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

TL;DR

This paper develops Bayesian optimization methods to efficiently determine optimal server locations in complex spatial queueing systems, accounting for server availability and dependencies, with proven convergence and superior real-world performance.

Contribution

It introduces novel parametric and nonparametric Bayesian optimization techniques for the NP-hard server location problem in spatial queues, with theoretical guarantees and practical validation.

Findings

Both methods achieve sublinear regret and converge to optimal solutions.

Numerical experiments show the approaches outperform baseline methods.

Real-world case study confirms effectiveness in emergency response system.

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.