Let (left(N_{t} ight)) be a birth-death process with only two states, 0 and 1 . Denote by (lambda) the birth rate at state 0 ,

Let \(\left(N_{t}\right)\) be a birth-death process with only two states, 0 and 1 . Denote by \(\lambda\) the birth rate at state 0 , and \(\mu\) the death rate at state 1 .

(a) Write the Kolmogorov forward equations for \(P_{00}(t)\) and \(P_{11}(t)\).

(b) Solve the equations obtained in part (a).

(c) A pay phone may be either engaged (state 1 ) or unengaged (state 0 ). Assuming that a birth-death process is an appropriate model, find the long-run proportion of time that the phone is engaged.