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Optimizing Server Locations in Spatial Queues: Parametric and Nonparametric Bayesian Optimization

Cheng Hua, Arthur J. Swersey, Wenqian Xing, Yi Zhang

TL;DR

This paper addresses the problem of optimally locating servers in a spatial hypercube queueing framework, where server availability and inter-server dependencies affect performance. It introduces two Bayesian optimization approaches: (i) SparBL, a sparse Bayesian linear surrogate with second-order interactions and submodular relaxation for efficient acquisition, and (ii) GP-$p$M, a Gaussian-process surrogate with a $p$-Median prior mean and a two-level search within feasible trust regions. The authors establish NP-hardness, derive bounds by linking to the $p$-Median and 1-Median problems, and prove sublinear regret for both methods, with finite-time convergence guarantees for the GP-based approach. Empirical results on synthetic instances and a St. Paul, MN ambulance-location case demonstrate that the proposed methods consistently achieve the optimal solution and outperform baselines, particularly in high-utilization settings, underscoring practical value for emergency-service deployment.

Abstract

This paper presents a new model for solving the optimal server location problem in a spatial hypercube queueing model. Unlike deterministic location models, our approach accounts for server availability, varying utilization levels, and dependencies across servers. We prove that the problem is NP-hard and establish lower and upper bounds, as well as asymptotic results, by relating it to special cases of the classical $p$-Median problem. To address the computational challenge, we propose two Bayesian optimization approaches: (i) a parametric approach based on a sparse Bayesian linear model with second-order interactions, and (ii) a nonparametric approach using a Gaussian process surrogate with the $p$-Median objective as the prior mean function. We prove that both methods achieve sublinear regret and converge to the optimal solution. Numerical experiments and a case study using real-world data from the St. Paul, Minnesota, emergency response system show that our approaches consistently identify optimal solutions and outperform all baseline methods.

Optimizing Server Locations in Spatial Queues: Parametric and Nonparametric Bayesian Optimization

TL;DR

This paper addresses the problem of optimally locating servers in a spatial hypercube queueing framework, where server availability and inter-server dependencies affect performance. It introduces two Bayesian optimization approaches: (i) SparBL, a sparse Bayesian linear surrogate with second-order interactions and submodular relaxation for efficient acquisition, and (ii) GP-M, a Gaussian-process surrogate with a -Median prior mean and a two-level search within feasible trust regions. The authors establish NP-hardness, derive bounds by linking to the -Median and 1-Median problems, and prove sublinear regret for both methods, with finite-time convergence guarantees for the GP-based approach. Empirical results on synthetic instances and a St. Paul, MN ambulance-location case demonstrate that the proposed methods consistently achieve the optimal solution and outperform baselines, particularly in high-utilization settings, underscoring practical value for emergency-service deployment.

Abstract

This paper presents a new model for solving the optimal server location problem in a spatial hypercube queueing model. Unlike deterministic location models, our approach accounts for server availability, varying utilization levels, and dependencies across servers. We prove that the problem is NP-hard and establish lower and upper bounds, as well as asymptotic results, by relating it to special cases of the classical -Median problem. To address the computational challenge, we propose two Bayesian optimization approaches: (i) a parametric approach based on a sparse Bayesian linear model with second-order interactions, and (ii) a nonparametric approach using a Gaussian process surrogate with the -Median objective as the prior mean function. We prove that both methods achieve sublinear regret and converge to the optimal solution. Numerical experiments and a case study using real-world data from the St. Paul, Minnesota, emergency response system show that our approaches consistently identify optimal solutions and outperform all baseline methods.

Paper Structure

This paper contains 52 sections, 20 theorems, 123 equations, 7 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Location Models in Emergency Service Systems
  4. Spatial Hypercube Queueing Model
  5. Bayesian Optimization
  6. Model
  7. Problem Formulation
  8. Lower and Upper Bounds
  9. Asymptotic Results
  10. Parametric Surrogate: Sparse Bayesian Linear Model
  11. An Overview of the Parametric Approach
  12. Bayesian Linear Model with Horseshoe Prior
  13. Acquisition Function
  14. Submodular Relaxation.
  15. Cardinality Constraint.
  16. ...and 37 more sections

Key Result

Theorem 1

The $p$-Hypercube problem is NP-hard.

Figures (7)

  • Figure 1: An example of graph construction.
  • Figure 2: Illustration of the solution space.
  • Figure 3: Results for simulation experiments.
  • Figure 4: Results for St. Paul case study.
  • Figure 5: Results for St. Paul case study under varying arrival rates.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 4: Vanishing load
  • Theorem 5: Heavy load
  • Example 6
  • Example 7
  • Theorem 10: Regret Bound of SparBL
  • Lemma 11
  • Lemma 12
  • ...and 13 more