Reliability Analysis of Two Unit Standby Model with Controlled and Uncontrolled Failure of Unit and Replacement Facility Available in the System

Planning a highly reliable and efficient system has always been a primary interest for reliability engineers by devising the powerful design strategy and employing effective repair and replacement policy. Keeping in view this, the basic aim of this paper is to analyze the reliability of a system which comprised of two units A and B in which unit A is functional and B is held standby. Unit A after failure may be controlled or uncontrolled. The failed unit undergoes for repair in the controlled unit. If repair of a unit is not controlled then it is replaced by a new one. Upon breakdown of operational unit A, unit B come becomes active instantaneously. Unit B after failure is repaired by regular repairmen. System failure takes place when both the units quit serving. The unit serves as good as a fresh after preventive repair and replacement policy. The regenerative point technique has been used to obtain the expression for several reliability measures. Finally, the graphical behavior of MTSF and profit of the present model has been observed for arbitrary values of parameters and costs.

may be either controlled or uncontrolled. If failure of the unit A is controlled then usual repairperson attends the system to do the repair activities of the failed unit. However if the breakdown of unit A is uncontrolled then unit A is replaced by new one. Moreover unit B after failure is repaired by regular repairperson. The system may work as fresh as new after getting repaired and replaced. Further the failure time distributions of both the units are taken as exponential. Also, the times to repair distribution of both the units are assumed as general.

Notations and Symbols:
1 = failure rate of unit A when A is in controlled state. 2 = failure rate of unit A when A is in uncontrolled state. γ 1 = replacement rate of A. 1 = failure rate of B. ( )= repair rate of A when A is in controlled state. ( )= repair rate of A when A is in uncontrolled state. ( )= repair rate of B. 0 = Unit A is normal and operative. 0 = Unit B is normal and operative. = Unit B is standby. = Unit A fails, which is controllable and under repair. = Unit A fails, and is under repair. = Unit A is in failure mode and is under replacement. = Unit B waits for repair.

States of the System:
The possible states of the system are: TRANSITION DIAGRAM System Description: The system can be in any of the following states under given assumptions: State 0: Initially unit A is operational and unit B is kept standby. State 1: Unit A is in failed state (uncontrolled) , it goes for repair while Unit B becomes operational and the system is working and hence in upstate. State 2: Upon A is in failure mode (controllable) and is under repair. Unit B comes in operation and the system is in upstate. State 3: Unit A is in failure mode and is continuously under repair from past state. Unit B has also failed and is waiting for repair. The system is in down state. State 4: Unit A is in operational mode. Unit B has failed and repairman is busy in repairing the failed unit. Stage 5: The uncontrolled failure unit under repair is replaced by a new unit ( ). Unit B comes is in operational mode. The system is in upstate. State 6: The uncontrolled Unit A is under repair while as Unit B has also failed and is waiting for repair. The system is in down state.

Mean Sojourn Times:
If T i represents the sojourn time in state S i then mean sojourn time Ψ i in state S i is:

Mean Time To System Failure:
To get the expressions for MTSF denoted by π i (t) for different values of i, the arguments of regenerative point processes has been used, taking the Laplace transform and solving the resultant set of equations for 0 * ( ), we have On taking → 0 and using the relation ̌( ) → , we have Thus 1 (0) = 1 (0) showing that ̃0(0) = 1. Hence 0 ( ) is a proper cdf. Therefore, mean time to system failure when the initial state is 0, is given by where, The steady state Availability will be given by A 0 = lim t→∞ A 0 (t) = lim s→0 s A 0 * (s) = 2 (0)/ 2 ′ (0) Where 2 (0) = 0 + 01 1 + 02 2 + ( 01 16 + 02 23 ) 4 + 01 15 5 3 ( ) = 2 ( ) is same as in availability analysis which is given by (1) In the steady state, the probability that the regular repairman will be busy is given by

Graphical Study of the System Model:
In this section, the behavior for MTSF and profit function has been studied with respect to failure and repair rates. Giving the arbitrary values to the parameters the numerical results for MTSF and profit function are obtained to show the graphical behavior with respect to different parameters, keeping other value of parameters fixed. The behavior of measures is shown in Fig. 1,2,3,4. Fig. 1 gives the graphical behavior of MTSF with respect to failure rate 1 , keeping the other values of parameters 2 , 1 , 2 , 1 , γ 1 fixed. It is seen that as failure rate 1 increases the MTSF of the system decreases. So we infer that with decrease in the failure rate of the unit which is in failure mode 1 , the expected life of the system increases. Fig. 2 demonstrates MTSF v/s Repair rate. We plot MTSF with respect to 1 and fixed values of parameters 1 , 2 , 2 , 1 , γ 1 . We see that MTSF of the system increases as repair rate increases. So we conclude that with increase in repair rate of a unit in the repair mode 1 , the expected life of the system increases. Fig. 3 depicts the graphical behavior of profit with respect to failure rate 1 . It is examined that profit of the system decreases as failure rate increases regardless of other parameters. So we see that with the decrease in failure rate of unit in failure mode 1 , the expected life of the system increases. In Fig. 4, we plot profit with respect to repair rate 2 . It is clear from the graph that there is an increase in the profit of the system with increase in the repair rate regardless of the other parameters. So we infer that with increase in repair rate of the unit in repair mode 2 , the expected life of the system increases. for different values of θ 2 , β 1 , β 2 , α 1 , γ 1 , k 0 , k 1 , k 2 values of θ 2 , θ 1 , β 2 , α 1 , γ 1 , k 0 , k 1 , k 2

Conclusion:
It can be observed that with increase in the failure rate of the system, the MTSF, profit analysis of the system decreases uniformly regardless of other parameters fixed. However, we see that MTSF and profit analysis increases with increase in repair rates. Hence the study winds up that the system is more profitable by decreasing failure rate of the system in failed state and speeding up the repair rate of the unit of the system in repair mode which will in turn enhance the reliability and consequently efficiency of the system.