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Lévy-driven queuing networks in multi-scale light and heavy traffic

Krzysztof Dȩbicki, Nikolai Kriukov, Michel Mandjes

TL;DR

The paper develops a Lévy-driven queueing network model with a strictly upper-triangular routing structure and root input $J(\cdot)$, analyzing the joint stationary workload under multiscale service-rate scaling in light- and heavy-traffic regimes. By exploiting a Skorokhod-type representation and carefully controlling the Laplace exponent asymptotics of the Lévy input, it derives an explicit product-form limit for the joint workload LST, revealing asymptotic decoupling across blocks and Mittag-Leffler-type limits for downstream queues. The results are supported by concrete examples of input processes (compound Poisson, Gamma, and stable-sum inputs) and network topologies (two-layer tree and tandem), illustrating how the constants $\mathcal{A}_{\alpha,k}$, $\mathcal{C}_{\alpha,j}$, and $\mathcal{D}_{\alpha,j}$ govern the limiting distributions. This work extends multiscale queueing limit theory to Lévy-driven feedforward networks and provides tractable formulas for the stationary behavior under regime-specific scalings, with potential applications in communication networks and stochastic storage systems.

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 Lévy 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 Lévy 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.

Lévy-driven queuing networks in multi-scale light and heavy traffic

TL;DR

The paper develops a Lévy-driven queueing network model with a strictly upper-triangular routing structure and root input , analyzing the joint stationary workload under multiscale service-rate scaling in light- and heavy-traffic regimes. By exploiting a Skorokhod-type representation and carefully controlling the Laplace exponent asymptotics of the Lévy input, it derives an explicit product-form limit for the joint workload LST, revealing asymptotic decoupling across blocks and Mittag-Leffler-type limits for downstream queues. The results are supported by concrete examples of input processes (compound Poisson, Gamma, and stable-sum inputs) and network topologies (two-layer tree and tandem), illustrating how the constants , , and govern the limiting distributions. This work extends multiscale queueing limit theory to Lévy-driven feedforward networks and provides tractable formulas for the stationary behavior under regime-specific scalings, with potential applications in communication networks and stochastic storage systems.

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 Lévy 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 Lévy 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.
Paper Structure (14 sections, 5 theorems, 136 equations, 1 figure)

This paper contains 14 sections, 5 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model, assumptions and notation
  3. Model and assumptions
  4. Notation
  5. Main result
  6. Examples
  7. Examples of the input process
  8. Examples of the network structure
  9. Proof of Theorem \ref{['main_light']}
  10. Proof of Lemma \ref{['S_D_relations']}:
  11. Proof of Remark \ref{['rem:zero_D']}
  12. Proof of Lemma \ref{['lem:Laplace_for_networks']}:
  13. Proofs related to the example of a two-layer network
  14. Proof related to the example of a tandem network

Key Result

Lemma 2.1

For any $j,k\in\langle n\rangle$, $j\not = k$

Figures (1)

  • Figure 1: Example tree network with six nodes. The values of the $r_i(u)$ are given in the circles, the values of the $p_{ij}$ on top of the corresponding arrows.

Theorems & Definitions (10)

  • Lemma 2.1
  • Example 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 5.1
  • Remark 5.2
  • Remark 5.3
  • Lemma A.1