Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals

TL;DR

This paper introduces the first perfect sampling algorithms for queues with Hawkes process arrivals and for Hawkes processes themselves, improving computational efficiency and providing theoretical and numerical validation.

Contribution

It presents novel perfect sampling algorithms for queues with Hawkes input and for Hawkes processes, enhancing efficiency and theoretical understanding.

Findings

Successful development of perfect sampling algorithms for Hawkes queues and processes.

Algorithms validated through theoretical analysis and numerical tests.

Improved computational efficiency over existing methods.

Abstract

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.