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In this paper, we derive rate-distortion functions under proper magnitude-error fidelity criteria and study instrumentable data-compression schemes for Poisson processes. In particular, we derive information rates and obtain rate-distortion relationships for practical data-compression schemes, for the reproduction of the unordered sequence of Poisson event occurrences, for the reproduction of the sample functions of the Poisson counting process, and for the reproduction of the sequence of intervals between the event occurrences of a Poisson process. The reproducing processes are taken to be point (or jump) processes themselves. The performances of the various data-compression schemes presented here are compared with those of the ideal schemes (us presented by the rate-distortion functions) and are shown to be close to the latter over wide regions of distortion.