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In this paper, probabilistic models for three redundant configurations have been developed to analyze and compare some reliability characteristics. Each system is connected to a repairable supporting external device for operation. Repairable service station is provided for immediate repair of failed unit. Explicit expressions for mean time to system failure and steady-state availability for the three configurations are developed. Furthermore, we compare the three configurations based on their reliability characteristics and found that configuration II is more reliable and efficient than the remaining configurations.

High system reliability and availability play a vital role towards industrial growth as the profit is directly dependent on production volume which depends upon system performance. Thus the reliability and availability of a system may be enhanced by proper design, optimization at the design stage and by maintaining the same during its service life. Because of their prevalence in power plants, manufacturing systems, and industrial systems, many researchers have studied reliability and availability problem of different systems (see, for instance, Ref [

The problem considered in this paper is different from the work of Ref [

We consider three redundant systems connected to an external supporting device for their operation as follows. The first system is a 2-out-of-3 system connected to a supporting device and has a repairable service station. The second is also a 2-out-of-3 system connected to supporting device and has two standby repairable service stations. The third system is a 3-out-of-4 system connected to a supporting device and has a repairable service station. We assume that switching is perfect and instantaneous. We also assume that two units cannot fail simultaneously. Whenever a unit fails with failure rate

For configuration I, we define

We obtain the following differential equations:

This can be written in the matrix form as

where

It is difficult to evaluate the transient solutions, the procedure to develop the explicit expression for

where

For configuration II, we define

The differential equations are expressed in the form

where

and

Using the procedure described in Subsection 3.1, the expected time to reach an absorbing state is

where

For configuration II, we define

The differential equations are expressed in the form

where

Using the procedure described in Subsection 3.1, the expected time to reach an absorbing state is

where

For the analysis of availability case of configuration I we use the same initial condition as in Subsection 3.1

The differential equations above are expressed in the form

The steady-state availability is given by

In the steady-state, the derivatives of the state probabilities become zero and therefore Equation (2) become

which is in matrix form

Using the following normalizing condition

Substituting (10) in the last row of (9) to compute the steady-state probabilities, the expression for steady- state availability is given by

For the analysis of availability case of configuration II we use the same initial condition as in Subsection 3.2

The differential equations are expressed in the form

The steady-state availability is given by

In the steady-state, the derivatives of the state probabilities become zero and therefore Equation (4) become

which is in matrix form

Using the following normalizing condition

Substituting (14) in the last row of (13) to compute the steady-state probabilities, the expression for steady- state availability is given by

For the analysis of availability case of configuration III we use the same initial condition as in Subsection 3.3

The differential equations are expressed in the form

The steady-state availability is given by

In the steady-state, the derivatives of the state probabilities become zero and therefore Equation (6) become

which is in matrix form

Using the following normalizing condition

Substituting (18) in the last row of (17) to compute the steady-state probabilities, the expression for steady- state availability is given by

In this section, we numerically compare the results for availability and MTSF for the developed models for the three configurations.

Case I:

We fix

Case II:

We fix

Case III:

We fix

Case IV:

We fix

From

clear from the

However, one can say that the results from

ty than configuration I as

Results from

II has higher availability than configuration I and III as

Simulations of MTSF for the three configurations depicted in Figures 7-9 show that MTSF increases as

In this paper, we studied the reliability characteristics of three dissimilar systems connected supporting device. We developed the explicit expressions for steady-state availability and mean time to system failure (MTSF) for each configuration and performed comparative analysis numerically to determine the optimal configuration. It is evident from Figures 1-6 that configuration II is optimal configuration using steady-state availability while using MTSF, the optimal configuration depends on the values of