I have just started studying the topic of queuing systems and I am having trouble with grasping the intuition that I need to start solving questions.
Consider a M/M/1 queue where arrivals occur as a Poisson Process with a rate of 4 customers per day (=24 hours). Then the service times are exponentially distributed with mean of 4 hours. I know that $\mu=\frac{1}{4}$ denotes the service rate. But since I have that $\lambda=4$, I have that the stability condition
\begin{equation}
\rho=\frac{\lambda}{\mu}<1,
\end{equation}
is clearly not met, which should be the case. My thinking is therefore that I need to rearrange the $\lambda$ in terms of either per hour or connecting to the service time, per 4 hours. I just can't decide which one. Could someone help me with figuring out how to interpret this situation?
Best Answer
It works out if you use consistent units for the rates:
You have arrivals at a rate of $4$ per day or $\lambda =\frac16$ an hour
and a service rate of $\mu =\frac14$ an hour
so $\rho= \frac{1/6}{1/4} =\frac23$ which is indeed less than $1$