Applying Correlated Bernoulli Processes in Stochastic Operations Research and Management Science Models: A Practical Guide

Applying Correlated Bernoulli Processes in Stochastic Operations Research and Management Science Models: A Practical Guide | 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 Applying Correlated Bernoulli Processes in Stochastic Operations Research and Management Science Models: A Practical Guide Zvi Drezner, Dawit Zerom This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9458638/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract In operations research and management science (ORMS), binomial modeling and Poisson distribution are routinely used to summarizethe number of ''successes'' in a list of $n$ events (wins in a season, stocks up-days in a window, defective units in a batch,customers reached by repeated advertising, etc.). The classical binomial and Poisson distributions assume (i) a constant success rateand (ii) independence between trials. In many ORMS computational statistics settings, neither assumption is fully credible: successes may be positivelyreinforced (heterogeneous skill, clustering, persistence) or negatively reinforced (sampling without replacement, mean reversion,capacity constraints). The purpose of this paper is to introduce to the ORMS community the Generalized Binomial (GBD) and the Correlated Poisson (CPD) distributions, which assume correlated events. They are useful tools for analyzing many ORMS stochastic models. Many applications apply the binomial or Poisson distributions even though the GBD or CPD may be more appropriate for such analyses because events are interdependent. If events are not correlated, the GBD and CPD will not be significantly different from the binomial and Poisson distributions because the binomial and Poisson distributions are special cases of the GBD and CPD. It is a good idea for researchers and practitioners to analyze a model by the GBD or CPD, and revert to the uncorrelated distributions when appropriate.An easy to use Excel file for calculating the GBD and CPD, find their optimal parameters, and the statistical analysis, is provided. Computational statistics Stochastic models Correlated events Generalized Binomial Distribution Correlated Poisson Distribution Full Text Additional Declarations No competing interests reported. Supplementary Files Correlated.xlsx Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 04 May, 2026 Editor assigned by journal 24 Apr, 2026 Submission checks completed at journal 21 Apr, 2026 First submitted to journal 18 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9458638","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":635713765,"identity":"ef28ac88-5c0a-4715-a5ce-f128e1b98ada","order_by":0,"name":"Zvi Drezner","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7ElEQVRIiWNgGAWjYFCCBAjFz3yA8cADmCAPMVok2xIYDgDZEsRrMThGrBb59tyHD3/usMk3PsZjcCCh4l4dv3QD44O3bbi1MPY8NzaQPJNmuQ2s5UyxhOScA8yGc/FoYZZIY5MwbDtsYHa/x+BAYluChMGNBDZpXjxa2CTS2H8ktv03MG4D2pL4D6yF/Tc+LTxAWxgOth0wMGADaWmA2MKMT4sEzzNmyca2ZAOJY2wFBxKOJUjOnJHYLDnnHG4t8u1pjB9/ttkZ8Lcxb3zwoSaBn18i+eCHN2W4tWADjA2kqR8Fo2AUjIJRgAEAwsFPENqIiHcAAAAASUVORK5CYII=","orcid":"","institution":"California State University, Fullerton","correspondingAuthor":true,"prefix":"","firstName":"Zvi","middleName":"","lastName":"Drezner","suffix":""},{"id":635713766,"identity":"6365f3bb-3c6d-4084-855d-b22d3f8e5dc5","order_by":1,"name":"Dawit Zerom","email":"","orcid":"","institution":"California State University, Fullerton","correspondingAuthor":false,"prefix":"","firstName":"Dawit","middleName":"","lastName":"Zerom","suffix":""}],"badges":[],"createdAt":"2026-04-19 01:53:47","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9458638/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9458638/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109054970,"identity":"616df600-2cb0-459e-9c53-1a10ca2c2ca0","added_by":"auto","created_at":"2026-05-12 07:29:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":857265,"visible":true,"origin":"","legend":"","description":"","filename":"Brenouli11.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9458638/v1_covered_e285f5ab-3132-4fb4-8531-8c59a1f4feee.pdf"},{"id":109054608,"identity":"d045b30b-c607-4555-9cd1-dc41847481a7","added_by":"auto","created_at":"2026-05-12 07:26:31","extension":"xlsx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":846156,"visible":true,"origin":"","legend":"","description":"","filename":"Correlated.xlsx","url":"https://assets-eu.researchsquare.com/files/rs-9458638/v1/ae6ea8ac2a3dea637a91fa9a.xlsx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Applying Correlated Bernoulli Processes in Stochastic Operations Research and Management Science Models: A Practical Guide","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"computational-management-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Computational Management Science](https://www.springer.com/journal/10287)","snPcode":"10287","submissionUrl":"https://submission.nature.com/new-submission/10287/3","title":"Computational Management Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Computational statistics, Stochastic models, Correlated events, Generalized Binomial Distribution, Correlated Poisson Distribution","lastPublishedDoi":"10.21203/rs.3.rs-9458638/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9458638/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In operations research and management science (ORMS), binomial modeling and Poisson distribution are routinely used to summarizethe number of ''successes'' in a list of $n$ events (wins in a season, stocks up-days in a window, defective units in a batch,customers reached by repeated advertising, etc.). 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