Generalized Hypercube Queueing Models with Overlapping Service Regions

TL;DR

This paper introduces a generalized hypercube queueing model for overlapping service regions, enabling accurate performance analysis of high-load, capacity-constrained service systems like police patrols using Markov models and efficient computations.

Contribution

It extends Larson's hypercube model to handle overlapping service regions with a richer state-space, providing a practical approximation method for high-load service systems.

Findings

Overlapping patrol regions reduce congestion in police systems.

The model accurately predicts system performance under high workload.

Simulation results validate the effectiveness of the approach.

Abstract

We present a generalized hypercube queueing model extending Larson's (1974) framework to overlapping service regions such as police beats. Traditional hypercube models effectively capture light-traffic systems by tracking each server's busy or idle state. However, modern service operations often operate near capacity, where each server handles only a subset of calls and queues may form. This saturation invalidates the simple binary representation and necessitates a richer state-space analysis. Our model addresses this by formulating a Markov model with non-negative integer-valued state vectors and introducing a truncated hyperlattice queueing approximation. Exploiting the sparsity of the transition matrix, we efficiently compute the steady-state distribution and demonstrate that it closely approximates the original system under canonical dispatching policies. The model enables accurate evaluation of system performance metrics and can be adapted to diverse service systems. We validate the approach through simulations on synthetic service networks and a case study of the Atlanta police operations system, which faces high workload, staffing shortages, and boundary effects. Using real 911 calls-for-service data, we find that allowing overlapping patrol regions significantly mitigates congestion and improves deployment efficiency. Although our focus is on police districting, the generalized hypercube framework applies broadly to mobile-server systems and high-load service environments.