Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Abstract

A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infiniteserver queues with multivariate Hawkes traffic.Organization of this paper. The rest of the paper is organized as follows. In Section 2, we formally introduce stationary linear Hawkes processes and review some of their properties. In Section 3, we state the main result on the functional central limit theorem for univariate stationary Hawkes processes with large baseline intensity µ and describe the properties of the limiting Gaussian process. In Section 4, we develop heavytraffic approximations for infinite-server queues with univariate Hawkes traffic. We also discuss in detail the special case when service times are exponentially distributed. In Section 5, we extend our results to multivariate stationary Hawkes processes and study infinite-server queues with multivariate Hawkes traffic. The proofs of all the results are collected in the Appendix.

Queues Driven by Hawkes Processes

“…This general type of service allows for accurate and robust modeling while preserving key characteristics for queues, such as the Markov property. Mathematically, this work is most similar to recent work by Gao and Zhu [9] and Koops et al [11]. Moreover, transient moments for infinite server queues with Markovian arrivals are also among the findings in Koops et al [11], an independent and concurrent work.…”

Queues Driven by Hawkes Processes

“…This general type of service allows for accurate and robust modeling while preserving key characteristics for queues, such as the Markov property. Mathematically, this work is most similar to recent work by Gao and Zhu [9] and Koops et al [11]. Moreover, transient moments for infinite server queues with Markovian arrivals are also among the findings in Koops et al [11], an independent and concurrent work.…”

Infinite-server queues with Hawkes input

“…Secondly, with regards to the functional equation (22), in case that J = ∞ (i.e., for the usual Hawkes process without departures), [15] commented that such relations are 'rather intractable'. Also in [10], it is mentioned that the problem of finding the probability mass function of the number of customers is generally 'numerically challenging'. We will now show, however, that Eqn.…”

“…[7,10] scaling limits are derived for queueing systems that allow for Hawkes input. Scaling limits for infinite-server queues designed specifically for Hawkes input are derived in [10]; it states that exact and numerical analysis of this model is 'challenging'. To the best of our knowledge, only [9] pays attention to exact (i.e., nonasymptotic) analysis of queues driven by Hawkes processes.…”