Baccelli, FrancoisMakowski, Armand M.An exponential martingale is associated with the Markov chain of the number of customers in the M/GI/1 queue. This together with renewal theory are shown to provide a unified probabilistic framework for deriving several well-known generating functions for the M/GI/1 queue, including the Pollaczek-Khinchine formula, the transient generating function of the number of customers at departure epochs and the generating function of the number of customers served in a busy period.en-USTechnical Report