  In this paper we develop the first perfect sampling algorithm for queues with
Hawkes input, i.e. single-server queues with Hawkes arrivals and i.i.d. service
times of general distribution. In addition to the stability condition, we also
assume the excitation function of the Hawkes process has a light tail and the
service time has finite moment generating function in the neighborhood of the
origin. In this procedure, we also propose a new perfect sampling algorithm for
Hawkes process with improved computational efficiency compared to the existing
algorithm. Theoretical analysis and numerical tests on the algorithms'
correctness and efficiency are also included.

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