{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T08:19:50Z","timestamp":1763367590610,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T00:00:00Z","timestamp":1670284800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tomsk State University Development Programme","award":["Priority-2030"],"award-info":[{"award-number":["Priority-2030"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, a retrial queueing system of the M\/M\/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server\u2019s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given.<\/jats:p>","DOI":"10.3390\/axioms11120699","type":"journal-article","created":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T04:00:37Z","timestamp":1670385637000},"page":"699","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Asymptotic Diffusion Analysis of Retrial Queueing System M\/M\/1 with Impatient Customers, Collisions and Unreliable Servers"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7016-492X","authenticated-orcid":false,"given":"Elena","family":"Danilyuk","sequence":"first","affiliation":[{"name":"Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 634050 Tomsk, Russia"}]},{"given":"Alexander","family":"Plekhanov","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 634050 Tomsk, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9285-1555","authenticated-orcid":false,"given":"Svetlana","family":"Moiseeva","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 634050 Tomsk, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5303-818X","authenticated-orcid":false,"given":"Janos","family":"Sztrik","sequence":"additional","affiliation":[{"name":"Faculty of Informatics, University of Debrecen, 4032 Debrecen, Hungary"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"218","DOI":"10.17516\/1997-1397-2020-13-2-218-230","article-title":"Asymptotic Analysis of Retrial Queueing System M\/M\/1 with Impatient Customers, Collisions and Unreliable Server","volume":"13","author":"Danilyuk","year":"2020","journal-title":"J. Sib. Fed. Univ. Math. Phys."},{"key":"ref_2","unstructured":"Jonin, G.L. (1982). Determination of probabilistic characteristics of single-line queues with double connections and repeated calls. Models of Systems of Distribution of Information and Its Analysis, Nauka."},{"key":"ref_3","unstructured":"Falin, G.I., and Sukharev, Y.I. (1985). On Single-Line Queues with Double Connections, All-Union Institute for Scientific and Technical Information."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/BF01163860","article-title":"Retrial queues with collision arising from unslotted CSMA\/CD protocol","volume":"11","author":"Choi","year":"1992","journal-title":"Queueing Syst."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Artajelo, J.R., and Gomez-Corral, A. (2008). Retrial Queueing Systems: A Computational Approach, Springer.","DOI":"10.1007\/978-3-540-78725-9"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1080\/03610926.2020.1758943","article-title":"An unreliable single server retrial queue with collisions and transmission errors","volume":"51","author":"Lakaour","year":"2020","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_8","first-page":"93","article-title":"On the M\/G\/1 queueing system with repeated orders and unreliable server","volume":"6","author":"Aissani","year":"1988","journal-title":"J. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2093","DOI":"10.1016\/0026-2714(93)90003-H","article-title":"Unreliable queueing with repeated orders","volume":"33","author":"Aissani","year":"1993","journal-title":"Microelectron. Reliab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/BF01158703","article-title":"A retrial queue with redundancy and unreliable server","volume":"17","author":"Aissani","year":"1994","journal-title":"Queueuing Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF01158474","article-title":"Retrial queues with server subject to breakdowns and repairs","volume":"7","author":"Kulkarni","year":"1990","journal-title":"Queueing Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3032","DOI":"10.1007\/BF01252189","article-title":"Diffusion approximation of systems with repeated calls and an unreliable server","volume":"72","author":"Anisimov","year":"1994","journal-title":"J. Math. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2136","DOI":"10.1134\/S0005117918120044","article-title":"Asymptotic Analysis of an Retrial Queueing System M\/M\/1 with Collisions and Impatient Calls","volume":"79","author":"Danilyuk","year":"2018","journal-title":"Autom. Remote Control"},{"key":"ref_14","first-page":"330","article-title":"Stability Analysis for Retrial Queue with Collisions and r-Persistent Customers","volume":"1605","author":"Nazarov","year":"2022","journal-title":"Commun. Comput. Inf. Sci."},{"key":"ref_15","first-page":"1","article-title":"Some Special Features of Finite-Source Retrial Queues with Collisions, an Unreliable Server and Impatient Customers in the Orbit","volume":"1391","author":"Sztrik","year":"2021","journal-title":"Commun. Comput. Inf. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s11134-018-9592-z","article-title":"On first-come, first-served queues with two classes of impatient customers","volume":"91","author":"Adan","year":"2019","journal-title":"Queueing Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.camwa.2008.10.084","article-title":"On the impact of customer balking, impatience and retrials in telecommunication systems","volume":"57","author":"Artalejo","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s11009-019-09707-7","article-title":"Analysis of queueing system with non-preemptive time limited service and impatient customers","volume":"22","author":"Kim","year":"2020","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.apm.2017.10.025","article-title":"On an unreliable-server retrial queue with customer feedback and impatience","volume":"55","author":"Chang","year":"2018","journal-title":"Appl. Math. Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1440009","DOI":"10.1142\/S0217595914400090","article-title":"Multiserver Retrial Queues with Two Types of Nonpersistent Customers","volume":"31","year":"2014","journal-title":"Asia-Pac. J. Oper. Res."},{"key":"ref_21","first-page":"233","article-title":"Asymptotic Diffusion Analysis of an Retrial Queueing System M\/M\/1 with Impatient Calls","volume":"1552","author":"Danilyuk","year":"2022","journal-title":"Commun. Comput. Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"225","DOI":"10.13189\/ms.2020.080219","article-title":"Stochastic Decomposition Result of an Unreliable Queue with Two Types of Services","volume":"8","author":"Choudhury","year":"2020","journal-title":"Math. Stat."},{"key":"ref_23","first-page":"35","article-title":"Unreliable Server Retrial Queue with Optional Service and Multi-Phase Repair","volume":"14","author":"Singh","year":"2017","journal-title":"Int. J. Oper. Res."},{"key":"ref_24","first-page":"319","article-title":"Virtual Waiting Time in Single-Server Queueing Model M|G|1 with Unreliable Server and Catastrophes","volume":"1391","author":"Kerobyan","year":"2021","journal-title":"Commun. Comput. Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9136","DOI":"10.53730\/ijhs.v6nS5.10690","article-title":"Markovian queueing model with catastrophe, unreliable and backup server","volume":"6","author":"Seenivasan","year":"2022","journal-title":"Int. J. Health Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1287\/opre.1040.0136","article-title":"A Diffusion Approximation for the G\/GI\/n\/m Queue","volume":"52","author":"Whitt","year":"2003","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s11134-005-3282-3","article-title":"A Diffusion Approximation for a GI\/GI\/1 Queue with Balking or Reneging","volume":"50","author":"Ward","year":"2005","journal-title":"Queueing Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1214\/105051604000000035","article-title":"Diffusion approximation for a processor sharing queue in heavy traffic","volume":"14","author":"Gromoll","year":"2004","journal-title":"Ann. Appl. Probab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1023\/A:1019108819713","article-title":"Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse","volume":"30","author":"Williams","year":"1998","journal-title":"Queueing Syst. Theory Appl."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Moiseev, A., Nazarov, A., and Paul, S. (2020). Asymptotic diffusion analysis of multi-server retrial queue with hyper-exponential service. Mathematics, 8.","DOI":"10.3390\/math8040531"},{"key":"ref_31","unstructured":"Nazarov, A., Phung-Duc, T., Paul, S., and Lizyura, O. (2020, January 23\u201327). Asymptotic-Diffusion Analysis of Retrial Queue with Two-Way Communication and Renewal Input. Proceedings of the 5th International Conference on Stochastic Methods, Moscow, Russia."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Nazarov, A., Phung-Duc, T., Paul, S., and Lizyura, O. (2022). Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls. Mathematics, 10.","DOI":"10.3390\/math10060948"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Nazarov, A., Phung-Duc, T., and Izmailova, Y. (2021). Asymptotic-Diffusion Analysis of Multi-server Retrial Queueing System with Priority Customers. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2020, Springer. Communications in Computer and Information Science.","DOI":"10.1007\/978-3-030-72247-0_18"},{"key":"ref_34","first-page":"230","article-title":"Asymptotic Analysis of Retrial Queueing System M\/GI\/1 with Collisions and Impatient Calls","volume":"1109","author":"Danilyuk","year":"2019","journal-title":"Commun. Comput. Inf. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/699\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:34:58Z","timestamp":1760146498000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/699"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,6]]},"references-count":34,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120699"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120699","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,6]]}}}