1. Introduction

Reliability is a vital factor in ensuring system effectiveness, productivity, product quality and generated profit. Efforts have been made to optimize the system reliability which tremendous impact on profit generated. Studies on availability and profit generated from the system using are becoming more and richer day by day due to the fact that numbers of researchers in the field of system reliability are making huge contributions. Availability and profit are good evaluations of a system’s performance. Expected profit is an important factor in economic evaluation of repairable systems. Expected profits have been obtained for different systems. Hajeeh[1] deals with availability of a system with different repair options. Long term performance of the system is investigated. Three special cases are derived to see the impact of imperfect repair, minimal repair or replacement at failure on system availability. Yusuf et al[2] obtained the availability and cost of a deterioration system. El-Damcese[3] analyzed the reliability availability of warm standby system with time varying failure and repair rates. Yusuf and Hussaini[4] obtained the reliability and availability characteristics of 2-out-of-3 system under a perfect repair condition. Expressions for steady-state availability and profit have been obtained. Bhardwaj and Chander[8], analyzed reliability models for 2-out-of-3 redundant system subject to conditional arrival time of the server. Chander and Bhardwaj [9] present reliability and economic analysis of 2-out-of-3 redundant system with priority to repair and Bhardwaj and Malik[10] studied MTSF and cost effectiveness of 2-out-of-3 cold standby system with probability of repair and inspection. Wang and kuo[11] studied the cost and probabilistic analysis of series system with mixed standby components while Wang et al[12] studied cost benefit analysis of series systems with warm standby components involving general repair time where the server is not subject to breakdowns. The failure time and repair time are assumed to have exponential distribution. Measures of system effectiveness such MTSF, steady-state availability, busy period and profit function are obtained. Yusuf and Bala[14] studied the evaluation of MTSF of 2-out-of-4 warm standby system attended by repair machines and repair men. Yusuf and Hussaini[15] studied the modeling of a redundant system with big and three small dissimilar units where various measures of system effectiveness have been obtained.

1.1. Objectives

In the present paper, we considered a series system and derived its corresponding mathematical models. Furthermore, we study reliability characteristics of system availability and profit generated model involving three types of failures using Kolmogorov forward equations method. The contribution of this paper is twofold. The first is to obtain explicit expression for availability, busy period and profit for the two configurations. The second is to capture the effect of both failure and repair rates on the measures of system effectiveness like availability and profit based on assumed numerical values given to the system parameters.

2. Methodology

2.1. Description of the System and Notations

2.1.1. Description of the System

In this paper, we consider a single series system exposes to three types of failures. Type I failure occurred to system when new, at this junction an imperfect repair is done to the system. From there the system continue to operate in a reduced capacity until type II a non repairable occurred. The entire system is replaced () at the occurrence of type II failure. The system can also transit to partial type III failure state with rate and the system is perfectly repaired ().

2.1.2. Notations

: The System’s initial (new) state

: The System is under imperfect repair

: The System is working in a reduced capacity

: The System is under non repairable failure

: The System is under partial failure

: Imperfect repair rate

: Replacement rate

: Perfect repair rate

: Type I failure rate

: Type II failure rate

: Type III failure rate

2.2. Models Formulation

Let be the probability row vector at time, then the initial conditions for this problem are as follows:

we obtain the following system of differential equations:

(1)

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

(2)

where

2.2.1. Availability and Busy Period Modeling

For the availability case of Fig. 1 using the initial condition in section 3 for this system,

The system of differential equations in (1) for the system above can be expressed in matrix form as:

Let be the time to failure of the system. The steady-state availability is given by

(3)

In steady state, the derivatives of state probabilities become zero, thus (2) becomes

(4)

which in matrix form is

using the normalizing condition

(5)

we substitute (5) in the last row of (4) following[5,6,7]. The resulting matrix is

The steady-state busy period is given by

(6)

2.2.2. Profit Modeling

Figure 1. Transition diagram of the system

The system is under corrective maintenance (repair) at failure as can be observed in states 1,3 and 4. From Fig. 1 the repairman is busy performing corrective maintenance action to the system at failure in states 1, 3 and 4. The expected profit per unit time incurred to the system in the steady-state is given by:

(7)

2.2.3. Mean time to System Failure Modeling

It is difficult to evaluate the transient solutions, hence we follow El-said[5], Haggag[6, 13], 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

(8)

Where

3. Results

4. Discussion and Conclusions

4.1. Discussion

In this section, we numerically obtained the results for system availability and profit function for all the developed models. For the models analysis, the following set of parameters values are fixed throughout the simulations for consistency:

1. ,,,, for Figures 2 and 11

2. ,,,, for Figures 3 and 12

3. ,,,, for Figures 4 and 13

4. ,,,, for Figures 5 and 14

5. ,,,,, , , for Figures 6,7,8 and 9

6. ,, , for Fig. 10

The impact of on system availability and busy period of repairman can be observed in Figures 2,9 and 11. From this Figures 2 and 9, it is evident that the availability and profit increases as increases while from Fig. 11 busy period decreases with increase in. Similar results can be observed in Figures 3, 8 and 12 of availability, profit and busy period with respect to. From these Figures 3 and 8, the availability and profit increases as increase while from Fig. 12 busy period decreases with increase in. Results of availability, profit and busy period with respect to are given in Figures 5, 6 and 13. It is evident from these Figures that as increases, the availability and profit decreases while busy period increases. Figures 4,7 and 14 reveal the effect of on availability, profit and busy period. The results obtained in these figures reveal similar to those of Figures 5, 6 and 13 that availability and profit decreases with increase in while busy period increases with increase in . Figure 10 shows that the mean time to system failure decreases with increase in failure rate.

Figure 2. Effect of on Availability

Figure 3. Effect of on Availability

Figure 4. Effect of on Availability

Figure 5. Effect of on Availability

Figure 6. Effect of on Profit

Figure 7. Effect of on Profit

Figure 8. Effect of on Profit

Figure 9. Effect of on Profit

Figure 10. Effect of on MTSF

Figure 11. Effect of on Busy period

Figure 12. Effect of on Busy period

Figure 13. Effect of on Busy period

Figure 14. Effect of on Busy period

4.2. Conclusions

In this paper, we considered a system exposes to three types of failures to study the availability, generated profit and mean time to system failure. Explicit expressions of mean time to system failure, busy period, steady-state availability and profit function were derived. We performed numerical investigation to see the effect of failure and repair rates on the system availability, busy period, mean time to system failure and generated profit. It is evident from the results obtained that repair rate increase the system availability, mean time to system failure and profit generated and decreases the busy period while failure rate decreases the system availability, mean time to system failure and generated profit and increases the busy period of repairman. It is evident from that the results obtained from this paper make a tremendous effect on measure of system effectiveness studied.

ACKNOWLEDGMENTS

The authors are grateful to the anonymous reviewers and handling for their constructive comments which have helped to improve the manuscript.