We consider a system consisting of a server alternating between two service points. At both service points, there is an infinite queue of customers that have to undergo a preparation phase before being served. We are interested in the waiting time of the server. The waiting time of the server satisfies an equation very similar to Lindley's equation for the waiting time in the GI/G/1 queue. We will analyze this Lindley-type equation under the assumptions that the preparation phase follows a phase-type distribution, whereas the service times have a general distribution. If we relax the condition that the server alternates between the service points, then the model turns out to be the machine repair problem. Although the latter is a well-known problem, the distribution of the waiting time of the server has not been studied yet. We derive this distribution under the same setting and we compare the two models numerically. As expected, the waiting time of the server is, on average, smaller in the machine repair problem than in the alternating service system, but they are not stochastically ordered.

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering. Also, it is shown that the marginal distributions of two upper orthant dispersive ordered random vectors are also dispersive ordered. Examples and applications are given.

Two well-known orders that have been introduced and studied in reliability theory are defined via stochastic comparison of inactivity time: the reversed hazard rate order and the mean inactivity time order. In this article, some characterization results of those orders are given. We prove that, under suitable conditions, the reversed hazard rate order is equivalent to the mean inactivity time order. We also provide new characterizations of the decreasing reversed hazard rate (increasing mean inactivity time) classes based on variability orderings of the inactivity time of k-out-of-n system given that the time of the (n − k + 1)st failure occurs at or sometimes before time t ≥ 0. Similar conclusions based on the inactivity time of the component that fails first are presented as well. Finally, some useful inequalities and relations for weighted distributions related to reversed hazard rate (mean inactivity time) functions are obtained.

The distribution of the linear combination αX + βY is derived when X and Y are independent Laplace random variables. Extensive tabulations of the associated percentage points are also given. The work is motivated by examples in automation, control, fuzzy sets, neurocomputing, and other areas of informational sciences.

In the article published earlier this year in Probability in the Engineering and Informational Sciences 19(1), Theorem 2 on p. 81 was stated incorrectly.

We compare and test statistical estimates of failure propagation in data from versions of a probabilistic model of loading-dependent cascading failure and a power system blackout model of cascading transmission line overloads. The comparisons suggest mechanisms affecting failure propagation and are an initial step toward monitoring failure propagation from practical system data. Approximations to the probabilistic model describe the forms of probability distribution of cascade sizes.

We propose the use of the cluster distribution, derived from a negative binomial probability model, to estimate the probability of high-order events in terms of number of lines outaged within a short time, useful in long-term planning and also in short-term operational defense to such events. We use this model to fit statistical data gathered for a 30-year period for North America. The model is compared to the commonly used Poisson model and the power-law model. Results indicate that the Poisson model underestimates the probability of higher-order events, whereas the power-law model overestimates it. We use the strict chi-square fitness test to compare the fitness of these three models and find that the cluster model is superior to the other two models for the data used in the study.

This paper addresses computer simulation of cascading failures in electric power systems. The paper analyzes the convergence rates of estimator variance in importance sampling and in random search strategies. A uniform search strategy based on the Metropolis algorithm is proposed.

It is known that Cournot game theory has been one of the theoretical approaches used more often to model electricity market behavior. Nevertheless, this approach is highly influenced by the residual demand curves of the market agents, which are usually not precisely known. This imperfect information has normally been studied with probability theory, but possibility theory might sometimes be more helpful in modeling not only uncertainty but also imprecision and vagueness. In this paper, two dual approaches are proposed to compute a robust Cournot equilibrium, when the residual demand uncertainty is modeled with possibility distributions. Additionally, it is shown that these two approaches can be combined into a bicriteria programming model, which can be solved with an iterative algorithm. Some interesting results for a real-size electricity system show the robustness of the proposed methodology.

It is widely accepted that medium-term generation planning can be advantageously modeled through market equilibrium representation. There exist several methods to define and solve this kind of equilibrium in a deterministic way. Medium-term planning is strongly affected by uncertainty in market and system conditions; thus, extensions of these models are commonly required. The main variables that should be considered as subject to uncertainty include hydro conditions, demand, generating units' failures, and fuel prices. This paper presents a model to represent a medium-term operation of the electrical market that introduces this uncertainty in the formulation in a natural and practical way. Utilities are explicitly considered to be maximizing their expected profits, and biddings are represented by means of a conjectural variation. Market equilibrium conditions are introduced by means of the optimality conditions of a problem, which has a structure that strongly resembles classical optimization of hydrothermal coordination. A tree-based representation to include stochastic variables and a model based on it are introduced. This approach to market representation provides three main advantages: Robust decisions are obtained; technical constraints are included in the problem in a natural way, additionally obtaining dual information; and large-size problems representing real systems in detail can be addressed.