Erlang Redux Resolved! (This time for real)

As I show in my Perl::PDQ book, the residence time at an M/M/1 queue is trivial to derive and (unlike most queueing theory texts) does not require any probability theory arguments. Great for Guerrillas! However, by simply adding another server (i.e., M/M/2), that same Guerrilla approach falls apart. This situation has always bothered me profoundly and on several occasions I thought I saw how to get to the exact formula—the Erlang C formula—Guerrilla style. But, on later review, I always found something wrong.

Although I've certainly had correct pieces of the puzzle, at various times, I could never get everything to fit in a completely consistent way. No matter how creative I got, I always found a fly in the ointment. The best I had been able to come up with is what I call the "morphing model" approximation where you start out with $m$ parallel queues at low loads and it morphs into a single $m$-times faster M/M/1 queue at high loads.

That model is also exact for $m = 2$ servers—which is some kind of progress, but not much. Consequently, despite a few misplaced enthusiastic announcements in the past, I've never been able to publish the fully corrected morphing model.