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Geometric lower bounds for the steady-state occupancy of processing networks with limited connectivity

Diego Goldsztajn, Andres Ferragut

TL;DR

This work analyzes load balancing in processing networks constrained by a bipartite compatibility graph between dispatchers and servers. It introduces two connectivity-based flexibility metrics, $\alpha_G$ and $\beta_G$, and proves geometric lower bounds on the steady-state occupancy via these metrics, establishing that large-scale performance cannot match Power-of-$d$ or JSQ unless the metrics diverge. The authors develop monotone network transformations to establish stochastic dominance and present two main lower bounds that depend on $\alpha_G$ (and $\theta_G$) and $\beta_G$, respectively. The results characterize fundamental limitations of locality-constrained load balancing and guide design principles for scalable systems where data locality and geographic constraints are pivotal.

Abstract

We consider processing networks where multiple dispatchers are connected to single-server queues by a bipartite compatibility graph, modeling constraints that are common in data centers and cloud networks due to geographic reasons or data locality issues. We prove lower bounds for the steady-state occupancy, i.e., the complementary cumulative distribution function of the empirical queue length distribution. The lower bounds are geometric with ratios given by two flexibility metrics: the average degree of the dispatchers and a novel metric that averages the minimum degree over the compatible dispatchers across the servers. Using these lower bounds, we establish that the asymptotic performance of a growing processing network cannot match that of the classic Power-of-$d$ or JSQ policies unless the flexibility metrics approach infinity in the large-scale limit.

Geometric lower bounds for the steady-state occupancy of processing networks with limited connectivity

TL;DR

This work analyzes load balancing in processing networks constrained by a bipartite compatibility graph between dispatchers and servers. It introduces two connectivity-based flexibility metrics, and , and proves geometric lower bounds on the steady-state occupancy via these metrics, establishing that large-scale performance cannot match Power-of- or JSQ unless the metrics diverge. The authors develop monotone network transformations to establish stochastic dominance and present two main lower bounds that depend on (and ) and , respectively. The results characterize fundamental limitations of locality-constrained load balancing and guide design principles for scalable systems where data locality and geographic constraints are pivotal.

Abstract

We consider processing networks where multiple dispatchers are connected to single-server queues by a bipartite compatibility graph, modeling constraints that are common in data centers and cloud networks due to geographic reasons or data locality issues. We prove lower bounds for the steady-state occupancy, i.e., the complementary cumulative distribution function of the empirical queue length distribution. The lower bounds are geometric with ratios given by two flexibility metrics: the average degree of the dispatchers and a novel metric that averages the minimum degree over the compatible dispatchers across the servers. Using these lower bounds, we establish that the asymptotic performance of a growing processing network cannot match that of the classic Power-of- or JSQ policies unless the flexibility metrics approach infinity in the large-scale limit.
Paper Structure (9 sections, 6 theorems, 53 equations, 4 figures)

This paper contains 9 sections, 6 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Steady-state occupancy
  4. Main results
  5. Monotone transformations
  6. Proof of Theorem \ref{['the: first lower bound']}
  7. Proof of Theorem \ref{['the: second lower bound']}
  8. Conclusion
  9. Proofs of several results

Key Result

Proposition 1

Let $\boldsymbol{X}$ be a simple and ergodic load balancing process with load $\rho$. Then the steady-state occupancy associated with $\boldsymbol{X}$ satisfies that

Figures (4)

  • Figure 1: Dispatchers and servers are represented by crossed white circles and black circles, respectively. Each server in $G_n^1$ is connected to all the dispatchers at its left and only one dispatcher at its right. The connected component on the right of $G_n^2$ consists of one dispatcher connected to $n$ servers, whereas each of the other $n$ connected components consists of one dispatcher and one server.
  • Figure 2: Edge simplification that removes the compatibility relation $(d, u)$ and incorporates server $v$ and the compatibility relation $(d, v)$. The servers $u$ and $v$ have the same potential departure process.
  • Figure 3: Bipartite graph $G_0$ obtained after performing an edge simplification at each edge of $G$. Each of the sets of servers $\{u_d, u_e\}$ and $\{v_d, v_e\}$ has a common potential departure process.
  • Figure 4: Schematic view of the bipartite graphs $G$, $G_0$ and $G_\gamma$. The circles represent sets of nodes and a thick line between two sets of nodes indicates that there may exist edges between the two sets. The set $\tilde{S}_\gamma$ is the set of servers incorporated through the edge simplifications. The potential departure processes of servers in $S_\gamma$ are coupled with those of servers in $\tilde{S}_\gamma$ but are mutually independent over $S_\gamma$.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Theorem 1
  • Theorem 2
  • Example 1
  • Theorem 3
  • proof
  • ...and 7 more