Exercise 11.61 Let L be a birth–death process with birth rate 1 and death rate 1. Suppose that L0 is a random variable having the Poisson distribution with parameter α. Show that the probability that the process is extinct by time t is exp[−α/(t + 1)]

If λ < μ, the queue has a unique steady-state distribution given by πk = 
1 −
λ
μ

λ
μ
k for k = 0, 1, 2, . . . . (11.75)
If λ ≥ μ, there is no steady-state distribution.