The Multiserver Numbers Game

In a previous post, I explained why \begin{equation} R_m \neq \dfrac{R_1}{m} \label{eqn:badest} \end{equation} and therefore doesn't work as an estimator of the mean residence time in an M/M/m multi-server queue. Although we expect the extra server capacity with m-servers to produce a shorter residence time ($R_m$), it is not m-times smaller than the residence time ($R_1$) for a single-server (i.e., $m = 1$) queue.

M/M/m multiserver queue

The problem is that eqn. \eqref{eqn:badest} grossly underestimates $R_m$, which is precisely the wrong direction for capacity planning scenarios. For that purpose, it's generally better to overestimate performance metrics. That's too bad because it would be a handy Guerrilla-style formula if it did work. You would be able do the calculation in your head and impress everyone on your team (not to mention performing it as a party trick).

Given that eqn. \eqref{eqn:badest} is a poor estimator, you might wonder if there's a better one, and if you'd been working for Thomas Edison he would have told you: "There's a better wsy. Find it!" Easy for him to say. But if you did decide to take up Edison's challenge, how would you even begin to search for such a thing?

Little's Lore

Guerrilla alumnus Paul P. has a penchant for sending me interesting things, and recently he sent me a piece on Little's law. Remarkably, it wasn't just another proof of L = λW, but a brief retrospective written by none other than John Little himself. I quote it here because it not only provides some unusual insight into how these things get done, but it is written in a charming and self-effacing style.