Cost Analysis of A System with Preventive Maintenance by Using the Kolmogorov’s Forward Equations Method

Abstract: This study deals with cost analysis of a twounit cold standby redundant system with preventive maintenance.The random failure occurs at random times which follow an exponential distribution and also the repair time are assumed to be exponentially distributed. Using the Kolmogorov’s forward equations method. Several reliability characteristics are obtained. The mean time to system failure (MTSF) and the profit function are studied graphically.


INTRODUCTION
Many authors [1,2] have studied the two unit redundant systems with two types of repair. [3] have studied stochastic analysis of a two -unit parallel system with partial and catastrophic failure and preventive maintenance. [4,5,6,7] have studied the cost analysis of different systems. In [8] by using the Kolmogorov's forward equations method. Evaluate the MTSF and availability of two different systems. This study devoted deals with cost analysis of a two-unit cold standby redundant system with two types of failure and preventive maintenance by using the Kolmogorov's forward equations. Initially one unit is operative and the other is kept as cold standby, i.e. it does not fail while standing by. Each unit works in two different types of failures. The both systems fail when both units fail totally. The failure and repair times are assumed to have exponential distribution. Using the special case study the effect of preventive maintenance on the system performance is shown by performing comparisons theoretically and graphically.
The following notations are adopted for the system: 1 α constant failure rate of type I. 2 α constant failure rate of type II. 1 β constant repair rate of type I. 2 β constant repair rate of type II.
) (  , we obtain the following differential equation: to evaluate the transient solution is too complex therefore we will restrict ourselves in calculating the MTSF. To calculate the MTSF we take the transpose matrix of Q and delete the rows and columns for the absorbing state the new matrix is called A. the expected time to reach an absorbing state is calculated from

. Availability analysis:
The initial conditions for this problem are the same as for the reliability case: , 0, 0, 0, 0, 0, 0, 0 = , the differential equations form can be expressed as: The steady state availability can be obtained using the following procedure. In the steady state, the derivatives of the state probabilities become zero. That allows us to calculate the steady state probabilities with.
we solve the equation (4.2) and the following normalizing condition:    The steady state busy period can be obtained using the following procedure. In the steady state, the derivatives of the state probabilities become zero. That allows us to calculate the steady state probabilities with.    The steady state, the expected frequency of preventive maintenance per unit time can be obtained using the following procedure. In the steady state, the derivatives of the state probabilities become zero. That allows us to calculate the steady state probabilities with.
2) or, in the matrix form  Cost analysis: The expected total profit per unit time incurred to the system in the steady-state is given by: Profit = total revenue -total cost PF: is the profit incurred to the system, C 0 : is the revenue per unit up-time of the system, C 1 : is the cost per unit time which the system is under repair C 2 : is the cost per preventive maintenance.
Special case: After study the system when the preventive maintenance is not allowed, we get The Mean Time to System Failure is given by MTSF = The steady state busy period is given by The steady state, the expected frequency of preventive maintenance per unit time is given by  The expected total profit per unit time incurred to the system in the steady-state is given by , C 0 =1000, C 1 = 100, C 2 = 100. The MTSF and the profit function of the system decrease for both systems with and without preventive maintenance. We conclude that the system with preventive maintenance is more grater than the system without preventive maintenance with respect to the MTSF and the the profit function incurred to the model.