= Pollaczek-Khinchine formula
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{wiki=Pollaczek–Khinchine_formula}
For a stable first-come first-served <M G 1 queue>, write $\rho=\lambda\mathbb ES<1$ and $\widetilde S(s)=\mathbb E e^{-sS}$. The stationary waiting-time <Laplace transform> is
$$
\widetilde W_q(s)=\frac{(1-\rho)s}{s-\lambda+\lambda\widetilde S(s)}.
$$
With 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.
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