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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.

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, [

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. Section 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.

: Minimal repair rate of.

: Failure rate of.

: Rate of going into reduced capacity of.

: Exchange rate of unit and in reduced capacity simultaneously.

: Minimal repair rate of unit and simultaneously.

: Failure rate of unit and simultaneously.

: Unit in full operation/reduced capacity/ failure/ standby.

1) 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.

, ,

, ,

,.

.

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:

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 [4-6], 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

For the availability case of

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

which in matrix form is

using the normalizing condition

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

We solve the system of linear equations in matrix above to obtain the state probabilities

Expression for thus is:

Computer programme (MATLAB) is used to develop the explicit expressions for the. The expression for the is lengthy to be shown here.

Using the same initial condition in Section 4.1 above as for the reliability case

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 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

using the normalizing condition

We substitute (6) in the last row of (5) (see [4-6]). The resulting matrix is

We solve the system of linear equations in matrix above to obtain the state probabilities

Expression for thus is:

Computer programme (MATLAB) is used to develop the explicit expressions for the. The expression for the is lengthy to be shown here.

The system/units are subjected to corrective maintenance at failure as can be observed in states 4, 5 and 6. From

Profit = total revenue generated – cost incurred for repairing the failed units.

where: is the profit incurred to the system;

: is the revenue per unit up time of the system;

: is the accumulated cost per unit time which the system is under repair and unit exchange.

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:, , , , , , , , , , for simulations in Figures 2-16.

Case II:, , , , , , , , for simulations in Figures 17-21.

The impact of on MTSF, steady-state availability, profit and busy period can be observed in Figures 3, 6, 14 and 19. From Figures 3, 6 and 14, it is evident that the MTSF, steady-state availability profit increases as increases while in

state availability and profit increases as increases while the busy period decreases with increase in from

In this paper, we constructed a linear consecutive 2-outof-3 repairable system operating in reduced capacity before failure. We have developed the explicit expressions for the MTSF, availability, busy period and profit function. We perform a parametric investigation of various system parameters on MTSF, system availability, busy period and profit function and captured their effect on MTSF, availability, busy period and profit function. These are the main contribution of the paper.