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

• Published in: European journal of operational research 2011-10, Vol.214 (2), p.298-307
• Main Authors: DELIA, Montoro-Cazorla, RAFAEL, Pérez-Ocón
• Format: Article
• Language: English
• Subjects: Applied sciences; Exact sciences and technology; Failure; Failure analysis; Fatal; Fracture mechanics (crack, fatigue, damage...); Friction, wear, lubrication; Fundamental areas of phenomenology (including applications); Machine components; Markov analysis; Markovian arrival process; Mathematical analysis; Mathematical models; Mathematics; Mechanical engineering. Machine design; Operational research and scientific management; Operational research. Management science; Phase-type distribution; Physics; Probability and statistics; Probability theory and stochastic processes; Random numbers; Reliability theory. Replacement problems; Repair; Repair & maintenance; Replacement; Sciences and techniques of general use; Shock and wear model; Shock and wear model Repair Replacement Markovian arrival process Phase-type distribution; Solid mechanics; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stands; Structural and continuum mechanics; Studies; Wear; Wear resistance
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2011.04.016

► We model two shocks and wear systems standing a finite number of shocks, random and fixed. ► Shocks and wear failures are repaired following severe and normal repair, respectively. ► Shocks arrive following a Markovian arrival process and lifetime following a phase-type distribution. ► Performance measures are calculated for these systems. ► Renewal processes associated to the replacements for fatal failures are constructed. 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.