Consider two machines. Machine i operates for an exponential time with rate i and then fails; its

Consider two machines. Machine i operates for an exponential time with rate λi and then fails; its repair time is exponential with rate μi,i = 1, 2. The machines act independently of each other. Define a four-state continuous-time Markov chain that jointly describes the condition of the two machines. Use the assumed independence to compute the transition probabilities for this chain and then verify that these transition probabilities satisfy the forward and backward equations.