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Functional limit theorems for Gaussian-fed queueing network in light and heavy traffic

Nikolai Kriukov, Krzysztof Dȩbicki, Michel Mandjes

TL;DR

This work analyzes a Gaussian-driven, strictly upper-triangular $n$-node queueing network to derive functional limit theorems for the joint stationary workload under light- and heavy-traffic scalings. By constructing appropriate time-space scalings via a regularly varying standard deviation function $oldsymbol{ ext{σ}}$ and a delta function $oldsymbol{δ}$, the authors show that the scaled workload converges to a linear transform of a Gaussian supremum process: in light traffic to $oldsymbol{ rak{Q}}_ ext{λ}(oldsymbol{t})=(I-P^oldsymbol{*})^ opoldsymbol{ rak{X}}_ ext{λ}(oldsymbol{t})$ driven by $oldsymbol{B}_ ext{λ}$, and in heavy traffic to $oldsymbol{ rak{Q}}_ ext{α}(oldsymbol{t})=(I-P^oldsymbol{*})^ opoldsymbol{ rak{X}}_ ext{α}(oldsymbol{t})$ driven by $oldsymbol{B}_ ext{α}$, with explicit covariance structures. A key insight is that, under mild growth/regular variation conditions, queues can decouple in the limit, yielding independent (or block-independent) limiting workloads across equivalence classes determined by the asymptotic routing rates. The proofs combine stochastic-process convergence with careful treatment of the supremum functional on $(- fty,t]$, extending prior Gaussian-input results to a network setting and offering a process-level analogue to product-form phenomena in heavy traffic.

Abstract

We consider a queueing network operating under a strictly upper-triangular routing matrix with per column at most one non-negative entry. The root node is fed by a Gaussian process with stationary increments. Our aim is to characterize the distribution of the multivariate stationary workload process under a specific scaling of the queue's service rates. In the main results of this paper we identify, under mild conditions on the standard deviation function of the driving Gaussian process, in both light and heavy traffic parameterization, the limiting law of an appropriately scaled version (in both time and space) of the joint stationary workload process. In particular, we develop conditions under which specific queueing processes of the network effectively decouple, i.e., become independent in the limiting regime.

Functional limit theorems for Gaussian-fed queueing network in light and heavy traffic

TL;DR

This work analyzes a Gaussian-driven, strictly upper-triangular -node queueing network to derive functional limit theorems for the joint stationary workload under light- and heavy-traffic scalings. By constructing appropriate time-space scalings via a regularly varying standard deviation function and a delta function , the authors show that the scaled workload converges to a linear transform of a Gaussian supremum process: in light traffic to driven by , and in heavy traffic to driven by , with explicit covariance structures. A key insight is that, under mild growth/regular variation conditions, queues can decouple in the limit, yielding independent (or block-independent) limiting workloads across equivalence classes determined by the asymptotic routing rates. The proofs combine stochastic-process convergence with careful treatment of the supremum functional on , extending prior Gaussian-input results to a network setting and offering a process-level analogue to product-form phenomena in heavy traffic.

Abstract

We consider a queueing network operating under a strictly upper-triangular routing matrix with per column at most one non-negative entry. The root node is fed by a Gaussian process with stationary increments. Our aim is to characterize the distribution of the multivariate stationary workload process under a specific scaling of the queue's service rates. In the main results of this paper we identify, under mild conditions on the standard deviation function of the driving Gaussian process, in both light and heavy traffic parameterization, the limiting law of an appropriately scaled version (in both time and space) of the joint stationary workload process. In particular, we develop conditions under which specific queueing processes of the network effectively decouple, i.e., become independent in the limiting regime.

Paper Structure

This paper contains 13 sections, 8 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Main results
  4. Light traffic.
  5. Heavy traffic
  6. Proof of Lemma \ref{['gauss_convergence']}
  7. Finite-dimensional distributions
  8. Tightness
  9. Proof of Lemma \ref{['sup_convergence']}
  10. Finite-dimensional distributions
  11. Tightness
  12. Discussion and concluding remarks
  13. Proofs of lemmas

Key Result

Theorem 3.1

Assume that $\boldsymbol{r}(u)\to \infty$ as $u\to\infty$. Let the model satisfy N1, N2, N3, L1 and L2. Then, as $u\to\infty$, in distribution in $C([0,\infty)\to\mathbb{R}^n)$.

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • ...and 7 more