Stochastic behavior of a cold standby system with maximum repair time

Article history: Received January 29, 2015 Received in revised format: April 28, 2015 Accepted May 7, 2015 Available online May 8 2015 The main aim of the present paper is to analyze the stochastic behavior of a cold standby system with concept of preventive maintenance, priority and maximum repair time. For this purpose, a stochastic model is developed in which initially one unit is operative and other is kept as cold standby. There is a single server who visits the system immediately as and when required. The server takes the unit under preventive maintenance after a maximum operation time at normal mode if one standby unit is available for operation. If the repair of the failed unit is not possible up to a maximum repair time, failed unit is replaced by new one. The failure time, maximum operation time and maximum repair time distributions of the unit are considered as exponentially distributed while repair and maintenance time distributions are considered as arbitrary. All random variables are statistically independent and repairs are perfect. Various measures of system effectiveness are obtained by using the technique of semi-Markov process and RPT. To highlight the importance of the study numerical results are also obtained for MTSF, availability and profit function. Growing Science Ltd. All rights reserved. 5 © 201


Introduction
Economic growth of a country totally depends on its industrial and mechanical development and this growth gives emergence to new technologies.These advance technologies developed many complex industrial systems with the composition of many simultaneous components.Complexity of the system also affects its performance, cost and reliability.But, consumers expect highly reliable systems at minimum cost.Now, system designers and researchers face a great challenge to develop such systems which operate continually and increase their performance, reliability and safety.Many researchers such as Moghaddass et al. (2011), Zhang and Wang (2009) and Wu and Wu (2011) analysed reliability of repairable systems under different assumptions.Goel and Sharma (1989), Cao and Wu (1989), Chandrasekhar et al. (2004), Gopalan and Nagarwall (1985), Gopalan andBhanu (1995), Chander (2005), Mahmoud and Moshref (2010), Mokaddis et al. (2008) and Kumar and Malik (2012) discussed several stochastic model under common assumptions that after a maximum operation time unit undergoes for preventive maintenance either standby unit available or not.Mokaddis et al. (2008) studied the effect of preventive maintenance on the systems of non-identical units.Jin et al. (2009) discussed option model for joint production and preventive maintenance.But, keeping in mind the economic aspects of the system sometimes this assumption is not practically feasible.It is also proved that preventive maintenance can slow the deterioration process of a repairable system and restore the system in a younger age or state.Malik and Nandal (2010) carried out the cost-benefit analysis of a stochastic model using the concept of preventive maintenance.Osaki and Asakura (1970) and Kapur and Kapoor (1974) developed reliability model for two-unit standby redundant system with repair and preventive maintenance.Thus, the method of preventive maintenance can be used to improve the reliability and profit of system.Nakagawa and Osaki (1975) analysed stochastic behaviour of a twounit parallel redundant system with preventive maintenance.
Further, it is a known fact that frequency of failure of standby systems can be reduced up to a desired extent by the method of redundancy.Therefore, keeping a system in cold standby has attracted the attention of many researchers.Barak and Malik (2014), Kumar and Malik (2015), Kumar et al. (2012) and Barak and Barak (2013) suggested some reliability models using the concept of redundancy.Recently, Malik and Munday (2014) developed stochastic model for a computer system by using the concept of hardware redundancy.
Further, the reliability of a system can be increased by making replacement of the components by new one in case repair time is too long i.e., if it extends to a pre-specific time.Singh and Agrafiotis (1995), Malik and Gitanjali (2012) Malik and Kumar (2011) and Kumar and Malik (2014) analyzed stochastically two-unit cold standby systems subject to maximum operation and repair time.
In view of the above importance and practical application of cold standby systems, a stochastic model is developed here by using the concept of preventive maintenance, priority and maximum repair time.For this purpose, a stochastic model is developed in which initially one unit is operative and other is kept as cold standby.There is a single server who visits the system immediately as and when required.The server takes the unit under preventive maintenance after a maximum operation time at normal mode if one standby unit is available for operation.If the repair of the failed unit is not possible up to a maximum repair time failed unit is replaced by new one.The failure time, maximum operation time and maximum repair time distributions of the unit are considered as exponentially distributed while repair and maintenance time distributions are considered as arbitrary.All random variables are statistically independent.Repairs are perfect.Various measures of system effectiveness such as transition probabilities, mean sojourn times, mean time to system failure, steady state availability, busy period of the server due to replacements, repair and preventive maintenance, expected number of repairs, replacements and preventive maintenance, expected number of visits by the server and expected profit earned by the system in (0, t) are obtained by using the technique of semi-Markov process and RPT.To highlight the importance of the study numerical results are also obtained for MTSF, availability and profit function.

Model Description
(i) Initially system consists of two identical units-one operative and other is kept as cold standby.(ii) Both units have three modes-normal, under repair due to failure and under preventive maintenance.(iii) The failed unit is replaced by new one if the repair of the unit is not possible up to a maximum repair time.(iv) The failure and maximum operation time distribution are exponentially distributed while repair and preventive maintenance times are distributed arbitrarily.(v) There is a single server who visits the system immediately as and when required.(vi) The switch devices, repairs and preventive maintenance are perfect.(vii)All random variables are statistically independent.

O
The unit is operative and in normal mode, Cs The  In view of the above notations and assumptions the system may be in one of the following states:

Transition Probabilities and Mean Sojourn Times:
Simple probabilistic considerations yield the following expressions for the non-zero elements The mean sojourn times (μi) of the state Si are:

Reliability and Mean Time to System Failure (MTSF)
Let ( ) i φ t be the cdf of first passage time from the regenerative state i S to a failed state.Regarding the failed state as absorbing state, we have the following recursive relations for ( ) where j S is an un-failed regenerative state to which the given regenerative state i S can transit and k S is a failed state to which the state i S can transit directly.Taking LST of Eq. ( 5) and solving for 0 ( ), The reliability of the system model can be obtained by taking inverse Laplace transform of Eq. ( 6).
The mean time to system failure (MTSF) is given by MTSF = s

Steady State Availability
Let Ai(t) be the probability that the system is in up-state at instant 't' given that the system entered regenerative state i S at t = 0.The recursive relations for Ai (t) are given as , where j S is any successive regenerative state to which the regenerative state i S can transit through n transitions.Mi(t) is the probability that the system is up initially in state i S E ∈ up at time t without visiting to any other regenerative state, we have 0 0 Taking LT from Eqs. ( 8) and solving for * 0 ( ), A s the steady state availability is given by * 0 0 0 ( ) lim ( ) , where where j S is any successive regenerative state to which the regenerative state i S can transit through n transitions.Wi(t) be the probability that the server is busy in state Si due to repair of the unit up to time t without making any transition to any other regenerative state or returning to the same via one or more non-regenerative states and so where j S is any successive regenerative state to which the regenerative state i S can transit through n transitions.Wi(t) be the probability that the server is busy in state Si due to repair of the unit up to time t without making any transition to any other regenerative state or returning to the same via one or more non-regenerative states and so + and 2 D is already defined.

Expected Number of Preventive Maintenances
Let p i R (t) be the expected number of preventive maintenances conducted by the server in (0, t] given that the system entered the regenerative state i S at t = 0.The recursive relations for p i R (t) are given as where j S is any regenerative state to which the given regenerative state i S transits and δj =1, if j S is the regenerative state where the server does job afresh, otherwise δj = 0. Taking LST of Eqs. ( 15) and solving for 0 p R (s)  .The expected numbers of preventive maintenances per unit time are given by 0 0 0 ( ) lim ( )

Expected Number of Repairs
Let r i R (t) be the expected number of repairs by the server in (0, t] given that the system entered the regenerative state i S at t = 0.The recursive relationships for r i R (t) are given as

Profit Analysis
The profit incurred to the system model in steady state can be obtained as  -3, it is revealed that MTSF, profit and availability decrease with the increase of failure rate ( λ ) and maximum operation time ( 0 α ), but the value of steady state availability and MTSF increase with the increase of repair rate (θ ) and preventive maintenance rate (α ).The profit of the system declines with the increases of preventive maintenance rate (α ).Thus finally it is concluded that a cod standby system in which a maximum repair time is given to server for repair can be made more reliable and profitable to use   IJSSST, 14(5), 55-62. Cao, J., & Wu, Y. (1989).Reliability analysis of a two-unit cold standby system with a replaceable repair facility.Microelectronics Reliability, 29(2), 145-150. Chander, S. (2005).Reliability models with priority for operation and repair with arrival time of the server.Pure and Applied Mathematika Sciences, 61(1-2), 9-22.Goel, L. R., & Sharma, S. C. (1989).Stochastic analysis of a 2-unit standby system with two failure modes and slow switch.Microelectronics Reliability, 29(4), 493-498.

T
the mean sojourn time in state Si E, i µ The mean Sojourn time in state Si this is given by 0 is the sojourn time in state Si, Ⓢ / © Symbol for Stieltjes convolution / Laplace convolution, ~ / * Symbol for Laplace Stieltjes Transform (LST) / Laplace Transform (LT).
23) where K0 = Revenue per unit up-time of the system K3 = Cost per unit time repair of the unit K1 = Cost per unit time for which server is busy due to repair K4 = Cost per unit time preventive maintenance of the unit K2 = Cost per unit time for which server is busy due to preventive maintenance K5 = Cost per unit time visit by the server 6 K = cost per unit time replacement of unit 13.Conclusion The graphical results for various reliability measures are obtained in Figs.1= 5000, K1 = 250, K2 = 300, K3 = 200, K4 = 190, K5 = 210, K6 = 160 and K0 = 150.From Figs. 1 more unit in cold standby,(ii)   By ignoring the concept of priority to operation over preventive maintenance, (iii) By increasing the repair rate, (iv) By increasing the maximum repair time.

FigFig. 2 .Fig. 3 .
Fig. 1.MTSF vs. Failure Rate ) be the probability that the server is busy in the replacement of the unit due to failure at an instant 't' given that the system entered state i S at t = 0.The recursive relations for