Uniform Moment Bounds for Generalized Jackson Networks in Multi-scale Heavy Traffic
Jin Guang, Xinyun Chen, J. G. Dai
TL;DR
This work addresses uniform control of steady-state queue-length moments for generalized Jackson networks under multi-scale heavy traffic, enabling rigorous analysis of limit stationary distributions. It develops a BAR-based approach, leveraging Palm representations and carefully crafted test functions to relate high-order and lower-order moments, and proves that under finite $(M+1)$-st moments of unitized interarrival and service times, the scaled queue lengths satisfy $\sup_{r\in(0,r_0)} \mathbb{E}[ ( r^k Z_k^{(r)} )^M ]\le C_k$ for all $1\le k\le \min(M,J)$. The analysis proceeds via a structured induction on moment orders, complemented by truncated test functions to ensure BAR applicability, and extends to non-integer moments through a Beta-generalization. The results provide a foundational step for establishing product-form limit behavior in multi-scale heavy traffic and underpin subsequent SSC and limit-distribution analyses in this regime.
Abstract
We establish uniform moment bounds for steady-state queue lengths of generalized Jackson networks (GJNs) in multi-scale heavy traffic as recently proposed by Dai et al. [2023]. Uniform moment bounds lay the foundation for further analysis of the limit stationary distribution. Our result can be used to verify the crucial moment state space collapse (SSC) assumption in Dai et al. [2023] to establish a product-form limit of GJN in the multi-scale heavy traffic regime. Our proof critically utilizes the Palm version of the basic adjoint relationship (BAR) as developed in Braverman et al. [2023].