This paper presents a proof of the rate distortion function of a Poisson process with a queuing distortion measure that is in complete analogy with the proofs associated with the rate distortion functions of a Bernoulli source with Hamming distortion measure and a Gaussian source with squared-error distortion measure. Analogous to those problems, the distortion measure that we consider is related to the logarithm of the conditional distribution relating the input to the output of a well-known channel coding problem, specifically the Anantharam and Verdu "Bits through Queues" coding problem. Our proof of the converse utilizes McFadden's point process entropy formulation and involves a number of mutual information inequalities, one of which exploits the maximum-entropy achieving property of the Poisson process. Our test channel uses Burke's theorem to prove achievability.