An Alternative Proof of a Theorem of Takács on the GI/M/1 Queue

An analytic proof is given of the fact that the stationary distribution for the imbedded Markov chain in a GI/M/1 queue is geometric. A generating function for the stationary transition probabilities is obtained as the unique solution to an integro-differential equation, which may be solved by reduction to a Wiener-Hopf equation.