1. Introduction

Preventive Maintenance schedules that minimize resource consumption or maximize availability can be determined through the use of quantitative decision-models, based on factual information such as time-to-failure distributions, cost of intervention (e.g. for inspection, repair or replacement) and consequences of failure. Most of engineering systems undergo either offline or online preventive. Under online Preventive maintenance is carried out when the system is operating and intends to slow down the wear process and reduce the frequency of occurrence of system or components failures.

There are systems of three/four units in which two/three units are sufficient to perform the entire function of the system. Example of such systems are 2-out-of-3,2-out-of-4, or 3-out-of-4 redundant systems which can be seen in modular communication amplifier system.

Many research results have been reported on reliability of redundant systems. For example, Chander and Bhardwaj^[1], analyzed reliability models for 2-out-of-3 redundant system subject to conditional arrival time of the server. Chander and Bhardwaj ^[2] present reliability and economic analysis of 2-out-of-3 redundant system with priority to repair. Bhardwaj and Malik ^[3] studied MTSF and cost effectiveness of 2-out-of-3 cold standby system with probability of repair and inspection. Wang et al ^[4] examined the cost benefit analysis of series systems with cold standby components and repairable service station. El-Said ^[5] and Haggag ^[6] examined the cost analysis of two unit cold standby system involving preventive maintenance respectively. Haggag ^[7] analyzed cost analysis of repairable k-out-of-n system with dependable failures and standby support. Wang and Kuo ^[8] studied the cost and probabilistic analysis of series system with mixed standby components. Wang et al ^[9] studied cost benefit analysis of series systems with warm standby components involving general repair time where the server is not subject to breakdowns. Yusuf ^[10] studied the availability and profit of 3-out-of-4 system.

Reference ^[10] studied the availability and profit of 3-out-of-4 system with preventive maintenance. In his study, it is not certain whether the preventive maintenance is offline or online. In fact most of studies on system reliability evaluation, did not mention which category of preventive maintenance the system will undergo. In our study, we assume that the system undergo online preventive maintenance where the system is operation while the preventive maintenance taking place. Also availability and profit cannot be use to judge the effectiveness of the system. The time interval from the time system is new to the time when the system experience its first failure (Mean time to system failure) is equally important in determining the effectiveness which has not been studied in Reference ^[10].

This paper is continuation of the work of Yusuf ^[10], a repairable 3-out-of-4 system under online preventive maintenance is considered and derived its corresponding mathematical model. The main contribution of this paper is three fold. First, is to develop the explicit expressions mean time to system failure. The second is to perform a parametric investigation of system parameters on mean time to system failure and capture their effect. The third to compare the mean time to system failure with and without online preventive maintenance to highlight the impact of online preventive maintenance on mean time to system failure.

The rest of the paper is organized as follows. Section 2 gives the notations and assumptions of the study. Section 3 is the description of the system. Section 4 deals with model formulation. The results of our numerical simulations are presented and discussed in Section 5. The paper is concluded in Section 6.

2. Notations and Assumptions

: Constant repair rates for type

: Constant failure rates for types

: Constant rate end of preventive maintenance

: Constant rate of taking the unit into preventive maintenance

: System transition rate matrix

: Mean time to system failure with online preventive maintenance

: Mean time to system failure without online preventive maintenance

1. The system is 3-out-of-4 system

2. The system can be in Operation, Fail state or online preventive maintenance

3. The system suffer four types of failures

4. The system is down when number of units failure goes beyond one

5. Failure and repair time follow exponential

6. Failure rates and repair rates are constant

7. The system is attended by one repairman

3. Description of the System

In this section, the 3-out-of-4 redundant system is considered. Through Markov assumption, the Chapman-Kolmogorov’s equations are obtained for the analysis of state probabilities. The system comprise of four units in which at least three units most be in operational for the system to work. Malfunctioning of two units lead the system to go down. The units can work consecutively or randomly as can be seen in the states of the system given below. The system transit to preventive maintenance before failure at the rate with corresponding rate of preventive maintenance. Unit fails with rate and is under minimal repair with rate and the standby unit is switch on. It is assumed that the switch from standby to operation is perfect. The system failed when two units failed. The states of the system according Markov chain is shown in Table 1 below.

Table 1. Transition rate table

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State of the System

State 0: Units 1,2 and 3 are working, unit 4 in standby, the system is working

State 1: units 1,2,and 4 are working; unit 3 is down and under repair, the system is working

State 2: units 1,3 and 4 are working, unit 2 is down and under repair, the system is working

State 3: units 2,3 and 4 are working, unit 1 is down and under repair, the system is working

State 4: units 1 is down, under repair, units 2 is down and waiting for repair, units 3 and 4 are idle, the system failed

State 5: unit 1 is down, under repair, unit 3 is down, waiting for repair, units 2 and 4 are idle, the system failed

State 6: unit 1 and 2 are idle, unit 3 is down, and waiting for repair, unit 4 is down, under repair, the system failed

State 7: unit 1 is down, waiting for repair, units 2 and 3 are idle, unit 4 is down, under repair, the system failed

State 8: units 1 and 4 are idle, unit 2 is down, and waiting for repair, unit 3 is down, under repair, the system failed

State 9: units 1 and 3 are idle, unit 2 is down, and waiting for repair, unit 4 is down, under repair, the system failed

State 10: all units are under online preventive maintenance, the system is working

4. Model Formulation

Let be the probability that the system is in state at time. Let be the probability row vector at time, then the initial conditions for this problem are as follows:

The corresponding set of kolmogorov’s differential equations obtained from Table 1 is as follows:

(1)

The differential equations in (1) above is transformed into matrix as

(2)

where

It is difficult to evaluate the transient solutions, hence we follow El-said ^[5], Haggag ^[6] and Wang ^[9], the procedure to develop the explicit expression for MTSF is to delete the rows and columns of an absorbing state in matrix and take the transpose to produce a new matrix, say . The expected time to reach an absorbing state is obtained from

(3)

Where

5. Results and Discussions

In this section, we numerically obtained the results for mean time to system failure. For the model analysis, the following set of parameters values are fixed throughout the simulations for consistency:

The simulations in Figure 1 and Figure 3 have shown that the mean time to system failure for both systems with and without preventive maintenance increase with increase in repair rate. It is evident from Figure 3 that the mean time to system failure of system with preventive maintenance increases more with respect to repair rate than the mean time to system failure of system of system without preventive maintenance. In Figure 2 and Figure 4 it is clear that the mean time to system failure of system decreases for both systems with and without preventive maintenance. Here also the mean time to system failure of system of system with preventive maintenance decreases more than the mean time to system failure of system of system without preventive maintenance.

Figure 1. Effect of α₁ on MTSF

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Figure 2. Effect of β₁ on MTSF

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Figure 3. Effect of α₁ on MTSF

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Figure 4. Effect of β₁ on MTSF

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6. Conclusion

In this paper, we developed the explicit expression for mean time to system failure of repairable 3-out-of-4 system in the presence of online preventive maintenance. In order to determine the effectiveness of the system under study, we performed numerical investigation to see the effect of failure and repair rates on mean time to system failure. It is evident from the results obtained that failure and repair rates decrease and increase the meant time to system failure of the system respectively. Through the analysis, we conclude that system with online preventive maintenance is more effective than system without online preventive maintenance.

References

[1]	Bhardwj, R.K. and Chander, S. (2007). Reliability and cost benefit analysis of 2-out-of-3 redundant system with general distribution of repair and waiting time. DIAS- Technology review- An Int. J. of business and IT. 4(1), 28-35.
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[2]	Chander, S. and Bhardwaj, R.K. (2009). Reliability and economic analysis of 2-out-of-3 redundant system with priority to repair. African J. of Maths and comp. sci, 2(11), 230-236.
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[3]	Bhardwj,R.K., and S.C. Malik. (2010). MTSF and Cost effectiveness of 2-out-of-3 cold standby system with probability of repair and inspection. Int. J. of Eng. Sci. and Tech. 2(1), 5882-5889.
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[4]	Wang, k. Hsieh, C. and Liou, C (2006). Cost benefit analysis of series systems with cold standby components and a repairable service station. Journal of quality technology and quantitative management, 3(1), 77-92.
	In article
[5]	El-Said, K.M., (2008). Cost analysis of a system with preventive maintenance by using Kolmogorov’s forward equations method. American Journal of Applied Sciences 5(4), 405-410.
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[6]	Haggag, M.Y., (2009). Cost analysis of a system involving common cause failures and preventive maintenance, Journal of Mathematics and Statistics 5(4), 305-310.
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[7]	Haggag, M.Y., (2009). Cost analysis of k-out-of-n repairable system with dependent failure and standby support using Kolmogorov’s forward equations method .Journal of Mathematics and Statistics 5(4), 401-407.
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[8]	Wang, K.H and Kuo, C.C. (2000). Cost and probabilistic analysis of series systems with mixed standby components. Applied Mathematical Modelling, 24, 957-967.
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[9]	Wang, K..C., Liou, Y.C, and Pearn W. L. (2005). Cost benefit analysis of series systems with warm standby components and general repair time. Mathematical Methods of operation Research, 61, 329-343.
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[10]	Yusuf, I. Availability and profit analysis of 3-out-of-4 repairable system with preventive maintenance, International Journal of Applied Mathematics Research, 1(4), 2012, 510-519.
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