International Journal of Computer Applications (0975 – 8887) Volume 40– No.3, February 2012 19 Profit Analysis of a System of Non-identical Units with Degradation and Replacement Jitender Kumar School of Mathematics and Computer Applications, Thapar University, Patiala (India) M. S. Kadyan Department of Statistics & O.R., Kurukshetra University, Kurukshetra (India) ABSTRACT This paper presents the profit analysis of a system of two non- identical units in which one unit is original which is initially operative and the other is duplicate kept in cold standby. The units may fail completely directly from normal mode. There is a single server who visits the system immediately when required. The original unit undergoes for repair upon failure while only replacement of the duplicate unit is made by similar new one. The original unit does not work as new after repair and so called degraded unit. The system is considered in up-state if any one of new/duplicate/degraded unit is operative. The server inspects the degraded unit at its failure to see the feasibility of repair. The failure time of the units are exponentially distributed whereas the distributions of inspection time, replacement time of the duplicate unit and repair time of the original/duplicate/degraded unit are taken as arbitrary with different probability density functions. Some reliability characteristics of the system model are evaluated using semi-Markov process and regenerative point technique. The numerical results for a particular case are also obtained to depict the behavior of Mean Time to System Failure (MTSF), availability and profit function graphically. Key Words: System of Non-Identical Units, Inspection, Degradation, Replacement and Profit Analysis. 2000 Mathematics Subject Classification: Primary 90 B25 and Secondary 60K10. 1. INTRODUCTION In many industrial processes the provision of a standby unit is necessary for very high reliability. But it is not always possible to keep a high cost unit on standby. Therefore, to improve the reliability of a system, an ordinary unit (called duplicate) unit may be kept as spare which is capable of performing the same nominal system function but with different degree of reliability and desirability. An example, of this situation is a system comprised of an electrical device and a battery operated device. The battery device is switched on as and when the electrical device is failed. The reliability models of standby systems have widely been studied by the engineers and scholars including Gopalan and Naidu (1984), Chung (1987) and Singh and Mishra (1994) under the assumptions that i) The unit works as new after repair. ii) Repair of the unit is always feasible. Infect, these assumptions cannot be imposed always on every system. Because the working capacity and efficiency of a repaired unit depends on the skilled knowledge of the repair facility used. In case of being repaired by an ordinary server, the chances of its failure may be high and thus such a unit may be considered as degraded. Malik et al. (2008) analyzed a system with inspection considering the concept of degradation of the unit after repair. In view of the above facts and observations, we in this paper analyzed a system of non-identical units- one unit is original which is initially operative and other is its duplicate kept in cold standby. There is a single server who visits the server immediately when required. The original unit undergoes for repair upon failure while the duplicate unit is replaced by similar new one. The original unit does not work as new after repair and so called degraded unit. The system is considered in up-state if any one of new/duplicate/degraded unit is operative. The server inspects the degraded unit at its failure to see the feasibility of repair. If repair of the degraded unit is not feasible, it is replaced by new one similar to the original unit in negligible time. The failure time of the units are exponentially distributed whereas the distributions of inspection time, replacement time of the duplicate unit and repair time of the original/duplicate/degraded unit are taken as arbitrary with different probability density functions. The random variables are mutually independent and uncorrelated. The expressions for some reliability characteristics such as mean sojourn times, Mean Time to System Failure, availability, busy period of the server, expected number of visits by the server and profit function are derived using semi- Markov process and regenerative point technique. The numerical results considering a particular case are also obtained to depict the graphically behavior of Mean Time to System Failure (MTSF), availability and profit of the system model. 2. NOTATIONS E :Set of regenerative states No :The unit is new and operative DUo :The unit is duplicate and operative Do :The unit is degraded and operative Ncs /DUcs/Dcs :The new/duplicate/degraded unit in cold standby q p :Probability that repair of degraded unit is feasible/not feasible 2 1 / /    :Constant failure rate of new/duplicate /degraded unit ) t ( G ) t ( g , g 1 (t)/G 1 (t) :pdf/cdf of repair time for new/degraded unit