A letnbspninbspbe the number of consecutive days during

Question

A system is composed of two components that operate at alternate times. When component f, for i = 1,2, starts to operate, it is active during X_i days, where X_i is an exponential random variable with parameter  and is independent of what happened previously. The state of the components is checked only at the beginning of each day. If we notice that component I is down, then we set the other component going, and component i will be repaired (in less than one day).

(a) Let N_i be the number of consecutive days during which component I is responsible for the functioning of the system, for i = 1,2. What is the probability distribution of ?

(b) Suppose that the two components are identical. That is, λ₁ = λ₂ := λ. At what rate do the components relieve each other (over a long period)?

(c) If λ₁ = 1/10 and λ₂ = 1/12, what proportion of time, when we consider a long period, is component 1 responsible for the functioning of the system?