{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T00:14:54Z","timestamp":1768436094855,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,2]],"date-time":"2024-11-02T00:00:00Z","timestamp":1730505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MDPI, vouchers for manuscripts review","award":["the list is uploaded along with the manuscript"],"award-info":[{"award-number":["the list is uploaded along with the manuscript"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we create a framework for the uniform algorithmic analysis of queueing systems with the Markov arrival process and the simultaneous service of a restricted number of customers, described by a multidimensional Markov chain. This chain behaves as the finite-state quasi-death process between successive service-beginning epochs, with jumps occurring at these epochs. Such a description of the service process generalizes many known mechanisms of restricted resource sharing and is well suited for describing various future mechanisms. Scenarios involving customers who cannot enter service upon arrival, access via waiting in an infinite buffer, and access via retrials are considered. We compare the generators of the multidimensional Markov chains describing the operation of queueing systems with a buffer and with retrials and show that the sufficient conditions for the ergodicity of these systems coincide. The computation of the stationary distributions of these chains is briefly discussed. The results can be used for performance evaluation and capacity planning of various queueing models with the Markov arrival process and a variety of different service mechanisms that provide simultaneous service to many customers.<\/jats:p>","DOI":"10.3390\/a17110493","type":"journal-article","created":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T09:52:54Z","timestamp":1730713974000},"page":"493","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Framework for Analysis of Queueing Systems with Correlated Arrival Processes and Simultaneous Service of a Restricted Number of Customers in Scenarios with an Infinite Buffer and Retrials"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2881-0227","authenticated-orcid":false,"given":"Alexander","family":"Dudin","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus"},{"name":"R&D Center, Baku Engineering University, Khirdalan City, Hasan Aliyev Str., 120, AZ0101 Absheron, Azerbaijan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergei","family":"Dudin","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Agassi","family":"Melikov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Baku Engineering University, Khirdalan City, Hasan Aliyev Str., 120, AZ0101 Absheron, Azerbaijan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6788-8783","authenticated-orcid":false,"given":"Olga","family":"Dudina","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,2]]},"reference":[{"key":"ref_1","unstructured":"Neuts, M.F. (1994). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation."},{"key":"ref_2","first-page":"731","article-title":"Phase-type distributions: Open problems and a few properties","volume":"15","year":"1999","journal-title":"Stoch. Model."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01182931","article-title":"Processor-sharing queues: Some progress in analysis","volume":"2","author":"Yashkov","year":"1987","journal-title":"Queueing Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1134\/S0005117907090202","article-title":"Processor sharing: A survey of the mathematical theory","volume":"68","author":"Yashkov","year":"2007","journal-title":"Autom. Remote Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1010","DOI":"10.1287\/mnsc.33.8.1010","article-title":"An Optimal Design Problem for Limited Processor Sharing Systems","volume":"33","author":"Yamazaki","year":"1987","journal-title":"Manag. Sci."},{"key":"ref_6","first-page":"245","article-title":"Analysis of queueing model with processor sharing discipline and customers impatience","volume":"5","author":"Dudin","year":"2018","journal-title":"Oper. Res. Perspect."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12651","DOI":"10.1007\/s12652-022-04233-w","article-title":"Priority queueing system with many types of requests and restricted processor sharing","volume":"14","author":"Apice","year":"2023","journal-title":"J. Ambient Intell. Humaniz. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kochetkova, I., Satin, Y., Kovalev, I., Makeeva, E., Chursin, A., and Zeifman, A. (2021). Convergence bounds for limited processor sharing queue with impatience for analyzing non-stationary file transfer in wireless network. Mathematics, 10.","DOI":"10.3390\/math10010030"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1145\/3199524.3199548","article-title":"Response time distribution of a class of limited processor sharing queues","volume":"45","author":"Telek","year":"2018","journal-title":"ACM SIGMETRICS Perform. Eval. Rev."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s11009-023-10042-1","article-title":"Analysis of a queueing system with mixed service discipline","volume":"25","author":"Dudin","year":"2023","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1081\/STM-100001400","article-title":"Sojourn times in non-homogeneous QBD processes with processor-sharing","volume":"17","year":"2001","journal-title":"Stoch. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1080\/00207543.2015.1056316","article-title":"Semi-open queuing networks: A review of stochastic models, solution methods and new research areas","volume":"54","author":"Roy","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.peva.2017.12.005","article-title":"Analysis of a semi-open queueing network with Markovian arrival process","volume":"120","author":"Kim","year":"2018","journal-title":"Perform. Eval."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105086","DOI":"10.1016\/j.cor.2020.105086","article-title":"A new solution approach for multi-stage semi-open queuing networks: An application in shuttle-based compact storage systems","volume":"125","author":"Kumawat","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1287\/opre.1080.0627","article-title":"Solving semi-open queuing networks","volume":"57","author":"Jia","year":"2009","journal-title":"Oper. Res."},{"key":"ref_16","first-page":"21","article-title":"The batch Markovian arrival process: A review and future work","volume":"1","author":"Chakravarthy","year":"2001","journal-title":"Adv. Probab. Theory Stoch. Process."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Chakravarthy, S.R. (2022). Introduction to Matrix-Analytic Methods in Queues 1: Analytical and Simulation Approach\u2014Basics, John Wiley and Sons.","DOI":"10.1002\/9781394165421"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Chakravarthy, S.R. (2022). Introduction to Matrix-Analytic Methods in Queues 2: Analytical and Simulation Approach\u2014Queues and Simulation, John Wiley and Sons.","DOI":"10.1002\/9781394174201"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/15326349108807174","article-title":"New results on the single server queue with a batch Markovian arrival process","volume":"7","author":"Lucantoni","year":"1991","journal-title":"Commun. Stat. Stoch. Model."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dudin, A.N., Klimenok, V.I., and Vishnevsky, V.M. (2020). The Theory of Queuing Systems with Correlated Flows, Springer Nature.","DOI":"10.1007\/978-3-030-32072-0"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gonzalez, M., Lillo, R.E., and Ramirez Cobo, J. (2024). Call center data modeling: A queueing science approach based on Markovian arrival processes. Qual. Technol. Quant. Manag.","DOI":"10.1080\/16843703.2024.2371715"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Falin, G.I., and Templeton, J.G.C. (1997). Retrial Queues, Chapman & Hall.","DOI":"10.1007\/978-1-4899-2977-8"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Artalejo, J.R., and Gomez-Corral, A. (2008). Retrial Queueing Systems, Springer.","DOI":"10.1007\/978-3-540-78725-9"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF01158472","article-title":"A survey of retrial queues","volume":"7","author":"Falin","year":"1990","journal-title":"Queueing Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF01158899","article-title":"A survey on retrial queues","volume":"2","author":"Yang","year":"1987","journal-title":"Queueing Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s10479-006-5298-4","article-title":"A bibliographical guide to the analysis of retrial queues through matrix analytic techniques","volume":"141","year":"2006","journal-title":"Ann. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/j.mcm.2009.12.011","article-title":"Accessible bibliography on retrial queues: Progress in 2000\u20132009","volume":"51","author":"Artalejo","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10479-015-2038-7","article-title":"A survey of retrial queueing systems","volume":"247","author":"Kim","year":"2016","journal-title":"Ann. Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"101","DOI":"10.5209\/rev_REMA.2002.v15.n1.16950","article-title":"Standard and retrial queueing systems: A comparative analysis","volume":"15","author":"Artalejo","year":"2002","journal-title":"Rev. Mat. Complut."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1287\/ijoc.2017.0759","article-title":"Space reduction for a class of multidimensional Markov chains: A summary and some applications","volume":"30","author":"He","year":"2018","journal-title":"INFORMS J. Comput."},{"key":"ref_31","unstructured":"Graham, A. (2018). Kronecker Products and Matrix Calculus with Applications, Courier Dover Publications."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0377-0427(00)00393-9","article-title":"The ubiquitous Kronecker product","volume":"123","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1023\/A:1015041602946","article-title":"A retrial BMAP\/PH\/N system","volume":"40","author":"Breuer","year":"2002","journal-title":"Queueing Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1002\/ett.1000","article-title":"Modeling the access to a wireless network at hot spots","volume":"16","author":"Breuer","year":"2005","journal-title":"Eur. Trans. Telecommun."},{"key":"ref_35","first-page":"419","article-title":"Independent Markov processes in parallel","volume":"1","author":"Ramaswami","year":"1985","journal-title":"Stoch. Model."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106933","DOI":"10.1109\/ACCESS.2021.3100561","article-title":"Mathematical model of operation of a cell of a mobile communication network with adaptive modulation schemes and handover of mobile users","volume":"9","author":"Kim","year":"2021","journal-title":"IEEE Access"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s10479-012-1254-7","article-title":"Computation of the steady state distribution for multi-server retrial queues with phase type service process","volume":"201","author":"Kim","year":"2012","journal-title":"Ann. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11134-006-7586-8","article-title":"A survey on discriminatory processor sharing","volume":"53","author":"Altman","year":"2006","journal-title":"Queueing Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/15326340701826872","article-title":"Discriminatory processor sharing queues and the DREB method","volume":"24","author":"Lian","year":"2008","journal-title":"Stoch. Model."},{"key":"ref_40","unstructured":"Boxma, O.J., Cohen, J.W., and Tijms, H.C. (1986). A simple telephone exchange with delayed feedbacks. Teletraffic Analysis and Computer Performance Evaluation, Elsevier Science."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/S0377-2217(00)00330-1","article-title":"Analysis of multiserver queues with constant retrial rate","volume":"135","author":"Artalejo","year":"2001","journal-title":"Eur. J. Oper. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/PL00013359","article-title":"On extreme values of orbit lengths in queues with constant retrial rate","volume":"23","year":"2001","journal-title":"OR-Spektrum"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"223","DOI":"10.2307\/3215189","article-title":"Steady state solution of a single-server queue with linear repeated requests","volume":"34","author":"Artalejo","year":"1997","journal-title":"J. Appl. Probab."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s11134-006-0300-z","article-title":"Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing. theory","volume":"54","author":"Klimenok","year":"2006","journal-title":"Queueing Syst."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (2012). Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9781139020411"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"112425","DOI":"10.1016\/j.cam.2019.112425","article-title":"Effective algorithm for computation of the stationary distribution of multi-dimensional level-dependent Markov chain with upper block-Hessenberg structure of the generator","volume":"366","author":"Dudin","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.cie.2013.06.020","article-title":"Retrial queuing system with Markovian arrival flow and phase-type service time distribution","volume":"66","author":"Dudina","year":"2013","journal-title":"Comput. Ind. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1016\/j.apm.2018.09.005","article-title":"Retrial multi-server queuing system with PHF service time distribution as a model of a channel with unreliable transmission of information","volume":"65","author":"Dudin","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_49","unstructured":"Kemeny, J.G., Snell, J.L., and Knapp, A.W. (2012). Denumerable Markov Chains: With a Chapter of Markov Random Fields by David Griffeath, Springer Science and Business Media."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Bini, D.A., Latouche, G., and Meini, B. (2005). Numerical Methods for Structured Markov Chains, OUP.","DOI":"10.1093\/acprof:oso\/9780198527688.001.0001"},{"key":"ref_51","first-page":"189","article-title":"Methods of judging the annoyance caused by congestion","volume":"4","author":"Palm","year":"1953","journal-title":"TELE"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1287\/opre.5.5.650","article-title":"Queuing with impatient customers and ordered service","volume":"5","author":"Barrer","year":"1957","journal-title":"Oper. Res."},{"key":"ref_53","first-page":"186","article-title":"Queueing with reneging","volume":"2","author":"Haight","year":"1959","journal-title":"Metr. Int. J. Theor. Appl. Stat."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1287\/opre.9.6.771","article-title":"Double queues and impatient customers with an application to inventory theory","volume":"9","author":"Sasieni","year":"1961","journal-title":"Oper. Res."},{"key":"ref_55","first-page":"173","article-title":"Queuing with balking and reneging in M\/G\/1 systems","volume":"12","author":"Rao","year":"1967","journal-title":"Metr. Int. J. Theor. Appl. Stat."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"688","DOI":"10.2307\/3213871","article-title":"A queueing system with impatient customers","volume":"22","author":"Tijms","year":"1985","journal-title":"J. Appl. Probab."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Wang, K., Li, N., and Jiang, Z. (2010, January 15\u201317). Queueing system with impatient customers: A review. Proceedings of the 2010 IEEE International Conference on Service Operations and Logistics, and Informatics, Qingdao, China.","DOI":"10.1109\/SOLI.2010.5551611"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1504\/IJMOR.2023.135546","article-title":"Queuing models with customers\u2019 impatience: A survey","volume":"26","author":"Sharma","year":"2023","journal-title":"Int. J. Math. Oper. Res."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"768","DOI":"10.2307\/1427473","article-title":"On queues with impatience","volume":"22","author":"Stanford","year":"1990","journal-title":"Adv. Appl. Probab."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1111\/itor.13277","article-title":"Optimal admission and queuing control with reneging behavior under premature discharge decisions","volume":"32","author":"Yang","year":"2025","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Dudin, A., Dudin, S., Klimenok, V., and Dudina, O. (2024). Stability of queueing systems with impatience, balking and non-persistence of customers. Mathematics, 12.","DOI":"10.3390\/math12142214"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/11\/493\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:27:18Z","timestamp":1760113638000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/11\/493"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,2]]},"references-count":61,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["a17110493"],"URL":"https:\/\/doi.org\/10.3390\/a17110493","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,2]]}}}