The Queue-Hawkes Process: Ephemeral Self-Excitement

In this paper we consider a self-exciting model that couples a Hawkes process and a queueing system which we call the Queue-Hawkes process. In this process the intensity responds to the queue: increasing at arrivals, dropping upon departures, and decaying between. Hence the influence of each arrival is ephemeral, as the excitement only lasts for the duration of the entity's time in system. We study this process both individually and by comparison to other processes, primarily the Hawkes process and the Affine Queue-Hawkes process, which we define as the zero-decay case of the Queue-Hawkes. Our results include showing that a batch-scaling of the Affine Queue-Hawkes process will converge to the Hawkes process. We also prove a strong law of large numbers for the non-identical and dependent inter-arrival times of the Queue-Hawkes, Hawkes, and Affine Queue-Hawkes processes. Additionally, we provide all moments of the Hawkes process and the Affine Queue-Hawkes process via a novel matrix structure. Finally, we prove fluid and diffusion limits for the Queue-Hawkes process using moment generating function techniques.