We consider a system composed of three components placed in parallel and operating independently. The lifetime X i (in months) of component I has an exponential distribution with parameter A, for i =...

We consider a system composed of three components placed in parallel and operating independently. The lifetime
X_i
(in months) of component
I
has an exponential distribution with parameter A, for
i
= 1,2,3. When the system breaks down, the three components are replaced in an exponential time (in months) with parameter μ. Let
X(t)
be the number of components functioning at time
t.
Then
{X(t),t
> 0} is a continuous-time Markov chain whose state space is the set {0,1,2,3}.

(a) Calculate the average time that the process spends in each state.

(b) Is the process
{X(t),t
≥ 0} a birth and death process? Justify.

(c) Write the Kolmogorov backward equation for P_o,o(t)

(d) Calculate the limiting probabilities of the process if λ = μ.