1. Introduction

With advancement of modern science and technology, complex systems have been manufactured to meet the demand of industries, economic growth and populace in general. Companies and organizations heavily rely on these systems to conduct their business. Due to their importance in promoting and sustaining industries and economy, reliability measures of such systems have become an area of interest. Among the reliability measures of interest are, the steady-state availability, busy period, profit function and mean time to system failure, (MTSF). Modelling the reliability and availability of the system is important because it will assist in diagnosing the best time to carry out a preventive maintenance which leads to increase in mean time to system failure and availability of a system and related profit. Reliability modeling and analysis of complex systems have drawn the attention different researchers. [3] Investigated multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint. [4] Proposed multi-objective particle swarm optimization method for solving reliability redundancy allocation problems.

Many research results have been reported on the analysis and comparison of reliability measures mention above. [1] studied MTSF and cost effectiveness of 2-out-of-3 cold standby system with probability of repair and inspection, [2] have analyzed the Availability of k-out-of-n: G systems with non identical components subject to repair priorities, [5] performed computational comparisons of confidence intervals for the steady-state availability of a repairable system, [6] performed comparative analysis between two unit cold standby and warm standby outdoor electric power systems in changing weather. [7] performed comparative analysis of availability between three systems with general repair times, reboot delay and switching failures. [8] performed comparative analysis of availability between two systems with warm standby units and different imperfect coverage. [9] Performed comparative analysis of some reliability characteristics between redundant systems requiring supporting units for their operation. [10] Have analyzed the mean time to system failure of 2-out-of-4 warm standby system attended repair machines and repairmen while [11] performed comparative analysis of profit between three dissimilar repairable redundant systems using supporting external device for operation.

The present paper is devoted to deal with mean time to system failure (MTSF) comparison between three dissimilar redundant systems that worked with the help of an external supporting device. Explicit expressions for mean time to system failure have been developed. Analytical and numerical investigation on the optimal configuration has been performed.

2. Notations, Description and States of the Systems

: Repair rate of unit for both systems,

: Repair rate of unit for both systems,

: Failure rate of unit for both systems,

: Failure rate of unit for both systems,

: Repair rate of the supporting unit for both systems

: Failure rate of the supporting unit for both systems

: Probability row vector

: Probability that the system is in state

3. Mean Time to System Failure Models Formulation

Let to be the probability that the systems at time are in the states for system I, for System II and for System III. Also let , be the probability row vector at time , we have the following initial conditions for system I, II and III respectively:

3.1. Mean Time to System Failure of System I

Let to be the probability that the systems at time are in the states and be the probability row vector for system I with the following differential equations.

(1)

where

It is difficult to evaluate the transient solutions, the procedure to develop the explicit expression for is to delete the fifth, sixth and seventh rows and fifth, sixth and seventh column of matrix and take the transpose to produce a new matrix, say . The expected time to reach an absorbing state is obtained from

(2)

where

3.2. Mean Time to System Failure Analysis of System II

Let to be the probability that the systems at time are in the states and be the probability row vector for system II with the following differential equations.

(3)

where

,

Following the procedure used in computing

(4)

3.3. Mean Time to System Failure Analysis of System III

Let to be the probability that the systems at time are in the states and be the probability row vector for system II with the following differential equations. be the probability row vector for system II with the following differential equations.

(5)

Following the procedure used in computing , the expected time to reach an absorbing state is obtained from

(6)

4. Comparative Analysis

The purpose of this section is to present analytical and numerical comparisons for the mean time to system failure between the systems using MAPLE and MSTLAB software. Five different cases are presented below: Cases I – IV are analytical comparison between the systems while case V is the numerical comparison.

Case I: Comparison when and

(7)

(8)

(9)

Equations (7), (8) and (9) are too spacious to be shown here. From (7), (8) and (9)

Case II: Comparison when

(10)

(11)

(12)

From (10), (11) and (12)

Case III: Comparison when

(13)

(14)

(15)

From (13), (14) and (15)

while the sign of depend on

Case IV: Comparison when and

(16)

(17)

(18)

From (16), (17) and (18)

Case V: The objective here is to present specific numerical comparisons for the mean time to system failure. In this part, we numerically compare the results for mean time to system failure for all the developed models. For each model the following set of parameters values are fixed throughout the simulations for consistency:

From Figures 1 the MTSF results for the three systems being studied against the repair rate . It is clear from the figure that it is clear that system I has higher MTSF with respect to as compared with the other two systems. The differences between the MTSF of system I and the other two systems widens as increases. These tend to suggest that system I is better than the other systems. Figure 2 depicts the MTSF calculations for the three systems against . The observations that can be made here are much similar to those made on Figure 1. Figure 3 shows the MTSF results for the three systems being studied against the repair rate . It is clear from the figure that it is clear that system I has higher MTSF with respect to as compared with the other two systems. The differences between the MTSF of system I and the other two systems widens as increases. There is slight difference between the MTSF of system II and that of system III with respect to . It is evident here that system I is has higher MTSF than systems II and III. From Figures 4, it can be seen that the MTSF of system I decreases much slower than those of the other two systems with increase in . By comparing systems II and III, it can be observed that there is slight difference between the two. System III is decreasing a little faster than system II with respect to . We can conclude as before that system I is better than the other two systems in all the three figures.

Figure 1. MTSF against

Figure 2. MTSF against

Figure 3. MTSF against

Figure 4. MTSF against

5. Conclusions

In this paper, we studied three dissimilar systems each consisting of two subsystems A and B each containing two units with supporting unit attached to the systems. We first formulated models three different repairable redundant systems that used an external supporting device for their operation and obtained the explicit expressions for mean time to system failure (MTSF) for each system and performed comparative analysis analytically and numerically to determine the optimal system. It is evident from case I to V that system I is better than the other two systems. The study of mean time to system measures will help the engineers and designers to develop sophisticated models and to design more critical system in interest of human kind.