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Queueing systems which map Poisson input processes to Poisson output processes have been well-studied in classical queueing theory. This paper considers two discrete-time queues whose analogs in continuous-time possess the Poisson-in-Poisson-out property. It is shown that when packets arriving according to an arbitrary ergodic stationary arrival process are passed through these queueing systems, the corresponding departure process has an entropy rate no less (some times strictly more) than the entropy rate of the arrival process. Some useful by-products are discrete-time versions of: (i) a proof of the celebrated Burke's (1956) theorem, (ii) a proof of the uniqueness, amongst renewal inputs, of the Poisson process as a fixed point for exponential server queues proposed by Anantharam (1993), and (iii) connections with the timing capacity of queues described by Anantharam and Verdu (1996).