I have been able to find, and understand reasonably well, expressions and derivations for the average wait times for (1) $s$ independent $M/M/1$ queues each with arrival rate $\lambda/s$ and service rate $\mu$, and (2) an $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$. (See this recent discussion for details. Motivation for this question is the anecdotal result that "one long line" is better than "many individual lines.")

But what if customers arrive with rate $\lambda$ and join the *shortest* of $s$ queues, each with service rate $\mu$? Is there an expression (preferably with derivation as well, I'm not looking for a recipe) for the average wait time in this case? Simulation of this approach suggests that the average wait time is worse than $M/M/s$... but not *that* much worse. I would like to figure out how to tackle this analytically.

Perhaps a side effect of an answer might simply be recommendation of a good queuing theory text. My challenge has been that there seems in some cases (online sources, etc.) to be no distinction, or at least confusion, between this "join the shortest line" situation and the $s$ x $M/M/1$ case. But simulation and my intuition at least suggests that they are very different.