On the input-output map of a G/G/1 queue

In this note, we consider GIG/I 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 GIMI1 queues with stationary and ergodic inputs, the MIMI/1 queue is the only queue whose