Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals

@article{Chen2021PerfectSO,
  title={Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals},
  author={Xinyun Chen},
  journal={Stochastic Systems},
  year={2021}
}
  • Xinyun Chen
  • Published 15 February 2020
  • Computer Science, Mathematics
  • Stochastic Systems
In this paper we develop to our best knowledge the first perfect sampling algorithm for queues with Hawkes input (i.e., single-server queues with Hawkes arrivals and independent and identically distributed 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… 

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References

SHOWING 1-10 OF 32 REFERENCES
Infinite-server queues with Hawkes input
TLDR
A system of differential equations is obtained that characterizes the joint distribution of the arrival intensity and the number of customers in the Hawkes arrival process and can be expressed in terms of the solution of a fixed-point equation.
Perfect sampling of GI/GI/c queues
TLDR
This work uses a coupled multi-server vacation system as the upper bound process and develops an algorithm to simulate the vacation system backward in time from stationarity at time zero, which has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.
Queues Driven by Hawkes Processes
TLDR
This paper analyzes an infinite server queueing system in which the arrivals are driven by the self-exciting Hawkes process and where service follows a phase-type distribution or is deterministic.
The Queue-Hawkes Process: Ephemeral Self-Excitement
TLDR
A self-exciting model that couples a Hawkes process and a queueing system which is defined as the zero-decay case of the Queue-Hawkes process is considered, and a strong law of large numbers is proved for the non-identical and dependent inter-arrival times of the queues.
Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues
TLDR
A functional central limit theorem is proved for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large and the limit is a non-Markovian Gaussian process with dependent increments.
Perfect simulation of Hawkes processes
TLDR
By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived and used to construct upper and lower processes for the perfect simulation algorithm.
Networks of ·/G/∞ queues with shot-noise-driven arrival intensities
TLDR
This work studies infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the departure rate is given by a shot-noise process, and derives heavy-traffic asymptotics for the number of jobs in the system by using a linear scaling of the shot intensity.
Rate of convergence to equilibrium of marked Hawkes processes
In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of
Steady-state simulation of reflected Brownian motion and related stochastic networks
This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first
Networks of $$\cdot /G/\infty $$·/G/∞ queues with shot-noise-driven arrival intensities
TLDR
This work studies infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the departure rate is given by a shot-noise process, and derives heavy-traffic asymptotics for the number of jobs in the system by using a linear scaling of the shot intensity.
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