A consecutive−k−out−of−n:F system is an n−vertex graph where the system fails if and only if some k consecutive vertices all fail. Assuming that the n vertices have, independently, the respective failure probabilities q1,…, qn, the problem is to find, subject to πqi = Q (a constant), a set of q1,…, qn so as to maximize the system reliability. In the case that the graph is a circle (or cycle), Chang and Hwang [1] conjectured that q1 =… = qn, = Q1/n is optimal if n and k are relatively prime. In this paper, it is shown that the conjecture is true for sufficiently small Q. It is also shown by counterexample that the conjecture is not true in general.

We show that natural estimators occurring in certain simulation settings have convergence rates less than the canonical rate usually associated with simulation. These natural estimators use replication schemes that attenuate bias. For some important examples, we find alternative estimators that converge at the canonical rate. The implications of these asymptotic comparisons for choosing good strategies when the computer-time budget is modest are discussed.

An important problem in computer communication systems is the design of finite buffers such that a small overflow probability is achieved. For the case of batch input and general service times, we derive an asymptotically exponential expansion for the overflow probability when the batch-arrival process is Poissonian. Next, the results are extended to handle applications with periodic service opportunities and random interruptions in service. A convenient model to handle these types of applications is the versatile queueing model with exceptional first services.

Simulated annealing is a probabilistic algorithm for approximately solving large combinatorial optimization problems. The algorithm can mathematically be described as the generation of a series of Markov chains, in which each Markov chain can be viewed as the outcome of a random experiment with unknown parameters (the probability of sampling a cost function value). Assuming a probability distribution on the values of the unknown parameters (the prior distribution) and given the sequence of configurations resulting from the generation of a Markov chain, we use Bayes's theorem to derive the posterior distribution on the values of the parameters. Numerical experiments are described which show that the posterior distribution can be used to predict accurately the behavior of the algorithm corresponding to the next Markov chain. This information is also used to derive optimal rules for choosing some of the parameters governing the convergence of the algorithm.

An estimator, based on a simulation, is given for the expected number of events by time t of a renewal process. The estimator is obtained by combining the techniques of control variates and conditional expectation.

Consider the standard network reliability model in which each edge of a given (n, m)-graph G is deleted, independently of all others, with probability q = 1– p (0 <p < 1). The pair-connectivity random variable X is defined to be the number of connected pairs of vertices that remain in G. The mean of X has been proposed as a measure of reliability for failure-prone communications networks in which the edge deletions correspond to failures of the communications links. We consider deviations from the mean, the law of large numbers, and the central limit theorem for X as n → ∞. Some explicit results are obtained when G is a tree using martingale difference sequences. Stars and paths are treated in detail.

The manipulations and basic results of stochastic decision theory are introduced. The manipulations of idempotence, transposition, and repetition, introduced for deterministic decision trees, can be used to manipulate stochastic trees. However, there are two major differences. First, in order to obtain a complete set of manipulations it is necessary to introduce an additional rule called indifference. Second, these identities must be treated as rules of inference. Not all the rules can be soundly applied in both directions; in particular, idempotence is a one-way rule.

A manipulation of a stochastic decision tree not only alters the structure of the tree, but also the probability distributions associated with the tree. This allows probability calculation to be viewed as structural manipulation. In particular, a retrieval corresponds to a conditional probability calculation. The algorithm for doing this calculation has, therefore, many applications. For example, the solution to the classical state-estimation problem and the retrieval of information from probabilistic or uncertain knowledge bases may both be viewed as an application of this algorithm.

The main result of this paper is that these manipulations are complete and sound. In order to prove this result, it is necessary to have a semantic setting for these theories. The setting chosen is the category of description spaces which is a generalization of the category of bounded measure spaces with maps which do not increase measure. The proof of this result exploits the retrieval properties of stochastic terms and its relationship to conditional probability calculations in the models.

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

To find zeros or locate maximum values of a regression function with noisy measurements, a commonly used algorithm is the RM or KW procedure. In various applications, the dimensionality of the problems involved might be quite large. As a result, enormous memory space and extensive computation time may be needed. Motivated by the recent progress in stochastic approximation methods for decentralized and distributed computing, a parallel stochastic approximation algorithm is developed in this paper. The essence is to take advantage of state-space decompositions, and to exploit the opportunities provided by parallel processing and asynchronous communication. In lieu of utilizing a single processor as in the classical cases, a number of parallel processors are employed to solve the underlying problem in a cooperative way. First, the large dimensional vector is partitioned into a number of subvectors with relatively small dimension, then each of the subvectors is assigned to one of the processors. The processors compute and communicate in an asynchronous manner and at random times. Under rather weak conditions, the global convergence of the parallel algorithm is obtained via the methods of randomly varying truncations.

The standard single-machine scheduling problem is modified to take into account unsuccessful job completions. We use a result due to Nash to analyze problems in which some jobs are alternative to one another in the sense that only one of a set of alternative jobs need be completed successfully. Conditions are proposed under which a nonpreemptive strategy is optimal for processing such a system.

We wish to schedule heterogeneous jobs on a single machine to stochastically maximize the number of jobs that are correctly completed before a random due date. Jobs arrive according to an arbitrary arrival process. Job i has probability ci, that it will be correctly completed, and a processing time that has distribution Fi. Jobs that are incorrectly processed cannot be reprocessed, and jobs that are not completed by the due date are assumed to be incorrectly processed. The machine is subject to breakdowns and repairs. Both premptive and non-preemptive policies are considered. We give conditions under which simple index rules are optimal. This generalizes and extends the results of Derman et al. [3], Pinedo and Rammouz [9], and Buyukkoc et al. [2], who derive policies that maximize the expected number of jobs correctly completed by the due date. We also consider policies that stochastically minimize the number of correct job completions. For single class systems, we give conditions under which FCFS (first-come-first-served) or LAST (least-attained-service-time) service disciplines stochastically minimize or maximize the number in the system.

This paper describes a highly efficient Monte Carlo sampling plan for evaluating the probability that the flow value in a stochastic flow network is greater than or equal to a prespecified level d. A stochastic flow network can characterize communication, transportation, and water or oil distribution systems. The paper first derives lower and upper bounds on the probability of interest and then describes how one can concentrate sampling in a specialized region of the arc capacity state space to increase the statistical efficiency of the resulting estimate. The paper also gives expressions for worst-case sample sizes needed to meet specified bounds on variances and coefficients of variation and illustrates the proposed sampling plan with an example.

Two problems are considered: (1) The expected waiting time for the partial sums of i.i.d. positive integer-valued random variables to be a multiple of k for the first time. For aperiodic distributions this expected value is shown to equal k. The periodic case is considered, as well as the waiting time for the partial sums to equal r modulo k. (2) For two independent sequences of partial sums of positive i.i.d. random variables, the expected value of the smallest common partial sum is derived. If both distributions are aperiodic, this expected value is the product of the two means. The periodic case is considered, as well as the case of more than two sequences.

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

For a two-station tandem system with a general arrival process, exponential ervice times, and blocking, we show that the distribution of the departure process does not change when the two stations are interchanged. Blocking here means that for some fixed b≥1, any customer completing service at the first station when b customers are at the second station cannot enter the second queue, and the first station cannot start serving a new customer until a service completion occurs at the second station. This result remains true if arrivals are lost when there are a(a≥0) customers in the system. We also show that when the sum of the service rates is held constant, each departure epoch is stochastically minimized if the two rates are equal. For a and b infinite, our results reduce to those given by Weber [14] and Lehtonen [8]. Our proof is based on a coupling method first used by Lehtonen. The same results hold for a blocking mechanism in which a customer completing its service at the first station must restart service when b customers are present at the second.

As the formal methods of proving correctness of a computer program are still very inadequate, in practice when a new piece of software is developed and all obvious errors are removed, it is tested with different (random) inputs in order to detect the remaining errors and assess its quality. We suppose that whenever the program fails the error causing the failure can be detected and removed correctly. Thus, the quality of the software increases as testing goes on. In this paper, we consider two different models and present the minimum variance unbiased estimators of the expected failure rate of the revised software at any time of testing t, based on the data generated up to that point.

A two-unit parallel system with cold standby, Markovian degradation of the working unit and one repair facility is considered. The calculation of the availability of this system under a control-limit repair policy is discussed. Iterative schemes for computing the availability in case of Erlangian repair-time distributions are presented along with numerical results.

We analyze the optimal preemptive sequencing of n jobs on two machines to minimize expected total flow time. The running times of the jobs are independent samples from the distribution Pr(X = 1) = p, Pr(X = κ + 1) = 1 − p. We verify that the shortest-expected-remaining-processing-time (SERPT) policy, which is optimal for independent and identically distributed (i.i.d.) running times with a monotone hazard-rate distribution, is not optimal for this distribution. However, we prove that if p ≥ 1/κ, then the number of decisions where SERPT and an optimal policy disagree is bounded by a constant independent of n. For p < 1/k, we prove that the expected number of such decisions has a similar bound. In addition, bounds on the expected increase in flow times under SERPT are derived; these bounds are also independent of n.

This paper analyzes a queueing system in which a constant-speed conveyor brings new items for service and carries away served items. The conveyor is a sequence of cells each able to hold at most one item. At each integer time, a new cell appears at the queue's input position. This cell holds an item requiring service with probability a, holds a passerby requiring no service with probability b, and is empty with probability (1– a – b). Service times are integers synchronized with the arrival of cells at the input, and they are geometrically distributed with parameter μ. Items requiring service are placed in an unbounded queue to await service. Served items are put in a second unbounded queue to await replacement on the conveyor in cells at the input position. Two models are considered. In one, a served item can only be placed into a cell that was empty on arrival; in the other, the served item can be placed into a cell that was either empty or contained an item requiring service (in the latter case unloading and loading at the input position can take place in the same time unit). The stationary joint distribution of the numbers of items in the two queues is studied for both models. It is verified that, in general, this distribution does not have a product form. Explicit results are worked out for special cases, e.g., when b = 0, and when all service times are one time unit (μ = 1). It is shown how the analysis of the general problem can be reduced to the solution of a Riemann boundary-value problem.

A continuum structure function (CSF) is a nondecreasing mapping from the unit hypercube to the unit interval. The Kim-Baxter definition of the reliability importance of component i in a CSF at system level α, Ri(α), say, is reviewed. Conditions under which Ri(α) is positive, under which Ri(α) is a continuous function of α, and under which Ri(α) ≥ Rj(α) uniformly in α are presented. A simple algorithm for evaluating Ri(α) is described.