Vol.:(0123456789) 1 3 Life Cycle Reliability and Safety Engineering https://doi.org/10.1007/s41872-018-00070-z ORIGINAL RESEARCH Stochastic behavior of dissimilar units cold standby system waiting for repair Pramendra Singh Pundir 1  · Rohit Patawa 1 Received: 30 May 2018 / Accepted: 13 December 2018 © Society for Reliability and Safety (SRESA) 2019 Abstract Investigation of the time behavior of repairable systems spans a very large class of stochastic processes. The repetition of the same events connects the theory of reliability with Markov and semi-Markov processes. Exploiting this theory, the present study deals with two repairable dissimilar units’ cold standby system waiting for repair facility after failure of system units. Also, the regenerative point technique has been employed to obtain various reliability measures for the assumed system. Next, a particular case with exponential failures, arbitrary waiting and arbitrary repair rate is simulated followed by conclu- sions in the last section. Keywords MTSF · Profit function · Exponential · Lindley · Availability · Maintenance Mathematics Subject Classification 90B25 · 37A50 · 44A10 · 91B70 1 Introduction The reliability and availability of the systems are the prob- lems of prime importance for the present business world. Redundant strategy is often used by engineers to ensure the reliability and availability of the systems and/or to improve these characteristics of the systems. Thus, a variety of standby systems have been designed and analyzed during the last few decades. The main objective of these studies has been to develop methods and tools for evaluation and to demonstrate the reliability, maintainability, availability, and safety of the systems (or components). But to make the system, cost and time effective, reliability engineering must be integrated with other related factors too like the project activities, support quality assurance and concurrent engi- neering efforts regarding repair maintenance. Here, “Repair maintenance” term is used for the “corrective maintenance” that returns the units into proper working stage after repair maintenance. Study of the stochastic behavior of repairable redundant systems has always been a problem of interest for engineers and reliability practitioners. Ionescu and Limnios (1999) defined regeneration cycle as a part of trajectory of a random process between two neighboring regenerative times of the process. In very simple terms, this random time is a hitting time of some fixed point where the sequence of such random times is made not to have limits in finite part of the time axis. This idea helps the practitioners a lot in deciding the state transition probabilities, i.e., the probabilities of transition of the system from one state to another. Various authors [Srinivasan and Gopalan (1973), Subra- manian et al. (1976), Cao and Wu (1989), Zhang and Wang (2007), Eryilmaz and Tank (2012), Manglik & Ram (2013), Pundir et al. (2018)] have analyzed the system models of two and more units by considering different repair policies assuming that a repairman always remains available at the system for the repairing of the failed unit and even when both the units are in normal mode. Recently, Liu et al. (2015) studied a cold standby repairable system with working vaca- tions for repairmen and vacation interruptions as needed. Jia et al. (2016) introduced the active switching policy in a two- unit standby system, in which the standby unit is activated at either a pre-fixed time or at the failure time of the active unit. Yang and Tsao (2018) analyzed the standby system under the assumptions of working vacations and the retrial of failed * Rohit Patawa rohitpatawa@gmail.com Pramendra Singh Pundir pspundir@gmail.com 1 Department of Statistics, University of Allahabad, Allahabad, India