{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:07:51Z","timestamp":1760144871903,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T00:00:00Z","timestamp":1716940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Research Council of Canada"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We consider two companies that are competing for orders. Let X1(n) denote the number of orders processed by the first company at time n, and let \u03c4(k) be the first time that X1(n)&lt;j or X1(n)=r, given that X1(0)=k. We assume that {X1(n),n=0,1,\u2026} is a controlled discrete-time queueing system. Each company is using some control to increase its share of orders. The aim of the first company is to maximize the expected value of \u03c4(k), while its competitor tries to minimize this expected value. The optimal solution is obtained by making use of dynamic programming. Particular problems are solved explicitly.<\/jats:p>","DOI":"10.3390\/g15030019","type":"journal-article","created":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T06:58:07Z","timestamp":1716965887000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Controlled Discrete-Time Queueing System as a Model for the Orders of Two Competing Companies"],"prefix":"10.3390","volume":"15","author":[{"given":"Mario","family":"Lefebvre","sequence":"first","affiliation":[{"name":"Department of Mathematics and Industrial Engineering, Polytechnique Montr\u00e9al, C.P. 6079, Succursale Centre-ville, Montr\u00e9al, QC H3C 3A7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1007\/s12597-022-00588-y","article-title":"Analysis of Markovian queueing system with second optional service operating under the triadic policy","volume":"60","author":"Laxmi","year":"2023","journal-title":"OPSEARCH"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1080\/01605682.2022.2053308","article-title":"Event-based optimization of service rate control in retrial queues","volume":"74","author":"Chen","year":"2023","journal-title":"J. Oper. Res. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/16843703.2022.2157695","article-title":"Equilibrium and social optimality in queues with service rate and customers\u2019 joining decisions","volume":"21","author":"Tian","year":"2024","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3819","DOI":"10.3934\/jimo.2022111","article-title":"Performance evaluation and bi-objective optimization for F-policy queue with alternating service rates","volume":"19","author":"Wu","year":"2023","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105001","DOI":"10.1016\/j.cor.2020.105001","article-title":"Optimal insertion of customers with waiting time targets","volume":"122","author":"Wen","year":"2020","journal-title":"Comput. Oper. Res."},{"key":"ref_6","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. Prob."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1108\/IJQRM-06-2019-0181","article-title":"Multi-objective optimization of Gas Station performance using response surface methodology","volume":"38","author":"Asadzadeh","year":"2021","journal-title":"Int. J. Qual. Reliab. Manag."},{"key":"ref_8","unstructured":"Whittle, P. (1982). Optimization over Time, Wiley."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1109\/9.83548","article-title":"Controlled wear process: Modeling optimal control","volume":"36","author":"Rishel","year":"1991","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_10","unstructured":"Lefebvre, M. (2023, January 19\u201321). An optimal control problem for a modified M\/G\/k queueing system. Proceedings of the Workshop on Intelligent Information Systems, Chi\u015fin\u01ceu, Republic of Moldova."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"342","DOI":"10.37394\/23203.2023.18.35","article-title":"Reducing the size of a waiting line optimally","volume":"18","author":"Lefebvre","year":"2023","journal-title":"WSEAS Trans. Syst. Control"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lefebvre, M., and Yaghoubi, R. (2024). Optimal control of a queueing system, Submitted for publication.","DOI":"10.1080\/02331934.2024.2422040"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Lefebvre, M. (2023). A discrete-time homing problem with two optimizers. Games, 14.","DOI":"10.3390\/g14060068"},{"key":"ref_14","unstructured":"Bellman, R. (1957). Dynamic Programming, Princeton University Press."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/3\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:50:03Z","timestamp":1760107803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/3\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,29]]},"references-count":14,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["g15030019"],"URL":"https:\/\/doi.org\/10.3390\/g15030019","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2024,5,29]]}}}