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

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 reveals the underlying dynamic mechanism that explains the asymptotic independence previously observed only in stationary distributions. The specific form of the limit process is shown to depend on the initial conditions. In this paper, we consider the matching-rate and the lowest-rate initial conditions. Although the corresponding limit processes have different laws, they have the same product-form exponential limiting distribution on the positive orthant as $t\to \infty$. 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.