Pollaczek-Khinchine formula

For a stable first-come first-served M/G/1 queue, write $\rho =\lambda \mathbb{E}S<1$ and $\tilde{S}(s)=\mathbb{E}{e}^{-sS}$. The stationary waiting-time Laplace transform isWith finite second service moment, differentiating at zero gives the Pollaczek-Khinchine mean formula. These are the transform and mean forms of the same queueing identity.

For a stable $M/G/1$ queue with traffic intensity $\rho =\lambda \mathbb{E}S<1$, the mean waiting time before service isEquivalently, the mean number in the system is