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The Service Rate Region Polytope

Gianira N. Alfarano, Altan B. Kilic, Alberto Ravagnani, Emina Soljanin

TL;DR

This work characterizes the polyhedral geometry of service rate regions arising in distributed coded storage systems. It models the region as the image of an allocation polytope under a linear map, and develops both outer bounds (via coding theory and knapsack-based optimization) and a discrete allocation framework to connect rational points with rational allocations. The paper proves that every rational point in the service rate region has a rational allocation and provides precise results for systematic MDS codes, including volume calculations in low dimensions. These insights guide design choices and allocation strategies in practical distributed storage by linking code properties to service-capacity geometry and enabling tractable approximations of the region. Key contributions include discretization results, rational-vertex structure, and explicit bounds and volumes for important code families.

Abstract

We investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. The service rate region of a distributed coded system describes the data access requests that the underlying system can support. In this paper, we study the polytope structure of the service rate region with the primary goal of describing its geometric shape and properties. We achieve so by introducing various structural parameters of the service rate region and establishing upper and lower bounds for them. The techniques we apply in this paper range from coding theory to optimization. One of our main results shows that every rational point of the service rate region has a so-called rational allocation, answering an open question in the research area.

The Service Rate Region Polytope

TL;DR

This work characterizes the polyhedral geometry of service rate regions arising in distributed coded storage systems. It models the region as the image of an allocation polytope under a linear map, and develops both outer bounds (via coding theory and knapsack-based optimization) and a discrete allocation framework to connect rational points with rational allocations. The paper proves that every rational point in the service rate region has a rational allocation and provides precise results for systematic MDS codes, including volume calculations in low dimensions. These insights guide design choices and allocation strategies in practical distributed storage by linking code properties to service-capacity geometry and enabling tractable approximations of the region. Key contributions include discretization results, rational-vertex structure, and explicit bounds and volumes for important code families.

Abstract

We investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. The service rate region of a distributed coded system describes the data access requests that the underlying system can support. In this paper, we study the polytope structure of the service rate region with the primary goal of describing its geometric shape and properties. We achieve so by introducing various structural parameters of the service rate region and establishing upper and lower bounds for them. The techniques we apply in this paper range from coding theory to optimization. One of our main results shows that every rational point of the service rate region has a so-called rational allocation, answering an open question in the research area.
Paper Structure (14 sections, 39 theorems, 76 equations, 7 figures, 1 table)

This paper contains 14 sections, 39 theorems, 76 equations, 7 figures, 1 table.

Table of Contents

  1. System Model and Problem Formulation
  2. Storage Model
  3. Service and Access Models
  4. Normalization and Integrality
  5. Service Rate Region and Problem Formulation
  6. The Service Rate Region and the Allocation Polytopes
  7. The Integer Allocation Model
  8. Fundamental Parameters of the Service Rate Region
  9. Outer Bounds
  10. Coding Theory Approach
  11. Optimization Approach
  12. Systematic MDS Codes
  13. Polytopes
  14. Error-Correcting Codes

Key Result

Proposition 1.6

Suppose that $\mathcal{R}=(\mathcal{R}_1,\ldots,\mathcal{R}_k)$ and $\mathcal{R}'=(\mathcal{R}'_1,\ldots,\mathcal{R}'_k)$ are $G$-systems with $\mathcal{R}'_i \subseteq \mathcal{R}_i$ for all $i \in \{1,\ldots,k\}$. Then $\Lambda(\mathcal{R},\mu) \supseteq \Lambda(\mathcal{R}',\mu)$. In particular,

Figures (7)

  • Figure 1: The service rate regions of the systems in Example \ref{['ex:different_examples']}.
  • Figure 2: Service rate region for the $G$-system in Example \ref{['ex:7']} and the point $P=(4/3, 2/3)$.
  • Figure 3: The service rate region of Example \ref{['ex:different_examples']}, Figure \ref{['subfig:2']}, is an example where the bound of Theorem \ref{['thm:2distance']} gives a better approximation than the bound of Theorem \ref{['thm:bounddualdis']}.
  • Figure 4: Service rate region and outer bounds for Example \ref{['ex:comparison']}.
  • Figure 5: Service rate region and bounding polytope for Example \ref{['ex:opt']}.
  • ...and 2 more figures

Theorems & Definitions (89)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6
  • Definition 1.7
  • Proposition 1.8
  • Remark 1.9
  • Example 1.11
  • Definition 2.1
  • ...and 79 more