Generalized Hypercube Queueing Models with Overlapping Service Regions

[2304.02824] Generalized Hypercube Queueing Models with Overlapping Service Regions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2304.02824 (math) [Submitted on 6 Apr 2023 ( v1 ), last revised 16 Aug 2026 (this version, v5)] Title: Generalized Hypercube Queueing Models with Overlapping Service Regions Authors: Wenqian Xing , Shixiang Zhu , Yao Xie View a PDF of the paper titled Generalized Hypercube Queueing Models with Overlapping Service Regions, by Wenqian Xing and 2 other authors View PDF HTML (experimental) Abstract: Motivated by the operations of the Atlanta Police Department, where heavy workloads and staffing shortages increasingly require units to patrol across overlapping service regions, we develop a generalized hypercube queueing model, extending Larson (1974), for spatial service systems with overlapping coverage. Designing effective service regions requires capturing both workload and the operations of mobile servers such as police units. The classical hypercube model, which tracks only whether each server is busy or idle, is well suited to light traffic but inadequate for congested systems with server-specific queues and restricted service regions. We model the system as a Markov chain on a nonnegative integer-valued state space and develop a sparse truncated hyperlattice approximation for efficient steady-state computation and performance evaluation. We further characterize the workloads attainable under stable dispatch policies, establish conditions for stabilizability, and identify dispatch policies that optimally balance workloads. We validate the model through simulation and apply it to the Atlanta Police Department, where rising workloads, staffing shortages, and boundary effects create significant operational challenges. Using real 911 calls-for-service data, our analysis indicates that a police operations system with permitted overlapping patrols can significantly mitigate these problems, leading to more effective deployment of the police force. Although the paper focuses on police districting applications, the generalized hypercube queueing model is applicable to other mobile server models in the general setup. Subjects: Probability (math.PR) Cite as: arXiv:2304.02824 [math.PR] (or arXiv:2304.02824v5 [math.PR] for this version) https://doi.org/10.48550/arXiv.2304.02824 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Wenqian Xing [ view email ] [v1] Thu, 6 Apr 2023 02:18:39 UTC (1,807 KB) [v2] Sun, 10 Dec 2023 19:56:01 UTC (9,859 KB) [v3] Fri, 12 Jan 2024 03:08:41 UTC (13,701 KB) [v4] Mon, 29 Sep 2025 04:58:52 UTC (12,532 KB) [v5] Sun, 16 Aug 2026 02:28:55 UTC (12,818 KB) Full-text links: Access Paper: View a PDF of the paper titled Generalized Hypercube Queueing Models with Overlapping Service Regions, by Wenqian Xing and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2023-04 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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