A birth and death process, {X{t), t ≥ 0}, has the following birth and death rates:
Moreover, the capacity of the system is equal to two individuals.
(a) Calculate, assuming that λ = μ, the average number of individuals in the system at a time instant t (large enough), given that the system is not empty at this time.
(b) Calculate the probability that the process will spend more time in state 0 than in state 1 on two arbitrary visits to these states.
(c) Suppose that μ1 = 0 and that, when X(t) = 2, the next state visited will be 0, at rate 2μ. Write the balance equations of the system, and solve them to obtain the limiting probabilities.