[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83618-en":3,"doc-seo-83618-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83618,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Markovian Arrival Process Parameter Estimation of Quasi-birth-death Queueing Systems with Utilization Data","Parameter estimation for queueing systems often relies on inter-arrival, waiting time, or queue-length observations, which are frequently unavailable in real computer systems. Utilization data, such as CPU busy fraction within monitoring intervals, is easier to collect but hides exact arrivals, service completions, phase transitions, and system states during unobservable periods. This work proposes an expectation-maximization (EM) algorithm to estimate Markovian arrival process parameters for MAP-driven quasi-birth-death (QBD) queues using utilization observations, enabling maximum-likelihood updates and AIC-based phase selection to reduce overfitting.","Markovian Arrival Process Parameter Estimation of Quasi-birth-death Queueing Systems with  \nUtilization Data  \nChen Li, Senior Member, IEEE, Junjun Zheng, Member, IEEE, Hiroyuki Okamura, Member, IEEE,  \nand Tadashi Dohi, Member, IEEE  \narXiv :2607 .0 19 14v 1 [ cs .PF] 2 Jul 2026  \nAbstract—Parameter estimation for queueing systems is commonly performed using inter-arrival times, waiting times, or queue-length observations. However, such detailed observations are often unavailable in practical computer systems, where utilization data, such as CPU utilization, is much easier to collect. Utilization data provides only the fraction of time during which the system is busy within each monitoring interval, while the exact arrivals, services, phase transitions, and system states in unobservable periods remain hidden. This paper proposesan expectation-maximization (EM) algorithm for estimating the parameters of Markovian arrival process (MAP)-driven quasibirth-death (QBD) queueing systems from utilization data. The proposed method formulates the underlying queueing dynamics as a QBD process and derives the expected sufficient statistics for sojourn times, phase transitions, arrivals, and services over both observable and unobservable intervals. These expectations are then used to iteratively update the MAP and service parameters under the maximum likelihood framework. In addition, Akaike’s information criterion is introduced to select the appropriate number of MAP phases and mitigate overfitting. The proposed framework enables MAP-based queueing parameter estimation from incomplete utilization observations and provides a practical modeling approach for systems where detailed event-level measurements are difficult to obtain.  \nIndex Terms—Utilization data, Markovian arrival process (MAP), Markov-modulated Poisson process (MMPP), Qusaibirth-death (QBD) process, Expectation-maximization (EM) algorithm, Maximum likelihood estimation (MLE).  \nNomenclature  \nAcronyms and Abbreviations  \nMAP Markovian arrival process.  \nMMPP Markov modulated Poisson process.  \nBMAP Batch Markovian arrival process.  \nCTMC Continuous-time Markov chain.  \nMLE Maximum likelihood estimation.  \nEM Expectation maximization.  \nPH Phase type.  \nHMM Hidden Markov chain.  \nC. Li is with the D3 Center, The University of Osaka, Japan. e-mail: [li.chen.d3c@osaka-u.ac.jp](li.chen.d3c@osaka-u.ac.jp)  \nJ. Zheng is with the Department of Information Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, Japan. e-mail: [jzhenghiroshima-u.ac.jp](jzhenghiroshima-u.ac.jp)  \nH. Okamura is with the Department of Information Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, Japan.  \ne-mail: [okamu@hiroshima-u.ac.jp](okamu@hiroshima-u.ac.jp)  \nT. Dohi is with the Department of Information Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, Japan.  \ne-mail: [dohi@hiroshima-u.ac.jp](dohi@hiroshima-u.ac.jp)  \nHPP Homogeneous Poisson process.  \nNHPP Non-homogeneous Poisson process.  \nFCFS First come first served.  \nLLF Log-likelihood function.  \nQBD Quasi-birth-death process.  \nAIC Akaike’s information criterion.  \nNotations  \n􀀠 Capacity of the 􀀢􀀖􀀥/􀀢/1/􀀠 queueing system.  \n􀁗 􀀢 􀀖􀀥 Infinitesimal generator matrix of MAP.  \n􀁊 0 Infinitesimal generator matrix in the case of no arrivals.  \n􀁊 1 Transition rate matrix when an arrival occurs.􀀣 (􀁃) Cumulative number of arrivals during the time  \ninterval [0,􀁃) .  \n􀀟 (􀁃) Phase process of the underlying CTMC at time  \n􀁃 .  \n􀀼 Maximum number of phases of MAP.  \n􀁀 􀀸, 􀀹 (􀀸 ≠ 􀀹) Transition rate from phase 􀀸 to phase 􀀹 .  \n􀁟􀀸, 􀀹 Arrival rate from phase 􀀸 to phase 􀀹 of MAP.  \nπ􀁂 Stationary probability vector of MAP.  \n˜􀁟 Fundamental arrival rate.  \n􀁖 􀀺 (􀁃) Matrix whose element indicates that the number  \nof arrivals is 􀀺 by time 􀁃 .π Initial probability vector for phases.  \n􀁟􀀸 Arrival rate of phase 􀀸 of MMPP.  \nD Observable data.  \n􀁄 􀀽 􀀽-th utilization sampl","cbCair7uS30RSjd4","https://ap.wps.com/l/cbCair7uS30RSjd4","pdf",255646,4,1,11,"English","en",105,"# Abstract\n# Index Terms\n# Nomenclature\n## Acronyms and Abbreviations\n## Notations","[{\"question\":\"Why is utilization data used for queueing parameter estimation instead of event-level observations?\",\"answer\":\"Utilization data is easier to collect in practice, while detailed measurements like exact arrivals, services, and phase transitions during unobservable periods are often unavailable. The method leverages busy/idle fractions within monitoring intervals to infer model parameters.\"},{\"question\":\"What modeling framework does the proposed method use for the queueing system?\",\"answer\":\"The queueing dynamics are formulated as a quasi-birth-death (QBD) process driven by a Markovian arrival process (MAP). The approach derives expected sufficient statistics over both observable and unobservable intervals.\"},{\"question\":\"How does the method estimate parameters and choose the number of MAP phases?\",\"answer\":\"An expectation-maximization (EM) algorithm iteratively updates the MAP and service parameters under a maximum-likelihood framework using expectations of sufficient statistics. Akaike’s information criterion (AIC) selects the appropriate number of MAP phases to mitigate overfitting.\"}]",1784189305,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"markovian-arrival-process-parameter-estimation-of-quasi-birth-death-queueing-systems-with-utilization-data","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/markovian-arrival-process-parameter-estimation-of-quasi-birth-death-queueing-systems-with-utilization-data/83618/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is utilization data used for queueing parameter estimation instead of event-level observations?","Question",{"text":75,"@type":76},"Utilization data is easier to collect in practice, while detailed measurements like exact arrivals, services, and phase transitions during unobservable periods are often unavailable. The method leverages busy/idle fractions within monitoring intervals to infer model parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What modeling framework does the proposed method use for the queueing system?",{"text":80,"@type":76},"The queueing dynamics are formulated as a quasi-birth-death (QBD) process driven by a Markovian arrival process (MAP). The approach derives expected sufficient statistics over both observable and unobservable intervals.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the method estimate parameters and choose the number of MAP phases?",{"text":84,"@type":76},"An expectation-maximization (EM) algorithm iteratively updates the MAP and service parameters under a maximum-likelihood framework using expectations of sufficient statistics. Akaike’s information criterion (AIC) selects the appropriate number of MAP phases to mitigate overfitting.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]