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**1 - 1**of**1**### 1The Rate-Distortion Function of a Poisson Process with a Queuing Distortion Measure

"... This paper characterizes 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 dis-to ..."

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This paper characterizes 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 dis-tortion 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 ” [1] coding problem. We show this rate-distortion problem to be equivalent to a standard rate-distortion problem due to: i) the independent increments property of the Poisson process ii) the numerical entropy rate of any finite-rate point process tending to 0, iii) the existence of a reproduction with finite expected distortion, iv) the additive structure and “honesty ” of the distortion measure. Our Shannon lower bound involves a number of mutual information inequalities, one of which exploits the maximum-conditional-entropy property of the exponential server timing channel. We also show that the rate-distortion functions pertaining to expected distortion and deviation probability are equivalent. We conclude with a comparison to other