Consider a standard M/M/N queue where the jobs' arrival rate is $N\lambda$, the service rate of each of the $N$ servers is $1$, and $\lambda < 1$.
When a new job arrives, assign it to an idle server with the longest idle time. I would like to ask, in the steady state, what is the expected idle time of a server once when it becomes idle? Specifically, does the expected idle time goes to zero as $N$ goes to infinity?