I am trying to make sense of the following setting.
M/M/1 queue. $\lambda$: arrival rate, $s=1/\mu$: average service time. Then, the mean waiting time in the queue is: $w_q={{\rho^2} \over {1-\rho}} \cdot {1 \over \lambda}$ (for $\rho = \lambda s$).
Now suppose we increase the arrival rate by a factor of $c$, but all the additional elements have zero service time. Denote the new parameters with $\hat{\square}$. Then, $\hat{s}=s/c$, $\hat{\lambda}=c\lambda$. So, $\hat{\rho}=\rho$. So, $\hat{w}_q=w_q/c$.
But why? Suppose that $\rho$ is near 1. So, the server is busy most of the time. So, the new elements will be essentially waiting just like the old ones. So, why does the average waiting time fall so dramatically?
What am I missing?
Thanks.