# Departure Process Of $M/M/S/K$ queue

Regarding to Burke Theorem it said that $M/M/S$ queue with Poisson arrival rate $\lambda$ will have departure process with rate $\lambda$. as far as i know it happen as the affect of time reversibility that actually applied for death and process .

My question is :

for finite capacity case $(M/M/S/K)$ with Poisson arrival rate $\lambda$ and exponential service time with rate $\mu$. how is the departure process rate will be? will it be Poisson rate $\lambda$ too like in Burke theorem or not?

by considering that time reversibility still apply as it is part of birth and dearh process i think it probably will have Poisson departure process with rate $\lambda$ but then i confused about the costumer who leave knowing that the queue is full. i hope any one can help me, thank you in advanced.

The departure rate cannot be $\lambda$ because in a finite buffer system some of the work is lost. The effective departure rate is then $\lambda P(Q<k)$, i.e., the rate multiplied by the probability of not finding the system full.