Problem 3. A service station in use at time t is designated stateX() -1, and X(t) = 0 if not in use. Assume that {X(t),1 > 0) is a two-state continuous-time Markov chain with states (0,1} and the matrix P() of transition probability functions is given as follows P(O) = ( Pot Pur(t) where 3 1 3 4 a) Suppose the service station is not in use at t=0, what is the probability it will in use at time t = 57 Pole) :**, Puce) = + b) Find the infinitesimal or the rate matrix R =PO = 400 901 e) Write down (no need to solve the Kolmogorov Backward Equations for Poſt). d) Find the stationary distribution (170, *} by solving the following equation 0-29), j = 0,1 k together with Ex= 1.