# 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.

## 1 Answer

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.

Furthermore, the departure process is no longer Poisson. In fact, even the arrival process of admitted customers (not including lost customers) is no longer Poisson because there is dependence between inter-admission times.