A Geometrically Convergent Solution to Spatial Hypercube Queueing Models

TL;DR

This paper introduces a geometrically convergent algorithm for solving heterogeneous hypercube queueing models, significantly improving computational efficiency for large-scale spatial queueing problems in emergency services.

Contribution

It extends the hypercube queueing model to heterogeneous service rates and develops a fast, convergent, parallel algorithm for large-scale problem solving.

Findings

Algorithm converges at a geometric rate.

Sequential algorithm is over 1,000 times faster than sparse solver.

Parallel algorithm achieves eightfold speedup with 12 processors.

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.