Cost-Benefit Analysis of a Redundant System with Server having Refreshment Facility Subject to Inspection

In this paper two units cold standby system has been discussed with the facility that server inspect the failed unit before repair/replacement of the unit and server may allow to take refreshment whenever needed. The operative unit may fail directly from normal mode and the cold standby unit may be failed owing to remain unused for a longer period of time. There is single server who serves the dual purpose of inspection and repair immediately whenever required. Also, after having refreshment the server may eventually perform the better service efficiently. The time to take refreshment and repair activity follows negative exponential distribution whereas the distributions of unit failure and server failure are taken as arbitrary with different probability density functions. The expressions of various stochastic measures are analyzed in steady state using semi-Markov process and regenerative point technique. The graphs are sketched for arbitrary values of the parameters to delineate the behavior of some important performance measures to check the efficacy of the system model under such situations.


INTRODUCTION
Redundancy is the provision of alternate means or parallel paths in a system for performing the given assignment to the system. Application of redundancy in the system design is found in almost all types of system due to its numerous advantages to improve reliability and availability of a system. Various forms of redundancyactive (hot) redundancy, standby (cold) redundancy, warm redundancy, component redundancy, system redundancy etc can be installed in a system, depending upon their feasibility. The use of a particular approach depends upon many factors such as the operating characteristics of components or systems, weight, size and initial cost. In literature, the stochastic behavior of cold standby system has been widely discussed by many researchers including Osaki and Nakagawa [1971] discussed a two-unit standby redundant system with standby failure. Nakagawa and Osaki [1975] analyzed stochastic behavior of a two-unit priority standby redundant system with repair. Subramanian et al [1976] explored reliability of a repairable system with standby failure. Gopalan and Naidu [1982] analyzed cost-benefit of a one-server system subject to inspection. Gopalan and Nagarwalla [1985] evaluated cost benefit of a one server two unit cold standby system with repair and age replacement. Singh and Sriniwas [1987] investigated stochastic analysis of a two unit cold standby system with preparation time for repair. Singh [1989] evaluated profit of a two-unit cold standby system with random appearance and disappearance time of the service facility. Dhillon [1992] evaluated reliability and availability analysis of a system with standby and common cause failures. Lam [1997] developed a maintenance model for two-unit redundant system. Kumar [2005] analyzed of reliability models with different types of failure and repair policies. Malik and Barak [2007] analyzed a single-server system operating under different weather conditions. Malik [2009] discussed reliability modeling and costbenefit analysis of a system-a case study. Bhardwaj et al [2014] have described semi-Markov approach for asymptotic performance analysis of a standby system with server failure. Malik et al [2015] analyzed performance of a stochastic system with standby failure and maintenance. Some of them have generally imagined the server to be always in good condition and it never fails while working. But this imagination seems to be quite impractical when a server has to work in varying environmental conditions. We may observe many cases where the server fails during his performance. Recently, Barak and Dhiraj [2016] investigated stochastic analysis of a cold standby system with server failure. In a cold standby redundant system whenever the operating unit fails, the standby unit takes its existence and the failed unit goes under repair. But it may be possible that the standby unit is already damaged owing to remain unused for a longer period of time or erosion etc. So keeping the view of above research work in mind, we developed a stochastic model of redundant standby system with server failure. The model consists of two identical units; one unit is in operative mode and other in cold standby. The cold standby unit becomes operative after failure of the operative unit. The failure of the server during any service activity can produce undesirable results in terms of safety as well as economic losses and server may go for refreshment to increase his efficiency whenever required. The server works afresh after taking refreshment with full efficiency. The time to take refreshment and repair activity follows negative exponential distribution whereas the distributions of unit and server failure are taken as arbitrary with different probability density functions. The expressions for various reliability measures such as transition probabilities, mean sojourn times, mean time to system failure, steady state availability are deduced by using semi-Markov process and regenerative point technique.

International Journal of Advanced Engineering Research and Science (IJAERS)
[ for these transition probabilities, it can be verified that:

Mean Sojourn Time
Let T denotes the time to system failure then the mean sojourn times (µ ) in the state Si are given by www.ijaers.com Taking L.S.T. of relation (5) Busy period of the server due to inspection of the failed unit Let ) (t B I i be the probability that the server is busy due to inspection of the failed unit at instant 't' given that the system entered the regenerative state Si at t = 0 .The recursive relations for ) (t B i are as follows : is the probability that the server is busy in state i S due to repairing of unit up to time 't' without making any transition to any other regenerative state or before returning to the same via one or more non-regenerative states so Taking L.T. of relation (10&11) and solving for ) ( * 0 s B I the time for which server is busy is given as Busy period of the server due to repair of the failed unit be the probability that the server is busy due to repair of the failed unit at instant 't' given that the system entered the regenerative state Si at t = 0 .The recursive relations for ) (t B i are as follows : is the probability that the server is busy in state i S due to repairing of unit up to time 't' without making any transition to any other regenerative state or before returning to the same via one or more non-regenerative states so Taking L.T. of relation (13 &14) and solving for B0 * (s), the time for which server is busy is given as

Expected Number of visits by the server due to repair of the unit
Let Ri(t) be the expected number of visits by the server in (0,t] , given that the system entered the regenerative state Si at t=0 .The recursive relations for Ri(t) are as follows:

Expected Number of refreshments given to server
be the expected number of treatments given to server in (0,t] such that the system entered the regenerative state at t = 0.The recursive relations for ) (t T i are as follows: Busy period of the server due to inspection Busy period of the server due to Repair Expected Number of visits due to repair Expected Number of treatments given to Server

Cost -Benefit Analysis:
The profit occurred in the system model in steady state can be calculated as where K0=(5000) Revenue per unit up-time of the system. K1= (600) Cost per unit time for which server is busy due to inspection . 2 K = (650) Cost per unit time for which server is busy due to repair K3= (450) Cost per unit visits by the server K4 = (300) Cost per unit time treatment given to server.

II.
DISCUSSION In this study the effect of various parameters on performance measure of system model is envisioned. Table-1 reflects that the availability of the system increase when server failure rate ' ' decrease. So we can improve the system availability by checking failure of the server. The third column of the table clearly shows that availability of the system again increase after making increment in the replacement rate ' ' after inspection Effect of increasing the rate ' ' (by which the unit goes for repair after inspection) clearly express that the availability of the system is also in increasing manner but slightly less than the previous values . At last, the availability of the system is increasing when increasing from .1 to 1.0 with the possible change of the other parameters. Table-2 also reflects that profit is also increasing with increasing of repair rate  from .1 to 1.0. By comparing column one and column two of the table-2 it is found that profit of the function increasing whenever the server failure rate µ declined from 0.48 to 0.35. It is observe that form column third the replacement rate  after inspection increased from 0.35 to 0.45 then the profit of the system also rapidly increased. In the forth column clearly shows that when the rate '' (by which unit goes for repair after inspection) increased from .40 to .50 the profit of the system less than the other cases but still in the trend of increasing.

III. CONCLUSION
The idea of provide refreshment to the server which improves the efficiency of the server is more beneficial and economical for smooth functioning of the system.