Burke's theorem says that the output process of an $M/M/1$, an $M/M/C$, and a $M/M/\infty$ queue with arrival rate $\lambda$ and service rate $\mu$ follows a Poisson with parameter $\lambda$.
I was able to derive the proof for the M/M/1 easily using Laplace transforms:
For the busy period, the time spent in between departures is given by an exponential distribution with parameter $\mu$. For the non-busy period the inter-departure is given by a sum of an inter-arrival and one service time.
$\mathcal{L}_D(s)=\frac{\lambda}{\mu} \cdot \frac{\mu}{\mu+s}+(1-\frac{\lambda}{\mu})\cdot \frac{\lambda}{\lambda+s}\cdot\frac{\mu}{\mu+s}= \frac{\lambda}{\lambda+s} $
$D(t) = \lambda e^{-\lambda t}$
However, I am struggling immensely to derive the proof for the $M/M/\infty$ case. Can someone point out where I am going wrong?
So far I have for the busy period: $\sum_{i=1}^{\infty} e^{-\lambda/\mu}\cdot \frac{i\mu}{i\mu+s}$. I am not even sure if this is correct. Do we need to account for the possibility of new arrivals before the departure too? How one would go with that?
Also, how would I set it up for the non-busy period? And how do I deal with this $\frac{i\mu}{i\mu+s}$ term in the busy one?
Thanks!
Well the study of the output process is always not an easy thing to do. Even for the $M/M/1$ queue your proof is not enough to describe the nature of the output process. What is being missed? For example, you did not say anything about the joint distribution of the two consequtive inter-departure times. If the departure process is Poisson, the two consequtive inter-departure times must be independent. Well, even this is answer would not be enough to fully characterize the output process. Please see https://www.jstor.org/stable/1425911 .
The step you made for the $M/M/1$ queue you can make for the $M/M/\infty$ as well. Firstly find the joint stationary distribution of the total number of customers in the queue (it is has Poisson distribution, see Section 6.1 in P.P. Bocharov, C. D'Apice, A.V. Pechinkin and S. Salerno: Queueing Theory, Modern Probability and Statistics, VSP, The Netherlands, 2004.). Then by conditioning on the number of customers, you can calculate time the next departure instant (in terms of Laplace transform). Or course, when the system is busy the next departure instant will have exponential distribution but with the parameter, depending on the current number of customers in the system. Thus, of coursem you have to track whether there was a new arrival or not before the next departure.