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Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes

Part of: Stochastic processes Operations research and management science Limit theorems Stochastic analysis Special processes Markov processes

Published online by Cambridge University Press:  07 August 2023

Bo Li  and
Guodong Pang
Bo Li*
Affiliation:
Nankai University
Guodong Pang*
Affiliation:
Rice University
*
*Postal address: School of Mathematics and LPMC, Nankai University, Tianjin, 300071 China. Email address: libo@nankai.edu.cn
**Postal address: Department of Computational Applied Mathematics and Operations Research, George R. Brown School of Engineering, Rice University, Houston, TX 77005, USA. Email address: gdpang@rice.edu
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Abstract

We consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.

Keywords

Hawkes processrandom splitting/samplingfunctional central limit theoremparallel single-server queuesmultidimensional reflected Brownian motionmultidimensional reflected Ornstein–Uhlenbeck diffusion

MSC classification

Primary: 60G55: Point processes 60K25: Queueing theory 60F17: Functional limit theorems; invariance principles 90B22: Queues and service
Secondary: 60H10: Stochastic ordinary differential equations 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 490 - 514
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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