A system fails after a random lifetime L. Then it waits a random time W for renewal. A renewal takes another random time Z. The random variables L, W and Z have exponential distributions with...

A system fails after a random lifetime L. Then it waits a random time W for renewal. A renewal takes another random time Z. The random variables L, W and Z have exponential distributions with parameters λ, ν and μ, respectively. On completion of a renewal, the system immediately resumes its work. This process continues indefinitely. All life, waiting, and renewal times are assumed to be independent. Let the system be in states 0, 1 and 2 when it is operating, waiting or being renewed..

(1) Draw the transition graph of the corresponding Markov chain {X(t), t ≥ 0}.

(2) Determine the point and the stationary availability of the system on condition