We consider a system made up of two subsystems placed in parallel. The first subsystem is composed of two components (components nos. 1 and 2) placed in parallel, while the second subsystem comprises a single component (component no. 3). Let Sk denote the lifetime of component no. k, for k = 1,2,3. The continuous random variables Skare assumed to be independent.
(a) Suppose that components nos. 1 and 2 operate at the same time, from the initial time, whereas component no. 3 is in standby and starts operating when the first subsystem fails. When the system breaks down, the three components are replaced by new ones. Let N(t)., for t > 0, be the number of system failures in the interval [0,t]. Then{N(t),t ≥ 0} is a renewal process. Let r be the time between two consecutive renewals. Calculate the mean and the variance of r if Sk ~ U(0,1), for k = 1,2,3.
(b) Suppose that we consider only the first subsystem and that the two components are actually placed in series. When this subsystem fails, the two components are replaced by new ones. As in (a), the process {N(t),t ≥ 0} is a renewal process. Calculate the renewal function mN(t), for 0 k ~ U(0,1), for k: = 1,2
Indication. The general solution of the differential equation
(c) Suppose that in (b) we replace only the failed component when the subsystem breaks down and that Sk ~ Exp(2), for k = 1,2.
