Analysis of k-Out-of-N-Systems with Different Units under Simultaneous Failures: A Matrix-Analytic Approach

An N-system with different units submitted to shock and wear is studied. The shocks cause damage and, eventually, simultaneous failures of several units. The units can also fail due to internal failures. At random times, the system is inspected, and the down units are simultaneously replaced by identical ones. The arrival of shocks is governed by a Markovian arrival process. The operational times and the interarrival times between inspections follow phase-type distributions. The generator of the multidimensional Markov process modeling the system is constructed. This is performed introducing indicator functions for the different transition rates among the units using the algorithm of Kronecker. This is a general Markov process that can be applied for modeling different reliability systems depending on the structure of the units and how the systems operate. The general model is applied to the study of k-out-of-N systems, calculating the main performance measures. A practical example is presented showing the approximation of the model to a system with units following different Weibull distributions

Optimizing Costs in a Reliability System under Markovian Arrival of Failures and Reposition by K-Policy Inspection

This paper presents an N warm standby system under shocks and inspections governed by Markovian arrival processes. The inspections detect the number of down units, and their replacement is carried out if there are a minimum K of failed units. This is a policy of the type (K, N) used in inventory theory. The study is performed via the up and down periods of the system (cycle); the distribution of these random times and the expected costs for each period comprising the cycle are determined on the basis of individual costs due to maintenance actions (per inspection and replacement of every unit) and others due to operation or inactivity of the system, per time unit. Intermediate addressed calculus are the distributions of the number of inspections by cycle and the expected cost involving every inspection, depending on the number of replaced units. The system is studied in transient and stationary regimes, and some reliability measures of interest and the cost rate are calculated. An optimization of these quantities is performed in terms of the number K in a numerical example. This general model extends to many others in the literature, and, by using the matrix-analytic method, compact and algorithmic expressions are achieved, facilitating its potential application

Two shock and wear systems under repair standing a finite number of shocks

A shock and wear system standing a finite number of shocks and subject to two types of repairs is considered. The failure of the system can be due to wear or to a fatal shock. Associated to these failures there are two repair types: normal and severe. Repairs are as good as new. The shocks arrive following a Markovian arrival process, and the lifetime of the system follows a continuous phase-type distribution. The repair times follow different continuous phase-type distributions, depending on the type of failure. Under these assumptions, two systems are studied, depending on the finite number of shocks that the system can stand before a fatal failure that can be random or fixed. In the first case, the number of shocks is governed by a discrete phase-type distribution. After a finite (random or fixed) number of non-fatal shocks the system is repaired (severe repair). The repair due to wear is a normal repair. For these systems, general Markov models are constructed and the following elements are studied: the stationary probability vector; the transient rate of occurrence of failures; the renewal process associated to the repairs, including the distribution of the period between replacements and the number of non-fatal shocks in this period. Special cases of the model with random number of shocks are presented. An application illustrating the numerical calculations is given. The systems are studied in such a way that several particular cases can be deduced from the general ones straightaway. We apply the matrix-analytic methods for studying these models showing their versatility.Shock and wear model Repair Replacement Markovian arrival process Phase-type distribution

MAINTENANCE OF SYSTEMS BY MEANS OF REPLACEMENTS AND REPAIRS: THE CASE OF PHASE-TYPE DISTRIBUTIONS

We present two models for studying a system maintained by means of imperfect repairs before a replacement or a perfect repair is allowed. The operational and repair times follow phase-type distributions. Imperfect repair means that successive operational times decrease and successive repair times increase. Under these assumptions, models that govern the systems are Markov processes, whose structures are determined, and several performance measures are calculated in transient and stationary regime. These models extend other previously studied in the literature. The incorporation of phase-type distributions allows to apply the model to many other distributions. A numerical example illustrates the calculations and allows a comparison of the results.Reliability, Markov process, phase-type distribution, imperfect repair, replacement, Primary 90B25, Secondary 62N05

An LDQBD process under degradation, inspection, and two types of repair

A warm standby n-system with operational and repair times following phase-type distributions is considered. The online unit goes through degradating levels, determined by inspections. Two types of repairs are performed, preventive and corrective, depending on the degradation level. The standby units undergo corrective repair. This systems is governed by a level-dependent-quasi-birth-and-death proces (LDQBD process), whose generator is constructed. The availability, rate of occurrence of failures, and other quantities of interest are calculated. A numerical example including an optimization problem and illustrating the calculations is presented. This system extend other previously studied in the literature.

A Longitudinal Study of the Bladder Cancer Applying a State-Space Model with Non-Exponential Staying Time in States

A longitudinal study for 847 bladder cancer patients for a period of fifteen years is presented. After the first surgery, the patients undergo successive ones (recurrences). A statemodel is selected for analyzing the evolution of the cancer, based on the distribution of the times between recurrences. These times do not follow exponential distributions, and are approximated by phase-type distributions. Under these conditions, a multidimensional Markov process governs the evolution of the disease. The survival probability and mean times in the different states (levels) of the disease are calculated empirically and also by applying the Markov model, the comparison of the results indicate that the model is well-fitted to the data to an acceptable significance level of 0.05. Two sub-cohorts are well-differenced: those reaching progression (the bladder is removed) and those that do not. These two cases are separately studied and performance measures calculated, and the comparison reveals details about the characteristics of the patients in these groups