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Infinite-server systems with Hawkes arrivals and Hawkes services.
Dharmaraja Selvamuthu1, Paola Tardelli2
1Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 India.
Queueing Systems
|May 4, 2022
Summary
This study analyzes customer numbers in infinite-server systems using Hawkes processes. Researchers derived the Markov property and characterized system dynamics for improved queueing theory understanding.
Area of Science:
- Queueing Theory
- Stochastic Processes
- Applied Probability
Background:
- Hawkes processes model self-exciting point events.
- Infinite-server systems are fundamental in performance analysis.
- State-dependent Hawkes (sdHawkes) processes introduce complexity in service times.
Purpose of the Study:
- To investigate the number of customers in infinite-server systems driven by Hawkes processes.
- To derive the Markov property for these systems under specific conditions.
- To characterize the joint time-dependent distribution of system state variables.
Main Methods:
- Utilizing Hawkes and state-dependent Hawkes (sdHawkes) processes for arrival and service.
- Deriving the Markov property for the system.
- Characterizing the joint time-dependent distribution using a system of differential equations.
Main Results:
- The Markov property was derived for the system.
- A system of differential equations was established to characterize the joint time-dependent distribution.
- Time-dependent results were deduced for the system.
Conclusions:
- The study provides a mathematical framework for analyzing queueing systems with self-exciting arrivals and services.
- The derived results offer insights into the dynamic behavior of customer populations in complex stochastic systems.
- This research contributes to a deeper understanding of queueing models with self-exciting point processes.
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