaluate derivatives; derivatives of time averages of the original process are time averages of the augmented process. Thus, through this augmentation we are able to apply standard renewal theory results to the convergence of derivatives.

The zero capacity M/M/1 queue with returning customers is examined. Performance measures are calculated and an imposed optimal return rate is determined for customers who return until served. The optimal rate is then calculated for an arbitrary customer as a function of the return rate of the other customers, and limiting cases are analyzed. The other customers may then adjust their return rate each time the arbitrary customer adjusts his/hers until a fixed point solution is obtained. A comparison between systems is made.

We consider the problem of controlling a two-server Markovian queueing system with heterogeneous servers. The servers are differentiated by their service rates and reliability attributes (i.e., the slower server is perfectly reliable, whereas the faster server is subject to random failures). The aim is to dynamically route customers at arrival, service completion, server failure, and server repair epochs to minimize the long-run average number of customers in the system. Using a Markov decision process model, we prove that it is always optimal to route customers to the faster server when it is available, irrespective of its failure and repair rates, if the system is stable. For the slower server, there exists an optimal threshold policy that depends on the queue length and the state of the faster server. Additionally, we analyze a variant of the main model in which there are multiple unreliable servers with identical service rates, but distinct reliability characteristics. For that case it is always optimal to route customers to idle servers, and the optimal policy is insensitive to the servers’ reliability characteristics.

We consider a Markovian stochastic fluid flow model in which the fluid level has a lower bound zero and a positive upper bound. The behavior of the process at the boundaries is modeled by parameters that are different than those in the interior and allow for modeling a range of desired behaviors at the boundaries. We illustrate this with examples. We establish formulas for several time-dependent performance measures of significance to a number of applied probability models. These results are achieved with techniques applied within the fluid flow model directly. This leads to useful physical interpretations, which are presented.

A class of Markov chains we call successively lumbaple is specified for which it is shown that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a(typically much) smaller state space and this yields significant computational improvements. We discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Finally, we will study applications of successively lumbaple Markov chains to classical reliability and queueing models.

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to investigate the conditions on the parameters that enable one to establish several stochastic comparisons of general p-spacings for a subclass of generalized order statistics in the likelihood ratio and the hazard rate orders. Preservation properties of the logconvexity and logconcavity of p-spacings are also given.

In this paper, a moment-iteration method is introduced. The method is used to solve Lindley's integral equation for the GI/G/l queue. From several forms of this integral equation, we derive the first two moments of the waiting-time distribution, the waiting probability, and the percentiles of the conditional waiting time. Numerical evidence is given that the method yields excellent results. The flexibility of the method provides the opportunity to solve the GI/G/l queue for all interarrival time distributions of practical interest. To show that the moment-iteration method is generally applicable, we give some results for an (s, S)-model with order-size-dependent lead times and finite production capacity of the supplier.

Let X1,..., Xn be independent exponential random variables with possibly different scale parameters. Kochar and Korwar [J. Multivar. Anal. 57 (1996)] conjectured that, in this case, the successive normalized spacings are increasing according to hazard rate ordering. In this article, we prove this conjecture in the case of a single-outlier exponential model when all except one of the parameters are identical. We also prove that the spacings are more dispersed and larger in the sense of hazard rate ordering when the vector of scale parameters is more dispersed in the sense of majorization.

Consider a single-commodity inventory system in which the demand is modeled by a sequence of independent and identically distributed random variables that can take negative values. Such problems have been studied in the literature under the name cash management and relate to the variations of the on-hand cash balances of financial institutions. The possibility of a negative demand also models product returns in inventory systems. This article studies a model in which, in addition to standard ordering and scrapping decisions seen in the cash management models, the decision-maker can borrow and store some inventory for one period of time. For problems with back orders, zero setup costs, and linear ordering, scrapping, borrowing, and storage costs, we show that an optimal policy has a simple four-threshold structure. These thresholds, in a nondecreasing order, are order-up-to, borrow-up-to, store-down-to, and scrap-down-to levels; that is, if the inventory position is too low, an optimal policy is to order up to a certain level and then borrow up to a higher level. Analogously, if the inventory position is too high, the optimal decision is to reduce the inventory to a certain point, after which one should store some of the inventory down to a lower threshold. This structure holds for the finite and infinite horizon discounted expected cost criteria and for the average cost per unit time criterion. We also provide sufficient conditions when the borrowing and storage options should not be used. In order to prove our results for average costs per unit time, we establish sufficient conditions when the optimality equations hold for a Markov decision process with an uncountable state space, noncompact action sets, and unbounded costs.

The concept of rainflow cycles is often used in fatigue of materials for analyzing load processes. Methods are developed for computation of the rainflow matrix for a random load that is changing properties over time due to changes of the system dynamics; for example, for a random vehicle load it could reflect different driving conditions. The random load is modeled by a switching process with Markov regime; that is, the random load changes properties according to a hidden (not observed) Markov chain.

An algorithm is developed for a switching process where each part of the load is modeled by a Markov chain. As only the local extremes are of importance for rainflow analysis, another approach is to model the sequence of turning points by a Markov chain. The main result of this paper is an algorithm for computation of the rainflow matrix for a switching process where each part is described by a Markov chain of turning points. The algorithms are illustrated by numerical examples.