Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. In this article we present a simple Laplace transform inversion algorithm that can compute the desired function values for a much larger class of Laplace transforms than the ones that can be inverted with the known methods in the literature. The algorithm can invert Laplace transforms of functions with discontinuities and singularities, even if we do not know the location of these discontinuities and singularities a priori. The algorithm only needs numerical values of the Laplace transform, is extremely fast, and the results are of almost machine precision. We also present a two-dimensional variant of the Laplace transform inversion algorithm. We illustrate the accuracy and robustness of the algorithms with various numerical examples.

This article deals with estimations of probabilities of rare events using fast simulation based on the splitting method. In this technique, the sample paths are split into multiple copies at various stages in the simulation. Our aim is to optimize the algorithm and to obtain a precise confidence interval of the estimator using branching processes. The numerical results presented suggest that the method is reasonably efficient.

Motivated by various applications in queuing theory, this article is devoted to the monotonicity and convexity of some functions associated with discrete-time or continuous-time denumerable Markov chains. For the discrete-time case, conditions for the monotonicity and convexity of the functions are obtained by using the properties of stochastic dominance and monotone matrix. For the continuous-time case, by using the uniformization technique, similar results are obtained. As an application, the results are applied to analyze the monotonicity and convexity of functions associated with the queue length of some queuing systems.

Suppose that there are n families with children attending a certain school and that the number of children in these families are independent and identically distributed random variables, each with probability mass function P{X = j} = pj, j ≥ 1, with finite mean μ = [sum ]j≥1jpj. If a child is selected at random from the school and XI is the number of children in the family to which the child belongs, it is known that limn→∞P{XI = j} = jpj /μ,j ≥ 1. Here, asymptotic expansions for P{XI = j} are developed under the condition E|X|3 < ∞.

Compared to the known univariate distributions for continuous data, there are relatively few available for discrete data. In this article, we derive a collection of 16 flexible discrete distributions by means of conditional Poisson processes. The calculations involve the use of several special functions and their properties.

Consider a discrete-time insurance risk model with risky investments. Under the assumption that the loss distribution belongs to a certain subclass of the subexponential class, Tang and Tsitsiashvili (Stochastic Processes and Their Applications 108(2): 299–325 (2003)) established a precise estimate for the finite time ruin probability. This article extends the result both to the whole subexponential class and to a nonstandard case with associated discount factors.

This article presents two methods for predicting weather-related overhead distribution feeder failures. The first model is based on linear regression, which uses a regression function to determine the correlation between the weather factors and overhead feeder failures. The second method is based on a one-layer Bayesian network, which uses conditional probabilities to model the correlation. Both methods are discussed and followed by tests to assess their performance. The results obtained using these methods are discussed and compared.

This paper valuates generation assets within deregulated electricity markets. A new framework for modeling electricity markets with a Markov chain model is proposed. The Markov chain model captures the fundamental economic forces underlying the electricity markets such as demand on electricity and supplied online generation capacity. Based on this new model, a real option analysis is adopted to valuate generation assets. The Markov chain model is combined with a binomial tree to approximate the stochastic movement of prices on both electric energy and ancillary services, which are driven by the market forces. A detailed example is presented. This method is shown to provide optimal operation policies and market values of generation assets. This method also provides means to analyze the impacts of demand growth patterns, competition strategies of competitors, and other key economic forces.

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 control and operation of an electric power system is based on the ability to determine the state of the system in real time. State estimation (SE) has been introduced in the 1960s to achieve this objective. The initial implementation was based on single-phase measurements and a power system model that was assumed to operate under single-frequency, balanced conditions, and a symmetric system model. These assumptions are still prevalent today. The single-frequency, balanced, and symmetric system assumptions have simplified the implementation but have generated practical problems. The experience is that the SE problem does not have 100% performance; that is, there are cases and time periods for which the SE algorithm will not converge. There are practical and theoretical reasons for this and they are explained in the paper. Recent mergers and mandated regional transmission organizations (RTOs) as well as recent announcements for the formation of mega-RTOs will result in the application of the SE in systems of unprecedented size. We believe that these practical and theoretical issues will become of greater importance. There are scientists who believe that the SE problem is scalable, meaning that it will work for the mega-RTOs the same way that it performs now for medium–large systems. There are scientists who believe that this is not true. The fact is that no one has investigated the problem, let alone performed numerical experiments to prove or disprove any claims. This paper identifies a number of issues relative to the SE of mega-RTOs and provides some preliminary results from numerical experiments for the relation between the SE algorithm performance and the power system size.

This paper investigates the effects of measurement and parameter errors on the calculation of power transfer distribution factors (PTDFs). Calculation of PTDFs depends mainly on two factors: the operating conditions and the network topology. Both of these factors change in real time and are monitored through the use of a state estimator. The role of the state estimator in providing not only the state information but also the real-time network model is shown to influence power market operations via PTDF-based decisions such as congestion management and pricing. The IEEE 14 bus test system is used for the illustration of these influences.

Optimal levels of preventive maintenance performed on any system ensures cost-effective and reliable operation of the system. In this paper a component with deterioration and random failure is modeled using Markov processes while incorporating the concept of minor and major preventive maintenance. The optimal mean times to preventive maintenance (both minor and major) of the component is determined by maximizing its availability with respect to mean time to preventive maintenance. Mathematical optimization programs Maple 7 and Lingo 7 are used to find the optimal solution, which is illustrated using a numerical example. Further, an optimal maintenance policy is obtained using Markov Decision Processes (MDPs). Linear Programming (LP) is utilized to implement the MDP problem.