Consider a 1 -out-of- 2 system, i.e., the system is operating when at least one of its two subsystems is operating. When a subsystem fails, the other one continues to work. On its failure, the joint renewal of both subsystems begins. On its completion, both subsystems resume their work at the same time. The lifetimes of the subsystems are identically exponential with parameter \(\lambda\). The joint renewal time is exponential with parameter \(\mu\). All life- and renewal times are independent of each other. Let \(X(t)\) be the number of subsystems operating at time \(t\).

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

(2) Given the initial condition \(P(X(0)=2)=1\), determine the time-dependent state probabilities \(p_{i}(t)=P(X(t)=i), \quad i=0,1,2\), and the stationary state distribution.

Consider separately the cases \((\lambda+\mu+v)^{2}(=)(<)(>) 4(\lambda \mu+\lambda v+\mu v)\).