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

Question:

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.

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