Kyle Siegrist

Reliability of systems with Markov transfer of control, II

Software/hardware systems are considered which can be decomposed into a finite number of modules. It is assumed that control of the system is transferred among the modules according to a Markov process. Each module has an associated reliability which gives the probability that the module will operate correctly when called and will transfer control successfully when finished. The system will eventually either fail or complete its task successfully and enter a terminal state. The reliability of the system is studied in terms of the module reliabilities and the transition probabilities. Improved methods of predicting system reliability, allocating module reliability, and determining module sensitivity are developed. Special branching and sequential systems are studied in detail. >

On the nonexistence of uniformly optimal graphs for pair-connected reliability

We consider probabilistic graphs G = (V, E) in which each edge xy ∈ E fails independently with probability q. The reliability measure studied is pair-connectivity, the expected number of pairs of connected vertices. We examine how the coefficients of the pair-connected reliability polynomial are determined by the subgraph structure of G, and we use these results to show that in most cases there does not exist a uniformly optimal n-vertex, m-edge graph.

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ∞) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.

Expected Value Expansions in Random Subgraphs with Applications to Network Reliability

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

On uniformly optimally reliable graphs for pair‐connected reliability with vertex failures

Let G be a probabilistic (n,m) graph in which each vertex exists independently with fixed probability p, 0 < p < 1. Pair-connected reliability of G, denoted PCv(G;p), is the expected number of connected pairs of vertices in G. An (n,m) graph G is uniformly optimally reliable if PCv(G;p) ≧ PCv(H;p) for all p, 0 < p < 1, over all (n,m) graphs H. It is shown that there does not exist a uniformly optimally reliable (n,m) graph whenever n ≦ m < ∼2n2/9. However, such graphs do exist for some other values of m. In particular, it is established that every complete k-partite pseudoregular graph on n vertices, 2 ≦ k < n, is uniformly optimally reliable.

The optimal unicyclic graphs for pair-connected reliability

Abstract We consider the standard network reliability model in which each edge of a graph fails, independently of all others, with probability q = 1 − p (0 ≤ p ≤ 1). The pair-connected reliability of the graph is the expected number of pairs of vertices that remain connected after the edge failures. The optimal graphs for pair-connected reliability in the class of unicyclic graphs (connected ( n , n ) graphs) are completely characterized. The limiting behavior of the intervals of optimality are studied as n → ∞.

Random Finite Subsets With Exponential Distributions

Let S denote the collection of all finite subsets of **. We define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural home for probability distributions with exponential properties, such as the memoryless and constant rate properties. We show that there are no exponential distributions on S, but that S can be partitioned into subsemigroups, each of which supports a one-parameter family of exponential distributions. We then find the distribution on S that is closest to exponential, in a certain sense. This work might have applications to the problem of selecting a finite sample from a countably infinite population in the most random way.

Exponential and gamma distributions on positive semigroups, with applications to Dirichlet distributions

In this paper, we give new characterizations of exponential distributions on positive semigroups and we study the corresponding gamma distributions that govern semigroup products of independent and identically distributed exponential variables. We apply these results to the positive integers under multiplication to obtain new interpretations of results on Dirichlet distributions.

A Debugging Model with Independent Flaws and Random Masking

A system with a Poisson-distributed number of flaws is undergoing a sequence of trials designed to detect and remove the flaws. On each trial, independently and with probability q, masking occurs that blocks the detection of flaws on that trial. If masking does not occur on a trial then each flaw in the system, independently, is detected with probability p on that trial. Each flaw detected on a trial, independently, is removed with probability r before the next trial. This note derives distributions and measures of quality for this problem.

Estimation and optimal stopping in a debugging model with masking

A system has an irremovable failure source and a number of removable flaws. The system undergoes a sequence of trials designed to detect and remove the flaws. On each trial, the irremovable failure source may cause failure which in turn may block the detection of any flaws. If not, then each flaw in the system, independently, is detected on that trial with a certain probability and each detected flaw, independently, is removed from the system with a certain probability before the next trial. Distributions of the outcomes of the trials are obtained. Point estimates of the parameters, based on accumulated trial data, are given. Assuming that certain costs are associated with trials, optimal stopping rules and a cost-benefit analysis are given. INHERENT FAILURE; DESIGN FAILURE; MAXIMUM LIKELIHOOD ESTIMATOR