Performance Analysis of Probabilistic Matching with Retrials and ImpatientCustomers via Level-Dependent QBD Analysis

Performance Analysis of Probabilistic Matching with Retrials and ImpatientCustomers via Level-Dependent QBD Analysis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Performance Analysis of Probabilistic Matching with Retrials and ImpatientCustomers via Level-Dependent QBD Analysis Heng Li Liu, Sherif I. Ammar, Yousef F. AlHarbi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9494137/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract Exact stationary analysis of a two-sided matching system with infinitely many phases is possible, despite level-dependence that forecloses standard matrix-geometric methods. The model captures a behavioral asymmetry common to ride-hailing and freelance platforms: demand-side customers retry failed matches actively, while supply-side participants wait passively and abandon when patience expires. The resulting level-dependent quasi-birth-and-death process is analyzed through UL-type RG-factorization, yielding the exact stationary distribution and closed-form performance measures for congestion, abandonment, throughput, and mean waiting times. Positive recurrence holds for all parameter values without balance conditions, a consequence of linear abandonment-rate growth that dominates constant arrivals. Computationally, the factorization scales as $O(J_{\max}d^{3})$ versus $O((J_{\max}d)^{3})$ for direct truncation, reducing wall-clock time from minutes to seconds at state-space sizes relevant for real-time platform monitoring. Comparative statics on the cost function show faster retrials reduce cost monotonically with diminishing returns, and that supply-side retention dominates demand-side retention by factors of $1.5$ to $7\times$ across every tested demand--supply ratio. Discrete-event simulation confirms the analytical predictions and establishes robustness under Weibull-distributed patience, non-stationary arrivals, and state-dependent matching probabilities. Matched queue retrial customers impatient customers quasi-birth-and-death process RG-factorization cost optimization simulation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 17 May, 2026 Reviewers agreed at journal 15 May, 2026 Reviewers invited by journal 01 May, 2026 Editor assigned by journal 22 Apr, 2026 Submission checks completed at journal 22 Apr, 2026 First submitted to journal 22 Apr, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Comparative statics on the cost function show faster retrials reduce cost monotonically with diminishing returns, and that supply-side retention dominates demand-side retention by factors of $1.5$ to $7\\times$ across every tested demand--supply ratio. 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