The Lagrange-Sylvester interpolation polynomial approach provides a simple, eigenvector-free representation for finite diagonalizable matrices. This paper discusses the Lagrange-Sylvester methodology and applies it to skip free to the right Markov chains. It leads to relatively simple, eigenvalue-based expressions for first passage time distributions and transition probabilities.

In this work, the discrete event systems called Stochastic Marked Graphs (SMGs) are investigated from a stability point of view. Being a special class of Timed Petri Nets with stochastic firing times, they are studied under general assumptions of stationarity and ergodicity of the firing times and ergodicity of flows of their free strongly connected components. The values of the flows of tokens in an SMG are specified as functions of the intrinsic rates of its free strongly connected components, and various stability issues are discussed.

We consider tandem queueing systems with a general arrival process and exponential service distribution. The queueing system consists of several stations with finite intermediate buffer capacity between the stations. We address the problem of determining the optimal arrangement for the stations. We find that considering the last two stations, the departure process is stochastically faster if the slower station is last. Our results are consistent with the “bowl shape” phenomenon that has been observed in serial queueing systems with zero buffer capacity.

A communication system of n stations is studied. Poisson arrival streams of messages enter the stations and require transmission to a destination station. Each station is equipped with k channels and has a waiting room for messages. For transmission of a message, one channel is needed both at the arrival and at the destination station. The transmission time is exponentially distributed. Messages are lost (possibly after a waiting time) if, at the destination station, all channels are busy. The queueing network model is of the nonproduct type. Sufficient conditions for the existence of an equilibrium distribution are given. For determining the losses at the n stations, a heuristical approximation that requires us to solve a system of nonlinear equations is developed. For this, an efficient iterative numerical algorithm is given. Simulation studies show that the approximation is accurate enough for larger n.

Sensitivity analysis is an integral part of virtually every study of system reliability. This paper describes a Monte Carlo sampling plan for estimating this sensitivity in system reliability to changes in component reliabilities. The unique feature of the approach is that sample data collected on K independent replications using a specified component reliability vector p are transformed by an importance function into unbiased estimates of system reliability for each component reliability vector q in a set of vectors Q. Moreover, this importance function together with available prior information about the given system enables one to produce estimates that require considerably less computing time to achieve a specified accuracy for all |Q| reliability estimates than a set of |Q| crude Monte Carlo sampling experiments would require to estimate each of the |Q| system reliabilities separately. As the number of components in the system grows, the relative efficiency continues to favor the proposed method.

The paper shows the intimate relationship between the proposal and the method of control variates. Next, it relates the proposal to the estimation of coefficients in a reliability polynomial and indicates how this concept can be used to improve computing efficiency in certain cases. It also describes a procedure that determines the p vector, to be used in the sampling experiment, that minimizes a bound on the worst-case variance. The paper also derives individual and simultaneous confidence intervals that hold for every fixed sample size K. An example illustrates how the proposal works in an s-t connectedness problem.

Since the introduction of the concept of coherent systems and the description of the reliability of such systems in terms of the reliabilities of the components, the concept of importance of a component has created a new and fruitful area of research. Two distinct concepts of importance can be found in the literature. We take the view that the importance of a component or a module that is part of a system can be derived directly from the role of the component or the module in the failure of the system. Here again, it is possible that there will be several definitions of role. In this paper we define the role of a module (or component) to be the probability that the module is among all the modules (or components) that failed at the time of system failure. The role of a module depends on the structure of the system in terms of the modules, the structure of the module in terms of its components and the distribution of lifetimes of the components. In this paper we study the role of a module under several structures and distributions for lifetimes. We establish various monotonicity properties and indicate applications of these properties to optimal allocation. Another quantity that describes the nature of the components in sustaining the system is the number of components that fail at the time of the failure of the system. We establish monotonicity properties for the expected number of failed components and also indicate applications to optimal allocation.

Probabilistic influence diagrams are a useful stochastic modeling tool. To calculate probabilities of interest relative to a probabilistic influence diagram efficiently, it will be helpful for us to use an associated decomposable-directed graph. We first explore and discuss some graph-theoretic and conditional independence properties of decomposable probabilistic influence diagrams. These properties are helpful in providing an efficient algorithm for obtaining a posterior decomposable probabilistic influence diagram given the state of one or more observed nodes. The connection between Shachter's “sequential creation of conditionally barren nodes” concept and Lauritzen and Spiegeihalter's “moralization and triangulation” algorithm for calculating probabilities relative to a probabilistic influence diagram is made explicit. We also discuss how to use wisely the concepts of “sequential creation of conditionally barren nodes” and “merging nodes” together with the graph-theoretic properties of decomposable directed graphs to compute probabilities relative to probabilistic influence diagrams.

In this paper we consider stochastic flow-shop scheduling with reference to certain lateness-related performance measures. We show that for various assumptions on the distribution of job-processing times of a flow shop, certain scheduling policies following the stochastic analogy of the Earliest Due Date (EDD) rule yield optimal results.