Steady-State Analysis and Online Learning for Queues with Hawkes Arrivals

Recent empirical studies found that arrivals in many real queueing systems exhibit a clustering or self-exciting behavior; that is, an arrival may increase the possibility of new arrivals. In some cases, such clustering behavior is intrinsic to the underlying system. For example, in the stock market, it is a common practice to split a large order into small child orders to reduce transaction cost. As a consequence, one observed arriving order may be followed by a sequence of other child orders (Abergel and Jedidi, 2015). As a natural extension of the classic Poisson process, Hawkes process has been used to model arrivals with self-excitement such as order flow in stock market (Abergel and Jedidi, 2015), infected patients during pandemic (Bertozzi et al., 2020), and the internet traffic in social media (Zhao et al., 2015). To understand the impact of self-excitement in the arrival process on the long-run performance of service systems, Koops et al. (2018) and Daw and Pender (2018) provided analytic solutions to steady-state moments on the number of people in system for different infinite-server systems with Hawkes arrivals.