This study investigates two random threshold shock models for a repairable deteriorating system with nonnegligible maintenance times, with and without a spare via a phase-type geometric process. The system fails whenever the intershock arrival time is less than a random threshold. The provision of stochastic lead time is incorporated in Model II so that an ordering policy N−1 and a replacement policy N based on the number of failures of the system are also considered. An explicit expression of the average cost rate is derived for both models and the optimal replacement policy N* is obtained by minimizing the long-run average cost rate analytically. The numerical illustrations and sensitivity analysis provided therein conform to the observations made in the study. KEYWORDS phase-type geometric process, random threshold, stochastic lead time Appl Stochastic Models Bus Ind. 2018;34:407-422. wileyonlinelibrary.com/journal/asmb 407 408SARADA AND SHENBAGAM this model, Lam studied two kinds of replacement policies, ie, one based on the working age T of the system (T-policy) and the other based on the failure number N of the system (N-policy). The objective is to choose the optimal replacement policies T* and N*, respectively, such that the long-run average cost per unit time is minimized. Other works on the geometric process model in maintenance analysis include those of Lam, 12 Stanley, 13 Pérez-Ocón and Torres-Castro, 14 Wang and Zhang, 15 Liang et al, 16 Zong et al, 17 and Cheng et al 18Under the assumption of geometric repair times, Lam and Zhang 9 and Tang and Lam 19 studied the -shock model, in which shocks arrive according to a Poisson process and renewal process, respectively, whereas Lam 20 introduced a geometric process -Shock model in a repairable deteriorating system and obtained an optimal replacement policy (N*). Liang et al 16 spelt an optimal replacement policy N* in a geometric process model with a -shock for a deteriorating and improving system. Retaining the geometric repair times modeling and assuming the shock arrival process to be a Poisson process with the threshold value to be an exponential random variable, Zong et al 17 dealt with the determination of the optimal replacement policy N* to minimize the long-run average cost rate. However, in realistic systems, phase-type distribution, introduced by Neuts, 21,22 plays an important role and has been applied in telecommunication, reliability, queueing, and inventory theory. It is generally assumed that the system replacement is immediate on failure, which may not be true always, as it would be expensive to maintain a spare, consequently hiking the system operating cost (eg, the work of Yu et al 23,24 ).From a thorough review of the existing literary works done, it is observed that the -shock model with a random threshold through stochastic lead time via phase-type geometric processes has not been studied from the viewpoint of deteriorating systems. Thus, in an attempt to establish the importance of the simplified computational implementation of phas...

Purpose Technological advancements and growing complexity of many real-time systems, namely, communication, transportation, defense systems, etc., necessitate the importance to adopt a well-planned repair process such as phase type quasi-renewal process contributing to an improved system performance. Further, in an attempt to boost the role of maintenance as a financial benefactor, repairman’s multiple vacation policy is incorporated. Also, the significance of the degree of repair is illustrated while indicating the suitability of the matrix-analytic approach via the phase type quasi-renewal operating/repair times in reliability. The paper aims to discuss these issues. Design/methodology/approach The optimal replacement policy is obtained by employing the matrix-analytic method and minimum average cost rate. Findings The considered models make a significant contribution towards establishing that the matrix-analytic method, using the phase type quasi-renewal process, aids in reducing the computations and also fills the gap in the literature in the study of deteriorating systems. Availability and rate of occurrence of failures are evaluated in transient and steady-state regime. Originality/value This model differs from the existing models, in that, a repairman’s multiple vacation, delayed repair time and representation of the failure occurrence by a mixed Poisson process have been incorporated into the analysis. Also, time-dependent case and N-policy have been adopted to explore the optimality issues using phase type quasi-renewal process analytically. The numerical illustrations warrant that the maintenance policy proposed in this paper produces a considerably lower cost.

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This study investigates two warranty models: fixed and extended warranty models with inspections, for a repairable deteriorating system. An alternating phase type quasi-renewal process is employed to model the operating and repair times. Failures occur at random instants of time. The condition of the system after repair is not as good as new. While the fixed warranty model is examined using the expected cost rate and a bi-criterion cost, an explicit expression for the long-run average cost rate is obtained for the extended warranty model by adopting a [Formula: see text]-policy. In addition, the extended warranty model is examined to include the downtime cost in the analysis while avoiding inspections. Numerical illustrations provided therein conform to the observations made in the study.

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