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Fluid limits for interacting queues in sparse dynamic graphs

Diego Goldsztajn, Sem C. Borst, Johan S. H. van Leeuwaarden

TL;DR

The paper analyzes a network of $n$ single-server queues connected by a dynamic, permutation-invariant graph resampled at rate $\mu_n$, with arrivals at rate $\lambda_n$ per node and dispatching to the shortest queue in the local neighborhood. In the truly sparse regime where $\lambda_n/n \to \lambda$ and $\mu_n \to \infty$, it proves a fluid limit: the occupancy process converges to a deterministic system of differential equations whose evolution depends only on $\lambda$ and the limiting degree distribution via its generating function $\varphi$, not on detailed graph structure. The authors show that the stationary distribution converges to a fluid-equilibrium $q^*$, and they derive phase-transition-type bounds on $q^*$ that reflect whether the degree distribution assigns mass to zero or concentrates on positive degrees; these bounds imply tail behavior of the equilibrium under different degree laws. The results extend load-balancing theory to truly sparse dynamic graphs, providing tractable, law-of-large-numbers-type descriptions for large-scale, time-varying networks while highlighting the impact of the degree distribution on performance.

Abstract

Consider a network of $n$ single-server queues where tasks arrive independently at each server at rate $λ_n$. The servers are connected by a graph that is resampled at rate $μ_n$ in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as $n \to \infty$ when $λ_n / n \to λ$ and $μ_n \to \infty$, and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on $λ$ and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across $n$, which is natural in load balancing systems.

Fluid limits for interacting queues in sparse dynamic graphs

TL;DR

The paper analyzes a network of single-server queues connected by a dynamic, permutation-invariant graph resampled at rate , with arrivals at rate per node and dispatching to the shortest queue in the local neighborhood. In the truly sparse regime where and , it proves a fluid limit: the occupancy process converges to a deterministic system of differential equations whose evolution depends only on and the limiting degree distribution via its generating function , not on detailed graph structure. The authors show that the stationary distribution converges to a fluid-equilibrium , and they derive phase-transition-type bounds on that reflect whether the degree distribution assigns mass to zero or concentrates on positive degrees; these bounds imply tail behavior of the equilibrium under different degree laws. The results extend load-balancing theory to truly sparse dynamic graphs, providing tractable, law-of-large-numbers-type descriptions for large-scale, time-varying networks while highlighting the impact of the degree distribution on performance.

Abstract

Consider a network of single-server queues where tasks arrive independently at each server at rate . The servers are connected by a graph that is resampled at rate in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as when and , and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across , which is natural in load balancing systems.
Paper Structure (29 sections, 30 theorems, 233 equations, 4 figures, 1 table)

This paper contains 29 sections, 30 theorems, 233 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of the main result
  3. Related work
  4. Some basic notation
  5. Organization of the paper
  6. Model description
  7. Fluid limit
  8. Properties of fluid trajectories
  9. Existence, uniqueness and monotonicity
  10. Global attractivity
  11. Equilibrium behavior
  12. Convergence of stationary distributions
  13. Phase transition for $q^*$
  14. Lower bound for $q^*$
  15. Proof of the fluid limit
  16. ...and 14 more sections

Key Result

Proposition 1

Suppose that $\lambda_n / n \to \lambda$ as $n \to \infty$ and there exist $\left\{\kappa_n \in \mathbbm{N} : n \geq 1\right\}$ and $\left\{\mu_n > 0 : n \geq 1\right\}$ such that the processes $\mathcal{R}_n$ satisfy one of the following conditions. Also, assume that there exist constants $\left\{d_n^- \geq 0 : n \geq 1\right\}$ such that in the system with $n$ servers the indegree of the server

Figures (4)

  • Figure 1: Equilibrium point for $\lambda = 0.9$ and distinct limiting outdegree distributions with mean $d = 5$: for $q_1^*$ the limiting outdegree distribution has mass only at $0$ and $2d$, for $q_2^*$ a uniform distribution on outdegrees between $0$ and $2d$ was used, a Poisson distribution was used for $q_3^*$ and a deterministic distribution for $q_4^*$. The tail of $\log q^*(i)$ decays almost linearly for the limiting outdegree distributions with mass at zero.
  • Figure 2: From left to right: the ring, the disjoint triangles and the double-star for $n = 12$.
  • Figure 3: Load balancing on static graphs with the topologies depicted in Figure \ref{['fig: graphs']}. In all the cases the system starts empty, $n = 1500$ and $\lambda_n = 9 n / 10$. The plot on the left shows time averages computed over the second half of the simulation and the equilibrium point $q^*$ defined in Proposition \ref{['prop: fixed point']}. The plot on the right concerns the double-star topology. It shows the maximum and minimum numbers of tasks across the central servers and the number of servers that have more tasks than, or as many tasks as, the central server with the fewest tasks. The maximum and minimum numbers of tasks across the central servers remain very close together and thus are difficult to distinguish from each other.
  • Figure 4: Solution of \ref{['eq: fluid dynamics']} and sample paths of $\boldsymbol{q}_n$ for dynamic graphs. In all the cases the system starts empty, $\lambda_n = 9 n / 10$ and the resampling process is Poisson. The dashed lines depicted in the two plots on the bottom correspond to time averages computed over the interval $[40, 100]$.

Theorems & Definitions (74)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Remark 5
  • Lemma 1
  • Proposition 2
  • Remark 6
  • ...and 64 more