Profit comparison of a cold standby system with Priority to repair over preventive maintenance and server failure during repair

Introduction

In the modern scenario of global competition that the industries become fully conscious to produce reliable system which give us the maximum profit. Most of the researcher Kadyan et al. (2010) [4,5] and Sureria et al. (2012) [2] have carried out cost-benefit analysis of systems respectively with the idea of priority to repair and priority to software replacement over hardware repair. To strengthen the existing literature, the present paper is devoted to examine the effect of priority on profit incurred to a system of two identical units have been analyzed stochastically with the concepts of preventive maintenance of the unit, server failure and priority to repair over preventive maintenance. Initially, one unit is operative and the other unit is kept as spare in cold standby. The operative unit undergoes for preventive maintenance before its complete failure while repair of the unit is conducted at its complete failure. All repair activities are tackled by a single server who visits the system immediately when required. The system is observed at suitable regenerative epochs using semi-Markov process and regenerative point technique to derive the expressions for some reliability and economic measures such as mean time to system failure (MTSF),availability, expected number of repair, expected number of preventive maintenance of server and busy period analysis. The numerical results for a particular case are obtained to depict the behavior of MTSF, availability and profit of the system model. The profit of the present model has also been compared with that of the model discussed by Nandal and Malik (2017) [1] .

Notations

System Description

A stochastic model of a system having two identical units has been developed with the ideas of priority to repair over preventive maintenance and server failure during repair. The block diagram for the system model is shown in Fig. :1

The system mode1 (figure 1) has the following transition states:  Regenerative States: , , ,  Non-regenerative: ,, , , , , , We have the following expressions for transition probabilities:

For a perfect distribution

Mean Sojourn Times (MST)

Reliability and Mean Time to System Failure (MTSF)

Let i(t) be the cdf of first passage time from regenerative state Si to a failed state.

Regarding the failed state as absorbing state, we have the following recursive relations for i(t):

The expressions for are as follows:

Let us take LST of above relations (4) and solving for π

Expected Number of Preventive Maintenance (PM) of the unit in the Long Run

Let be the expected number of repairs by the server in (0, t] given that the system entered the regenerative state Si at t = 0. The recursive relations for are given as:

Expected Number of Treatments (ENT) Given to the Server in the Long Run

Let ( ) i T t be the expected number of treatments given to the server in (0, t] given that the system entered the regenerative state Si at t = 0. The recursive relations for ( ) i T t are given as:

Ⓢ

. , It is observed that profit of the system model go on decreasing with the increase of the failure rates of the unit and server, the rate by which unit under goes for preventive maintenance while there is a positive increment in their values with the increase of treatment rate of the server (α), repair rate (β) and preventive maintenance rate of the unit (γ) in fig. 4.

Graphical Representation of Reliability Measures

b) Comparative Study of the Profit of the System Models

The profit of the present model has been compared with that of the model in which no such priority is given. The graph of the profit difference as shown in figure 5 indicates that profit difference goes on increasing with the increase of failure rate of the unit while it declines with the increase of repair rate of the unit and the rate by which unit under goes for preventive maintenance. The study for the particular values of the parameters shows that present model is profitable. Hence, it is concluded that the idea of priority to repair over preventive maintenance is useful in making the system more effective and profitable to use.

References

1. Nandal Jyoti, Chauhan SK, Grewal AS. Reliability Measures of a Cold Standby System With Preventive Maintenance and Server failure During Repair. International Journal of Statistics and Reliability Engineering, 2017; 4(1):77-85,(ISSN (P): 2350-0174)
2. Sureria JK, Malik SC. Probabilistic Analysis of a Computer System with Arrival Time of the Server at H/w Failure and Priority for Replacement of the S/w over H/w Repair. International Journal of Mathematics and Applied Statistics. 2012; 3(2):115-128.
3. Sureria JK, Malik SC. Reliability Measures of a Computer System with Arrival Time of the Server and Priority to H/w Repair over S/w Up-gradation. International Journal of Computer Applications. 2012; 65(13). 44-48, 2013 (ISSN: 0975-8887)
4. Jitender Kumar, Kadyan MS, malik SC. Cost-analysis of a system under priority to repair and degradation. International Journal of Statistics and Systems. 2010; 5(1):1-10, (ISSN: 0973-2675)
5. Promila, Malik SC, Kadyan MS. Analysis of an Operating System with Inspection Subject To Degradation and Different Weather Conditions, International Journal of Engg. Science and Technology (IJEST), 2010; 2(11):6779-6787.