System composed of two subsystems placed in series

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We consider a system composed of two subsystems placed in series. The first subsystem comprises a single component (component no. 1), whereas the second subsystem comprises two components placed in parallel. Let S_kbe the lifetime of component no. k, for k = 1,2,3. We assume that the continuous random variables S_k are independent.

(a) Let N(t) be the number of system failures in the interval [0,t]. In what case(s) will the stochastic process {N(t),t ≥ 0} be a renewal process if the random variables Sk do not all have an exponential distribution? Justify.

(b) In the particular case when the variable Sk has an exponential distribution with parameter λ = 1, for all k, the process {N(t),t ≥ 0} is a renewal process. Calculate the probability density function of the time r between two consecutive renewals.

Reference no: EM131130653

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