Analysis of Stochastic Reliability Characteristics of a Repairable 2-out-of-3 System with Minimal Repair at Failure

In this paper, we study the reliability and availability characteristics of a repairable 2-out-of-3 system. Failure and repair times are assumed exponential. The explicit expressions of reliability and availability characteristics such as mean time to system failure (MTSF), steady-state availability, busy period and profit function are derived using Kolmogorov’s forward equations method. Various cases are analyzed graphically to investigate the impact of system parameters on MTSF, availability, busy period and profit function.


Introduction
During operation, the strengths of systems are gradually deteriorated, until some point of deterioration failure, or other types of failures.Maintenance policies are vital in the analysis of deterioration and deteriorating systems as they help in improving reliability and availability of the systems.Maintenance models assume perfect repair (as good as new), minimal repair (as bad as old) and imperfect repair which between perfect and minimal repair.There are systems of three/four units in which two/three units are sufficient to perform the entire function of the system.Examples of such systems are 2-out-of-3, 2-outof-4, or 3-out-of-4 redundant systems.These systems have wide application in the real world especially in industries.Many research results have been reported on reliability of 2-out-of-3 redundant systems.For example, [1] analyzed reliability models for 2-out-of-3 redundant system subject to conditional arrival time of the server.Reference [2] presented reliability and economic analysis of 2-out-of-3 redundant system with priority to repair, [3] studied MTSF and cost effectiveness of 2-out-of-3 cold standby system with probability of repair and inspection while [4] examined the cost benefit analysis of series systems with cold standby components and repairable service station.Reference [5,6] examined the cost analysis of two unit cold standby system involving preventive maintenance respectively.Reference [7] studied the cost and probabilistic analysis of series system with mixed standby components, and [8] 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.Reference [9] studied availability of a system with different repair options, while [10] evaluate the reliability of network flows with stochastic capacity and cost constraint.
In this paper, a 2-out-of-3 redundant system is constructed and derived its corresponding mathematical models.The main contribution of this paper is two fold.First, is to develop the explicit expressions for MTSF, system availability, busy period and profit function.The second is to perform a parametric investigation of various system parameters on MTSF, system availability and profit function and capture their effect on MTSF, availability, busy period and profit function.
The rest of the paper is organized as follows.Section 2 the notations, assumptions of the study, and the states of the system.Section 3 gives the states of the system.Sec-tion 4 deals with models formulation.The results of our numerical simulations are presented and discussed in Section 5.The paper is concluded in Section 6.   : Unit in full operation/reduced capacity/ failure/ standby.

Notations and Assumptions
The system is 2-out-of-3 system.
2) The system work in a reduced capacity before failure.
3) The systems have three states: normal, reduced and failure.
4) Unit failure and repair rates are constant.5) Repair is as bad as old (minimal).6) failure and repair time are assumed exponential.7) The system failed when two units have failed.8) The system is under the attention of two repairmen., ,

States of the System
, ,   , ,

Mean Time to System Failure for System
Let   P t be the probability row vector at time t , then the initial conditions for this problem are as follows: 0 0 , 0 , 0 , 0 , 0 , 0 , 0 1, 0, 0, 0, 0, 0, 0 we obtain the following system of differential equations: The above system of differential equations can be written in matrix form as where It is difficult to evaluate the transient solutions, hence we follow [4][5][6], the procedure to develop the explicit expression for MTSF is to delete the seventh row and column of matrix T and take the transpose to produce a new matrix, say A. The expected time to reach an absorbing state is obtained from where

System Availability Analysis
For the availability case of Figure 1 using the initial condition in Section 4.1 for this system, 0 0 , 0 , 0 , 0 , 0 , 0 1, 0, 0, 0, 0, 0, 0 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 In steady state, the derivatives of state probabilities become zero, thus (2) becomes   0 AP   (5) which in matrix form is using the normalizing condition we substitute (6) in the last row of ( 5) following [4][5][6].
The resulting matrix is We solve the system of linear equations in matrix above to obtain the state probabilities Computer programme (MATLAB) is used to develop the explicit expressions for the .The expression for the is lengthy to be shown here.

Busy Period Analysis
Using the same initial condition in Section 4.1 above as for the reliability case 0 0 , 0 , 0 , 0 , 0 , 0 1, 0, 0, 0, 0, 0, 0 and ( 5) and ( 6) the busy period is obtained as follows: In the steady state, the derivatives of the state probabilities become zero and this will enable us to compute steady state busy period due to failure: The system of differential equations in (1) for the system above can be expressed in matrix form as: Let   B  be the probability that the repair man is busy either repairing the failed unit or exchanging the degraded units with new ones.The steady-state busy period is given by In steady state, the derivatives of state probabilities become zero, thus (2) becomes which in matrix form is We substitute (6) in the last row of (5) (see [4][5][6]).The resulting matrix is We solve the system of linear equations in matrix above to obtain the state probabilities Computer programme (MATLAB) is used to develop the explicit expressions for the .The expression for the is lengthy to be shown here.

Profit Analysis
The system/units are subjected to corrective maintenance at failure as can be observed in states 4, 5 and 6.From Figure 1, the repairman is busy performing corrective maintenance action to the units at failure in states 4, 5 and 6.According to [4,5], the expected profit per unit time incurred to the system in the steady-state is given by: Profit = total revenue generated -cost incurred for repairing the failed units.

  
where : is the profit incurred to the system; PF 0 C C : is the revenue per unit up time of the system; 1 : is the accumulated cost per unit time which the system is under repair and unit exchange.

Results and Discussions
In this section, we numerically obtained the results for mean time to system failure, system availability, busy period and profit function for all the developed models.
For the model analysis, the following set of parameters values are fixed throughout the simulations for consistency: Case I: 1 0.

1 U 2 U 3  3 
: Failure rate of unit and U simultaneously.

Figure 20. Effect of 3  on busy period.
Results of MTSF, steady-state availability, profit and busy period with respect to 1 Figures 10, 15, and 20 and Fi res 11, 6 and 21 respectively.It is evident from Figures 10 and 15 that the steady-state availability and profit decreases as 3  .Simulation results of steady-state availability, profit and busy period can be observed in Fig- ures 11, 16 and 21.In Figures 11 and 16, the steadystate availability and profit increases as 3  from Figure 13.