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 \(\lambda, v\), and \(\mu\), 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. The transitions between the states are governed by a Markov chain \(\{X(t), t \geq 0\}\).

(1) Draw the transition graph of \(\{X(t), t \geq 0\}\) and set up a system of linear differential equations for the time-dependent state probabilities \(p_{i}(t)=P(X(t)=i), i=0,1,2\).

(2) Use this system to derive an algebraic system of equations for the stationary state probabilities \(\pi_{i}\) of \(\{X(t), t \geq 0\}\). Determine the stationary availability of the system.