Ilya Gertsbakh

Estimation of network reliability using graph evolution models

Monte Carlo techniques for estimating various network reliability characteristics, including terminal connectivity, are developed by assuming that edges are subject to failures with arbitrary probabilities and nodes are absolutely reliable. The core of the approach is introducing network time-evolution processes and using certain graph-theoretic machinery, resulting in a considerable increase in accuracy for Monte Carlo estimates, especially for highly reliable networks. Simulation strategies and numerical results are presented and discussed. >

Minimal Resources for Fixed and Variable Job Schedules

We treat the following problem: There are n jobs with given processing times and an interval for each jobs starting time. Each job must be processed, without interruption, on any one of an unlimited set of identical machines. A machine may process any job, but no more than one job at any point in time. We want to find the starting time of each job such that the number of machines required to process all jobs is minimal. In addition, the assignment of jobs to each machine must be found. If every job has a fixed starting time the interval is a point, the problem is well-known as a special case of Dilworths problem. We term it the fixed job schedule problem FSP. When the job starting times are variable, the problem is referred to as the variable job schedule problem VSP, for which no known exact solution procedure exists. We introduce the problems by reviewing previous solution methods to Dilworths problem. We offer an approximate solution procedure for solving VSP based on the entropy principle of informational smoothing. We then formulate VSP as a pure integer programming problem and provide an exact algorithm. This algorithm examines a sequence of feasibility capacitated transportation problems with job splitting elimination side constraints. Our computational experience demonstrates the utility of the entropy approach.

Parallel time scales and two-dimensional manufacturer and individual customer warranties

We deal with a system whose failures depend on several parallel effects, such as the time in use L and the mileage H. Manufacturer warranties are typically described by a two-dimensional region in the (L,H)-plane. A proper determination of the warranty limits must be based on a two-dimensional distribution of time to failure on this plane. The aim of this paper is to demonstrate the possibility of designing individual warranties for a “;nontypical” customer who has a very low or very high usage rate b=H/L, and to show a simple way to calculate warranty limits by minimizing the lifetime coefficient of variation. The latter is carried out by introducing the “best” combined time scale in the form K=(1−∈)L+∈H which provides the minimal lifetime coefficient of variation.

Multiple time scales and the lifetime coefficient of variation: engineering applications

We consider linear combinations of “natural” timescales and choose the “best” one which provides the minimum coefficient of variation of the lifetime. Our time scale is in fact a generalized Miner time scale because the latter is based on an appropriate weighting of the times spent on low and high level loadings. The suggested modus operandi for finding the“best” time scale has many features in common with the approach suggested by Farewell and Cox (1979) and Oakes (1995) which is devoted to multiple time scales in survival analysis.

The shorter queue problem: A numerical study using the matrix-geometric solution

Abstract We consider a service system with two similar servers in which a customer, on arrival, joins the shorter queue. The state of the system is described by a pair ( i , j ), j ⩾ i , where i and j are the number of customers in the shorter and the longer queue, respectively. The stationary probability vector and several performance characteristics are obtained using the matrix-geometric solution technique.

System state monitoring and lifetime scales. II

Abstract The first part of this paper [Kordonsky, Kh. B. & Gertsbakh, I. B., System state monitoring and lifetime scales—I. Reliab. Engng & System Safety , 47 (1995) 1–14] was devoted to finding a time-scale for system state monitoring. The Best Monitoring Scale (BMS) was defined as a linear combination of several observable ‘principal’ time scales like the operational time, number of cycles, etc. These ‘principal’ time scales were chosen in such a way that they capture the significant dimensions of failure behaviour. In practice, the key issue is the estimation of the BMS. In the first part we suggested an estimation procedure when all observations (in two time-scales) are complete. This is rarely the case in real-life situations where most of observations are censored. The second part is devoted to finding the BMS on the basis of incomplete observations. We consider two types of data: observations up to a certain time or up to the first failure, and observations of a renewable system. We preserve the notation, abbreviations, and the terminology of part I of this paper.

Choice of the best time scale for system reliability analysis

Abstract Several time scales (total operation time, the number of high and load operation cycles, the number of low load operation cycles, their linear combinations, etc.) are compared. Examples of test and field data are analyzed and it is shown that it is possible to find a linear combination of different scales which guarantees a considerable decrease in the coefficient of variation (c.v.) of the system lifetime. It means that, using this ‘optimal’ scale, an accurate reliability prediction can be done for a wide spectrum of loading conditions. A formal model for selecting an optimal (‘load invariant’) time scale is described. In this scale the e-quantiles of lifetime under various loadings are maximally ‘compressed’ in terms of their c.v. ‘Artificial’ time scales based on damage counters are suggested and some aspects of their use are discussed.

Maximum Likelihood Estimation in a Mininum-Type Model with Exponential and Weibull Failure Modes

Abstract The existence and some properties of maximum likelihood estimators (MLEs) are studied for a minimum-type distribution function corresponding to a minimum of two independent random variables having exponential and Weibull distributions. It is shown that if all three parameters are unknown, then there is a path in the parameter space along which the likelihood function (LF) tends to infinity. It is also proved that if the Weibull shape parameter is known, then the LF is concave, the MLEs exist, and they can be found by solving the set of likelihood equations. Properties of the MLEs for this case are illustrated by a Monte Carlo experiment. A sufficient condition for the existence of MLEs is given for the case of known Weibull scale parameter.

Characterization of the Weibull distribution by properties of the Fisher information under type-I censoring

It is proved that within the class of scale parameter families of lifetime distributions, subject to certain regularity conditions, the Weibull family is characterized by an information version of the classical lack-of-memory property or by a related property of the Fisher information in type-I censored experiments.

Estimation in a random censoring model with incomplete information: exponential lifetime distribution

Analysis of methods and simulation results for estimating the exponential mean lifetime in a random-censoring model with incomplete information are presented. The instant of an items failure is observed if it occurs before a randomly chosen inspection time and the failure is signaled. Otherwise, the experiment is terminated at the instant of inspection during which the true state of the item is discovered. The maximum-likelihood method (MLM) is used to obtain point and interval estimates for item mean lifetime, for the exponential model. It is demonstrated, using Monte Carlo simulation, that the MLM provides positively biased estimates for the mean lifetime and that the large-sample approximation to the log-likelihood ratio produces accurate confidence intervals. The quality of the estimates is slightly influenced by the value of the probability of failure to signal. Properties of the Fisher information in the censored sample are investigated theoretically and numerically. >