{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:09:45Z","timestamp":1762254585187,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T00:00:00Z","timestamp":1687564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for system stability are presented. In addition, the steady-state probability distribution of the queueing model was derived by employing the matrix geometric method. Furthermore, a few formulae were constructed to determine the model\u2019s performance indicators. Finally, the influence of system parameters was also investigated using some numerical examples.<\/jats:p>","DOI":"10.3390\/sym15071304","type":"journal-article","created":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T02:05:02Z","timestamp":1687745102000},"page":"1304","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Analysis of a Heterogeneous Queuing Model with Intermittently Obtainable Servers under a Hybrid Vacation Schedule"],"prefix":"10.3390","volume":"15","author":[{"given":"Divya","family":"Kothandaraman","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7765-094X","authenticated-orcid":false,"given":"Indhira","family":"Kandaiyan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,24]]},"reference":[{"key":"ref_1","unstructured":"Morse, P.M. (2004). Queues, Inventories and Maintenance: The Analysis of Operational Systems with Variable Demand and Supply, Courier Corporation."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1287\/opre.18.1.145","article-title":"Two-server Markovian queues with balking: Heterogeneous vs. homogeneous servers","volume":"18","author":"Singh","year":"1970","journal-title":"Oper. Res."},{"key":"ref_3","first-page":"145867","article-title":"An M\/M\/2 queueing system with heterogeneous servers including one with working vacation","volume":"2012","author":"Krishnamoorthy","year":"2012","journal-title":"Int. J. Stoch. Anal."},{"key":"ref_4","first-page":"205","article-title":"Queuing analysis of markovian queue having two heterogeneous servers with catastrophes using matrix geometric technique","volume":"12","author":"Indra","year":"2017","journal-title":"Int. J. Syst. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/S0166-5316(00)00041-9","article-title":"Waiting time distribution in a two-class two-server heterogeneous priority queue","volume":"43","author":"Leemans","year":"2001","journal-title":"Perform. Eval."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1080\/16843703.2021.1981529","article-title":"An M\/G\/1 queueing model with k sequential heterogeneous service steps and vacations in the transient state","volume":"19","author":"Mohammadi","year":"2022","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"012001","DOI":"10.1088\/1742-6596\/1724\/1\/012001","article-title":"Performance Analysis of Two Heterogeneous Server Queuing Model with Intermittently Obtainable Server Using Matrix Geometric Method","volume":"1724","author":"Seenivasan","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_8","unstructured":"Agarwal, N.N. (1965). Some Problems in the Theory of Reliability and Queues. [Ph.D.Thesis, Kurukshetra University]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sharda (1968). A queuing problem with intermittently available server and arrivals and departures in batches of variable size. ZAMM, 48, 471\u2013476.","DOI":"10.1002\/zamm.19680480707"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2332","DOI":"10.1016\/j.matpr.2021.11.567","article-title":"M\/M\/2 heterogeneous queueing system having unreliable server with catastrophes and restoration","volume":"51","author":"Seenivasan","year":"2022","journal-title":"Mater. Today Proc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1287\/mnsc.22.2.202","article-title":"Utilization of idle time in an M\/G\/1 queueing system","volume":"22","author":"Levy","year":"1975","journal-title":"Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF01149327","article-title":"Queueing systems with vacations\u2014A survey","volume":"1","author":"Doshi","year":"1986","journal-title":"Queueing Syst."},{"key":"ref_13","unstructured":"Takagi, H. (1991). Queueing analysis: A Foundation of Performance Evaluation, North-Holland. Vacation Priority Systems 1."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/S0166-5316(02)00057-3","article-title":"M\/M\/1 queues with working vacations (M\/M\/1\/WV)","volume":"50","author":"Servi","year":"2002","journal-title":"Perform. Eval."},{"key":"ref_15","first-page":"621","article-title":"The M\/M\/1 queue with single working vacation","volume":"19","author":"Tian","year":"2008","journal-title":"Int. J. Inf. Manag."},{"key":"ref_16","first-page":"3","article-title":"Recent developments in vacation queueing models: A short survey","volume":"7","author":"Ke","year":"2010","journal-title":"Int. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1080\/16843703.2022.2054088","article-title":"A multi-station unreliable machine model with working vacation policy and customers\u2019 impatience","volume":"19","author":"Bouchentouf","year":"2022","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"GnanaSekar, M.M.N., and Kandaiyan, I. (2022). Analysis of an M\/G\/1 Retrial Queue with Delayed Repair and Feedback under Working Vacation policy with Impatient Customers. Symmetry, 14.","DOI":"10.3390\/sym14102024"},{"key":"ref_19","first-page":"495","article-title":"Analysis for the M\/M\/1 queue with multiple working vacations and N-policy","volume":"19","author":"Zhang","year":"2008","journal-title":"Int. J. Inf. Manag."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mytalas, G.C., and Zazanis, M.A. (2022). Performance analysis for Bernoulli feedback queues subject to disasters: A system with batch Poisson arrivals under a multiple vacation policy. Qual. Technol. Quant. Manag., 1\u201334.","DOI":"10.1080\/16843703.2022.2092954"},{"key":"ref_21","unstructured":"Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models, JHU."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1504\/IJOR.2009.026941","article-title":"The M\/M\/1 queue with single working vacation and set-up times","volume":"6","author":"Xu","year":"2009","journal-title":"Int. J. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1080\/16843703.2021.1892907","article-title":"An M\/M\/1 queue subject to differentiated vacation with partial interruption and customer impatience","volume":"18","author":"Vijayashree","year":"2021","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1080\/16843703.2020.1755088","article-title":"Matrix-geometric solution of multi-server queueing systems with Bernoulli scheduled modified vacation and retention of reneged customers: A meta-heuristic approach","volume":"18","author":"Shekhar","year":"2021","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_25","first-page":"12052","article-title":"A multiphase queuing system with assorted servers by using matrix geometric method","volume":"12","author":"Aniyeri","year":"2017","journal-title":"Int. J. Appl. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1287\/opre.34.5.752","article-title":"Queueing system with service interruptions","volume":"34","author":"Federgrune","year":"1986","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apm.2009.03.019","article-title":"Working vacations queueing model with multiple types of server breakdowns","volume":"34","author":"Jain","year":"2010","journal-title":"Appl. Math. Model."},{"key":"ref_28","first-page":"113","article-title":"Multi-server retrial queuing system with unreliable server","volume":"7","author":"Kalyanaraman","year":"2010","journal-title":"Int. J. Comput. Cogn."},{"key":"ref_29","first-page":"917","article-title":"Matrix-Geometric Method for Queueing Model with Subject to Breakdown and N-Policy Vacations","volume":"5","author":"Chandrasekar","year":"2015","journal-title":"Int. J. Math. Aeterna"},{"key":"ref_30","first-page":"2379","article-title":"M\/M\/1 Retrial queueing System with Pre-emptive priority service","volume":"4","author":"Subramanian","year":"2010","journal-title":"Int. J. Comput. Appl."},{"key":"ref_31","first-page":"35","article-title":"A single-server Markovian feedback queuing system with discouraged arrivals and retention of reneged customers","volume":"4","author":"Sharma","year":"2013","journal-title":"Am. J. Oper. Res."},{"key":"ref_32","first-page":"259","article-title":"Cost Optimization of an Unreliable server queue with two stage service process under hybrid vacation policy","volume":"204","author":"Anshul","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Latouche, G., and Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. SIAM J. Comp.","DOI":"10.1137\/1.9780898719734"},{"key":"ref_34","first-page":"331","article-title":"Analysis of Heterogeneous Queueing Model with Unreliable Server and Working Vacation","volume":"881","author":"Seenivasan","year":"2022","journal-title":"Adv. Electr. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1304\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:59:34Z","timestamp":1760126374000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,24]]},"references-count":34,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071304"],"URL":"https:\/\/doi.org\/10.3390\/sym15071304","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,6,24]]}}}