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A Geometrically Convergent Solution to Spatial Hypercube Queueing Models

Cheng Hua, Jun Luo, Arthur J. Swersey, Yixing Wen

TL;DR

This work extends the hypercube queueing model to handle heterogeneous service rates by devising an exact solution through a birth-death process and an equivalent reformulation, and develops a parallel algorithm that leverages the convergence property and two structural features of the hypercube model, achieving more than 91% parallelization.

Abstract

The hypercube queueing model was initially developed to address spatial queueing problems and has found wide applications in emergency services, such as ambulance and police systems. While the model was originally designed for homogeneous service rates, we extend it to handle heterogeneous service rates by devising an exact solution through a birth-death process and an equivalent reformulation. We demonstrate that our algorithm converges to the exact solution at a geometric rate. Additionally, we developed a parallel algorithm that leverages the convergence property and two structural features of the hypercube model, achieving more than 91% parallelization. Numerical experiments on emergency medical service systems show that our sequential algorithm is over 1,000 times faster than the sparse solver and more than 500 times faster than discrete-event simulation, while maintaining high accuracy. The parallel algorithm further improves efficiency, achieving an approximately eightfold speedup with 12 processing units, with additional gains possible when more computational resources are available. Overall, the proposed algorithms improve computational efficiency and enable the solution of large-scale problems that are otherwise intractable using traditional approaches.

A Geometrically Convergent Solution to Spatial Hypercube Queueing Models

TL;DR

This work extends the hypercube queueing model to handle heterogeneous service rates by devising an exact solution through a birth-death process and an equivalent reformulation, and develops a parallel algorithm that leverages the convergence property and two structural features of the hypercube model, achieving more than 91% parallelization.

Abstract

The hypercube queueing model was initially developed to address spatial queueing problems and has found wide applications in emergency services, such as ambulance and police systems. While the model was originally designed for homogeneous service rates, we extend it to handle heterogeneous service rates by devising an exact solution through a birth-death process and an equivalent reformulation. We demonstrate that our algorithm converges to the exact solution at a geometric rate. Additionally, we developed a parallel algorithm that leverages the convergence property and two structural features of the hypercube model, achieving more than 91% parallelization. Numerical experiments on emergency medical service systems show that our sequential algorithm is over 1,000 times faster than the sparse solver and more than 500 times faster than discrete-event simulation, while maintaining high accuracy. The parallel algorithm further improves efficiency, achieving an approximately eightfold speedup with 12 processing units, with additional gains possible when more computational resources are available. Overall, the proposed algorithms improve computational efficiency and enable the solution of large-scale problems that are otherwise intractable using traditional approaches.
Paper Structure (26 sections, 6 theorems, 53 equations, 6 figures, 8 tables, 3 algorithms)

This paper contains 26 sections, 6 theorems, 53 equations, 6 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Spatial Hypercube Queueing Model
  3. Model
  4. Model Reformulation
  5. Algorithm and Convergence Analysis
  6. CPU Algorithm
  7. Convergence Analysis
  8. Complexity Analysis
  9. Parallel Computing Solution Algorithm
  10. Parallel CPU Algorithm
  11. Coefficient Generation
  12. Complexity Analysis
  13. Numerical Results
  14. Datasets
  15. Computation Time and Accuracy
  16. ...and 11 more sections

Key Result

Proposition 1

The conditional probability formulation is given, for $n=1, 2, \ldots, N-1$, and $m \in \mathscr{C}_n$, by and $p_{0}(B_0) = p_{N}(B_{2^N-1}) = 1$. The probability distribution of system states is given, for $m=0,1,\ldots, 2^N-1$, by $P\{B_m\} = p(n)\cdot p_{n}(B_m)$, where $n = w(B_m)$ and $p(n)$ follows This solution yields the same probability distribution of system states as the original mod

Figures (6)

  • Figure 1: The States and Transitions of Spatial Hypercube Queueing Models.
  • Figure 2: Reformulation to a Birth-Death Process.
  • Figure 3: Reformulation to a Birth-Death Process for a Finite Buffer System.
  • Figure 4: Task Assignment and Parallel Computing.
  • Figure 5: Percentage Time Savings and Errors for Heterogeneous Cases.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 1: Reformulation
  • Proposition 2: Reformulation, Finite Buffer
  • Remark 1
  • Lemma 1
  • Theorem 1: Geometric Convergence
  • Corollary 1: Homogeneous Setting
  • Lemma 2