> length of the queue becomes infinite If you simulated the queue forever and took a running average (for any finite length of time that's just an integral, an

> length of the queue becomes infinite If you simulated the queue forever and took a running average (for any finite length of time that's just an integral, and you take the limit as t->infinity) of the queue length, the result would "be infinite" -- for ever possible real number the result of such a limit of it existed would be greater. > is poisson special It's easiest to work out with elementary mathematical tools with a poisson process, but otherwise no. There's a decent overview [0] that derives the expected length, and showing that it grows without bound up to 100% is suggestive of the result (combine that with the fact that the mean length is strictly monotonic increasing where it exists and is infinite at >100% utilization, and that shores up the mathematical loose ends). For the poisson process in particular, the "balance principle" in that link I think will serve to provide the right intuition about the problem. > should converge on the number of outcomes in the middle That's the problem. We have infinite head and tail flips, but when the length is 0 only half of those matter. Adding up those one-sided differences still wouldn't have to be infinite per se, but 100% utilization is the threshold at which it gets pushed over the edge (large differences are "rare" to get to in the first place but you'll spend time inverse to that probability in their vicinity). [0] https://www.win.tue.nl/~iadan/que/h4.pdf https://www.win.tue.nl/~iadan/que/h4.pdf