In Example 6.20, we computed m(t) = E[O(t)], the expected occupation time in state 0 by time t for the two-state continuous-time Markov chain starting in state 0. Another way of obtaining this quantity is by deriving a differential equation for it.

(a) Show that m(t + h) = m(t) + P00(t)h + o(h)

(b) Show that m

(t) = μ

λ + μ

+

λ

λ + μe−(λ+μ)t

(c) Solve for m(t).