Infinite-Server Systems Driven by Finite-Dimensional Hawkes and Dynamic Contagion Arrival Processes

This paper investigates the infinite-server systems that use finite-dimensional Hawkes and dynamic contagion processes as arrival processes. Many real-world stochastic systems demonstrate arrival patterns that exhibit clustering behavior. In these systems, the arrival of one entity can lead to an increase in the arrival of others, which can occur through selfexcitation, multi-excitation, or external excitation. This interdependent behavior significantly influences the dynamics of the system. Consequently, under suitable conditions, we derive the Markov property of these systems. Additionally, we determine the joint time-dependent probability generating functions-Laplace transform for both the system size and the arrival intensities. We also present a recursive method to identify and derive both transient and stationary moments. Furthermore, we discuss and compare several cases of these arrival models to provide deeper insights into its structure and behavior.