I found the Q matrix, and I know that I will solve pi*Q=0. But I have difficulty for the second part, to find the probability Poo
The question is below:
A mail order company receives orders via an automated telephone answering service. The orders arrive according to a Poisson process with intensity λ>0. The answering machine is emptied at time points that form another Poisson process, which is independent of the arrivals and has intensity µ > 0, after which all received orders on the
answering machine are immediately treated. It is assumed that new calls can arrive immediately upon emptying the answering machine and that there are no orders waiting for service at time t = 0.
a) Find the unique stationary distribution for the number of customer orders on the answering machine.
b) Show that the probability that the answering machine is empty at time t ≥ 0 is given by (P_zero_zero_t)
Poo(t)=(µ / (λ+µ)) + (λ / (λ+µ))*e^-(λ+µ)t
The question is below:
A mail order company receives orders via an automated telephone answering service. The orders arrive according to a Poisson process with intensity λ>0. The answering machine is emptied at time points that form another Poisson process, which is independent of the arrivals and has intensity µ > 0, after which all received orders on the
answering machine are immediately treated. It is assumed that new calls can arrive immediately upon emptying the answering machine and that there are no orders waiting for service at time t = 0.
a) Find the unique stationary distribution for the number of customer orders on the answering machine.
b) Show that the probability that the answering machine is empty at time t ≥ 0 is given by (P_zero_zero_t)
Poo(t)=(µ / (λ+µ)) + (λ / (λ+µ))*e^-(λ+µ)t