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](http://possiblywrong.wordpress.com/2010/12/26/why-multiple-checkout-lines-are-not-all-bad/) 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.



Average wait time for multiple queues where arrivals enter shortest queue
There is also a paper by Vvedenskaya, Dobrushin, and Karpelevich "Queueing System with Selection of the Shortest of Two Queues: An Asymptotic Approach" (Problems of Information Transmission, 1996, 32:1, 15–27; a Russian original is freely available at <http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=298&option_lang=eng>).


They consider a policy in which each customer chooses two queues at random and then, among them, the shortest one. Considering the limit of a large number of servers they are then able to derive an equation for the asymptotic stationary distribution, which turns out to be a discrete approximation to the Hopf (inviscid Burgers) equation.


As far as I know there was some development of this work for the case of more than two queues, but it is better to ask Nikita Vvedenskaya directly (ndv AT iitp DOT ru; I've asked her permission to post this address).


