American Journal of Theoretical and Applied Statistics 2015; 4(4): 277-290 Published online June 29, 2015 (http://www.sciencepublishinggroup.com/j/ajtas) doi: 10.11648/j.ajtas.20150404.18 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online) Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes Medhat Ahmed El-Damcese 1 , Naglaa Hassan El-Sodany 2 1 Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt 2 National Accounts Department, Central Agency for Public Mobilization and Statistics, Cairo, Egypt Email address: meldamcese@yahoo.com (M. A. El-Damcese), naglaa_hassan17@yahoo.com (N. H. El-Sodany) To cite this article: Medhat Ahmed El-Damcese, Naglaa Hassan El-Sodany. Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes. American Journal of Theoretical and Applied Statistics. Vol. 4, No. 4, 2015, pp. 277-290. doi: 10.11648/j.ajtas.20150404.18 Abstract: This paper presents the reliability and availability measures of a two non-identical unit cold standby redundant system (unit-1 is operating, and unit-2 is cold standby) using semi-Markov process under discrete parametric Markov-Chain i.e. failure and repair times of a unit and time to PM and PM time are taken as discrete random variables assuming three different modes (normal (N) mode, partial failure (P) mode and total failure (F) mode) of each unit. The unit-1 is sent for preventive maintenance (PM) after its working for a random period of time assuming that the failure and repair times of a unit and time to PM and PM time are taken as discrete random variables having geometric distributions with different parameters. A single repairman is available with the system for PM of unit-1 and repair of both units. The system is considered in up-state if only one or two units are operative or in partial failure (P) mode. After some basic definitions and notations, we obtain various measures of system effectiveness; reliability, availability, mean time to failure, busy period of repairman due to PM of unit-1, busy period of repairman due to repair of unit-1 and unit-2 from total failure, and the expected profit function using regenerative point technique. The mathematical problem thus developed has next been solved numerically and graphically represented by the aid of Maple program. Keywords: Semi-Markov, Discrete-Time, Cold Standby, Reliability, Mean Sojourn Time, Regenerative Point Technique 1. Introduction Many reliability systems can be modeled using semi- Markov process. The idea of a semi-Markov process was proposed by [1].The essential developments of semi-Markov processes theory applications in reliability were proposed by many authors. [2] and [3] presented a comprehensive treatment of semi-Markov processes and their applications to reliability theory. Some concepts of a semi-Markov process theory was presented by [4]. [5] discussed the basic definitions and theorems from the semi-Markov processes theory and considered the semi-Markov model of the cold standby system with repair. [6] considered the semi-Markov model of multistate system. [7] presented the properties of the reliability function of an object with failure rate modeled by a semi-Markov process applying the renewal equations and obtained the Laplace-Stieltjes transform of the reliability function and its mean time to failure. [8], [9] and [10] studied the semi-Markov processes and their applications in reliability. In this paper, we are interested in the reliability and availability analysis of a two non-identical unit cold standby system with three modes (normal (N) mode, partial failure (P) mode and total failure (F) mode) with preventive maintenance based on discrete-time semi-Markov processes. In cold standby systems, only one component will be working at any given time, the others being standbys and not working. One of the standby components starts working only when the currently working component fails. Standby components do not fail when they are in standby. The total failure of the primary unit results in the cold standby unit being an operative unit and its failure rate becoming nonzero. The system works until all of its components fail. Two-unit cold standby redundant system models have been