Stochastic Analysis of a Compound Redundant System Involving Human Failure

This study deals with a compound standby redundant system consisting of three subsystems A, B, and C connected in series. The sub-system B consists of one main unit and the other is its standby redundant unit. These units further consist of two sub-units connected in series. The sub – units of main unit are connected to sub-units of standby unit through imperfect switching over device. Failure of all sub-systems and repair rate of switching over devices are exponential while repairs of all sub-systems are distributed quite generally. The various reliability parameters have been computed and analyzed by tabular and graphical illustrations.


INTRODUCTION
Reliability is an important concept at the planning, design and operation stages of various complex systems. As long as man has built things, he has wanted to make them as reliable as possible. In practice, we come across with a number of complex systems consisting of one or more parts, failure of any of the parts results in the reduction of efficiency of whole systems or the complete failure of the system and as a result of it, the reliability of the system reduces. The better maintenance of such parts originate better reliability and then only we can achieve the markets demands of reliability, functionality, price and performance of that system. On the other hand it may not be economical to obtain higher order of reliability always through any amount of maintenance. Thus introducing redundant parts and providing maintenance and repair at the time of need may achieve high degree of reliability. In a redundant system, some additional paths are created for the proper functioning of the system. If all the redundant parts start working together at the time of operation, then it is termed as parallel redundancy. A standby redundant system is the one in which one operating unit is followed by spare units called standbys. On the failure of the operating unit, a standby unit is switched on by perfect or imperfect switching device. In the present discussion, the authors have considered a compound system consisting of three sub-systems A, B and C. The sub-system B consists of two units, one is main and the other is its standby. These units further consist of two sub-units viz, (B11, B21) in main unit and (B12, B22) in its standby redundant system. The sub -system B also has two imperfect switching devices S1 and S2. S1 connects B11 and B22 and S2 connects B21 and B12. B12 and B22 sub-units are same as B11 and B21 respectively. Initially, in sub-system B, the sub-units B11 and B21 are assumed to be in operation. If B11 fails then B12 and B21 may begin to operate through a switching device S2. If B21 fails then B11 and B22 may begin to operate through a switching device S1. The failure in sub-system C requires waiting time for repair. The system will be in down state due to failure of subsystem A or C or occurrence of any human error. Also, due to failure of two sub-units of one unit and one subunit of other unit of sub-system B, the system suffers complete break down. The Laplace transforms of the time dependent probabilities of the system being in various states have been obtained by employing the supplementary variable technique. Various reliability parameters have been computed and some tabular and graphical illustrations are also given at the end of the paper. The state transition diagram of the system is shown in Fig. a. ( , ) devices and the failures of sub-systems are exponentially distributed. * All the units recover their functioning perfectly after repair. * All the repairs are perfect i.e., the repair facility never does any damage to the units. * A failed subsystem is repaired at a single service channel.
* At time t=0, B11 and B21 are operating and B12 and B22 are in standby mode. * As soon as the operating unit fails, it is replaced by it 's standby unit. * The system ceases to function due to failure of subsystem A or C or due to occurrence of any human error.
* The system suffers complete break down due to failure of sub-systems B21, B22 or B11, B12 or any three sub-units of sub-system B at a single state. * The sub-system C requires waiting time for repair. * In state S(7) of the system no priority is given to any sub-unit i.e., B11, B21 and repair rate of these sub-units is assumed to be identical. * Standby sub-units in sub-system B are assumed to be perfect as long as they are in standby mode. State probability description 0 ( ) P t : Probability that the system is in operable state S(0) at time 't'.
1 ( ) P t : Probability that the system is in failed state S(1) at time t. 2 ( , ) P z t ∆ : Probability that the system is in waiting state S(2) with elapsed repair time lying in the interval ( , ) z z + ∆ .

( ) P t
: Probability that the system is in failed state S(3) at any time't' 4 ( ) P t : Probability that the system is in failed state S(4) at any time't'. 5 ( , ) P x t ∆ : Probability that the system is in operable state S(5) with elapsed repair time lying in the interval ( , ) x x + ∆ . 6 ( , ) P x t ∆ : Probability that the system is in operable state S(6) with elapsed repair time lying in the interval ( , ) x x + ∆ . 7 ( , ) P x t ∆ : Probability that the system is in operable state S(7) with elapsed repair time lying in the interval ( , ) x x + ∆ . 8 ( , ) P y t ∆ : Probability that the system is in failed state S(8) with elapsed repair time lying in the interval ( , ) y y + ∆ . 9 ( , ) P z t ∆ : Probability that the system is in failed state S(9) with elapsed repair time lying in the interval ( , ) z z + ∆ . 10 ( , ) P z t ∆ : Probability that the system is in failed state S(10) with elapsed repair time lying in the interval ( , ) z z + ∆ . 11 ( , ) P y t ∆ : Probability that the system is in failed state S(11) with elapsed repair time lying in the interval ( , ) y y + ∆ . 12 ( , ) P z t ∆ : Probability that the system is in failed state S(12) with elapsed repair time lying in the interval ( , ) z z + ∆ .
x P x t dx y P y t dy z P z t dz z P x t dx z P z t dz z P z t dz z P z t dz y .

Reliability:
The reliability of the system is given as follows: 1.
ii. For repairable system, the profit function ) (t H in the interval (0,t) is given by                  Table 3.  Table 1exhibits that the operational availability of the system decreases with increase in time period. Table 2 shows that the reliability of the system decreases as the time period increases. Table 3 and 4 show the variation in profit function with the increase in time period for non -repairable and repairable system respectively. The series of curve in both the figures exhibits that the profit function decreases with the increase in service cost of the system. Table 5 depicts the variation of M.T.T.F. with the failure rate of subsystem B11 (or B12). The series of curve represents that M.T.T.F. decreases as the failure rate of sub -system A increases. Table 6 depicts the variation of M.T.T.F. with human failure rate. The series of curve represents that M.T.T.F. decreases as the failure rate of subsystem C increases. Table 7 depicts the variation of M.T.T.F. with the human failure rate. The series of curve represents that M.T.T.F. decreases as the failure rate of subsystem A increases.