Functional Limits of Generalized Jackson Networks in Multi-scale Heavy Traffic

TL;DR

This paper analyzes the behavior of generalized Jackson networks under multi-scale heavy traffic conditions, revealing that queue length processes converge to independent limit processes and introducing a blockwise multi-scale regime.

Contribution

It introduces a novel multi-scale heavy traffic regime for Jackson networks and characterizes the asymptotic independence and limit processes under these conditions.

Findings

Queue length processes converge to independent limit processes.

The limit process form depends on initial conditions.

Blockwise multi-scale regime leads to blockwise independence.

Abstract

We investigate the functional limits of generalized Jackson networks in a multi-scale heavy traffic regime where stations approach full utilization at distinct, separated rates. Our main result shows that the appropriately scaled queue length processes converge weakly to a limit process whose coordinates are mutually independent. This finding provides the fundamental dynamic mechanism that explains the asymptotic independence previously observed only in stationary distributions. The specific form of the limit processes is shown to depend on the initial conditions. Moreover, we introduce and analyze a blockwise multi-scale heavy traffic regime. In this regime, the network's stations are partitioned into blocks, where stations in different blocks approach the heavy traffic at different rates, while stations within the same block share a common rate. We obtain the functional limits in this regime as well, showing that the limit process exhibits blockwise independence.