L\'{e}vy-driven queuing networks in multi-scale light and heavy traffic

TL;DR

This paper analyzes a specialized Lévy-driven queueing network, deriving the limiting distribution of workloads under different traffic regimes and identifying conditions for workload independence in the limit.

Contribution

It characterizes the stationary workload distribution in Lévy-driven networks with specific routing, under light and heavy traffic scaling, revealing workload decoupling conditions.

Findings

Limiting workload distribution identified under scaling

Workloads become asymptotically independent in certain regimes

Conditions established for workload decoupling

Abstract

We study a queueing network with a strictly upper-triangular routing matrix, where each column contains at most one non-negative entry, and the root node receives input from a spectrally positive L\'{e}vy process. Our aim is to characterize the distribution of the multivariate stationary workload under a specific scaling of the service rates. Under mild conditions on the Laplace exponent of the driving L\'{e}vy process, we identify the limiting law of an appropriately scaled joint stationary workload in both light-traffic and heavy-traffic regimes. In particular, we establish conditions under which certain queueing workloads within the network asymptotically decouple, becoming independent in the limiting regime.