In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.
