{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T01:36:47Z","timestamp":1769391407843,"version":"3.49.0"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T00:00:00Z","timestamp":1611532800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T00:00:00Z","timestamp":1611532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Queueing Syst"],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player\u2019s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) <jats:italic>N<\/jats:italic>-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a <jats:inline-formula><jats:alternatives><jats:tex-math>$$2\\times 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-grid, and (iv) 2-player games on an <jats:inline-formula><jats:alternatives><jats:tex-math>$$A\\times B$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>B<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-grid with small differences in the service rates.\n<\/jats:p>","DOI":"10.1007\/s11134-020-09681-9","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T17:05:31Z","timestamp":1611594331000},"page":"279-301","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Non-cooperative queueing games on a network of single server queues"],"prefix":"10.1007","volume":"97","author":[{"given":"Corine M.","family":"Laan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9299-4775","authenticated-orcid":false,"given":"Judith","family":"Timmer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard J.","family":"Boucherie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,25]]},"reference":[{"issue":"17","key":"9681_CR1","doi-asserted-by":"publisher","first-page":"1552","DOI":"10.1016\/j.tcs.2008.12.035","volume":"410","author":"H Ackermann","year":"2009","unstructured":"Ackermann, H., R\u00f6glin, H., V\u00f6cking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theoret. Comput. Sci. 410(17), 1552\u20131563 (2009)","journal-title":"Theoret. Comput. Sci."},{"issue":"2","key":"9681_CR2","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1016\/j.cor.2004.06.005","volume":"33","author":"E Altman","year":"2006","unstructured":"Altman, E., Boulogne, T., El-Azouzi, R., Jim\u00e9nez, T., Wynter, L.: A survey on networking games in telecommunications. Comput. Oper. Res. 33(2), 286\u2013311 (2006)","journal-title":"Comput. Oper. Res."},{"issue":"4","key":"9681_CR3","doi-asserted-by":"publisher","first-page":"1602","DOI":"10.1137\/070680096","volume":"38","author":"E Anshelevich","year":"2008","unstructured":"Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM J. Comput. 38(4), 1602\u20131623 (2008)","journal-title":"SIAM J. Comput."},{"issue":"7","key":"9681_CR4","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1287\/mnsc.29.7.831","volume":"29","author":"CE Bell","year":"1983","unstructured":"Bell, C.E., Stidham, S.: Individual versus social optimization in the allocation of customers to alternative servers. Manag. Sci. 29(7), 831\u2013839 (1983)","journal-title":"Manag. Sci."},{"issue":"2","key":"9681_CR5","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1007\/s00224-008-9128-8","volume":"45","author":"HL Chen","year":"2009","unstructured":"Chen, H.L., Roughgarden, T.: Network design with weighted players. Theory Comput. Syst. 45(2), 302 (2009)","journal-title":"Theory Comput. Syst."},{"issue":"3","key":"9681_CR6","doi-asserted-by":"publisher","first-page":"730","DOI":"10.2307\/3214558","volume":"27","author":"JE Cohen","year":"1990","unstructured":"Cohen, J.E., Kelly, F.P.: A paradox of congestion in a queuing network. J. Appl. Probab. 27(3), 730\u2013734 (1990)","journal-title":"J. Appl. Probab."},{"issue":"1","key":"9681_CR7","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10479-009-0653-x","volume":"175","author":"F Facchinei","year":"2010","unstructured":"Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175(1), 177\u2013211 (2010)","journal-title":"Ann. Oper. Res."},{"issue":"3","key":"9681_CR8","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1287\/moor.1120.0543","volume":"37","author":"T Harks","year":"2012","unstructured":"Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res. 37(3), 419\u2013436 (2012)","journal-title":"Math. Oper. Res."},{"key":"9681_CR9","doi-asserted-by":"publisher","DOI":"10.1201\/b20014","volume-title":"Rational Queueing","author":"R Hassin","year":"2016","unstructured":"Hassin, R.: Rational Queueing. Chapman and Hall\/CRC, Boca Raton (2016)"},{"key":"9681_CR10","series-title":"International Series in Operations Research & Management Science","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0359-0","volume-title":"To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems","author":"R Hassin","year":"2003","unstructured":"Hassin, R., Haviv, M.: To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. International Series in Operations Research & Management Science. Kluwer Academic Publishers, Amsterdam (2003)"},{"key":"9681_CR11","volume-title":"Reversibility and Stochastic Networks","author":"FP Kelly","year":"1979","unstructured":"Kelly, F.P.: Reversibility and Stochastic Networks. Wiley, Chichester (1979)"},{"key":"9681_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/978-3-642-22012-8_43","volume-title":"Automata, Languages, and Programming. ICALP 2011","author":"K Kollias","year":"2011","unstructured":"Kollias, K., Roughgarden, T.: Restoring pure equilibria to weighted congestion games. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) Automata, Languages, and Programming. ICALP 2011. Lecture Notes in Computer Science, vol. 6756, pp. 539\u2013551. Springer, Berlin (2011)"},{"issue":"3","key":"9681_CR13","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1016\/j.ejor.2017.02.035","volume":"260","author":"CM Laan","year":"2017","unstructured":"Laan, C.M., van der Mijden, T., Barros, A.I., Boucherie, R.J., Monsuur, H.: An interdiction game on a queueing network with multiple intruders. Eur. J. Oper. Res. 260(3), 1069\u20131080 (2017)","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"9681_CR14","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1006\/game.1996.0027","volume":"13","author":"I Milchtaich","year":"1996","unstructured":"Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econ. Behav. 13(1), 111\u2013124 (1996)","journal-title":"Games Econ. Behav."},{"issue":"3","key":"9681_CR15","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/s00182-014-0443-9","volume":"44","author":"I Milchtaich","year":"2015","unstructured":"Milchtaich, I.: Network topology and equilibrium existence in weighted network congestion games. Int. J. Game Theory 44(3), 515\u2013541 (2015)","journal-title":"Int. J. Game Theory"},{"issue":"2","key":"9681_CR16","doi-asserted-by":"publisher","first-page":"286","DOI":"10.2307\/1969529","volume":"54","author":"J Nash","year":"1951","unstructured":"Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286\u2013295 (1951)","journal-title":"Ann. Math."},{"key":"9681_CR17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511800481","volume-title":"Algorithmic Game Theory","author":"N Nisan","year":"2007","unstructured":"Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, New York, (2007)"},{"issue":"6","key":"9681_CR18","doi-asserted-by":"publisher","first-page":"3251","DOI":"10.1109\/TSP.2010.2043138","volume":"58","author":"JS Pang","year":"2010","unstructured":"Pang, J.S., Scutari, G., Palomar, D.P., Facchinei, F.: Design of cognitive radio systems under temperature-interference constraints: a variational inequality approach. IEEE Trans. Signal Process. 58(6), 3251\u20133271 (2010)","journal-title":"IEEE Trans. Signal Process."},{"key":"9681_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-69291-1","volume-title":"Game Theory: A Multi-leveled Approach","author":"H Peters","year":"2008","unstructured":"Peters, H.: Game Theory: A Multi-leveled Approach, 1st edn. Springer, Berlin (2008)","edition":"1"},{"issue":"3","key":"9681_CR20","first-page":"520","volume":"33","author":"JB Rosen","year":"1965","unstructured":"Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econom. J. Econom. Soc. 33(3), 520\u2013534 (1965)","journal-title":"Econom. J. Econom. Soc."},{"issue":"1","key":"9681_CR21","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01737559","volume":"2","author":"RW Rosenthal","year":"1973","unstructured":"Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2(1), 65\u201367 (1973)","journal-title":"Int. J. Game Theory"},{"issue":"3","key":"9681_CR22","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1109\/MSP.2010.936021","volume":"27","author":"G Scutari","year":"2010","unstructured":"Scutari, G., Palomar, D.P., Facchinei, F., Pang, J.S.: Convex optimization, game theory, and variational inequality theory. IEEE Signal Process. Mag. 27(3), 35\u201349 (2010)","journal-title":"IEEE Signal Process. Mag."},{"key":"9681_CR23","doi-asserted-by":"publisher","DOI":"10.1201\/9781420010008","volume-title":"Optimal Design of Queueing Systems","author":"S Stidham Jr","year":"2009","unstructured":"Stidham Jr., S.: Optimal Design of Queueing Systems. Chapman & Hall\/CRC, Boca Raton (2009)"},{"issue":"1","key":"9681_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11134-012-9336-4","volume":"75","author":"J Timmer","year":"2013","unstructured":"Timmer, J., Scheinhardt, W.: Cost sharing of cooperating queues in a Jackson network. Queueing Syst. 75(1), 1\u201317 (2013)","journal-title":"Queueing Syst."},{"issue":"03","key":"9681_CR25","doi-asserted-by":"publisher","first-page":"1850002","DOI":"10.1142\/S0219198918500020","volume":"20","author":"J Timmer","year":"2018","unstructured":"Timmer, J., Scheinhardt, W.: Customer and cost sharing in a Jackson network. Int. Game Theory Rev. 20(03), 1850002 (2018)","journal-title":"Int. Game Theory Rev."},{"key":"9681_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/978-3-642-24829-0_22","volume-title":"Algorithmic Game Theory. SAGT 2011","author":"L Tran-Thanh","year":"2011","unstructured":"Tran-Thanh, L., Polukarov, M., Chapman, A., Rogers, A., Jennings, N.R.: On the existence of pure strategy Nash equilibria in integer-splittable weighted congestion games. In: Persiano, G. (ed.) Algorithmic Game Theory. SAGT 2011. Lecture Notes in Computer Science, vol. 6982, pp. 236\u2013253. Springer, Berlin (2011)"},{"issue":"3","key":"9681_CR27","doi-asserted-by":"publisher","first-page":"654","DOI":"10.1109\/TNS.2007.897829","volume":"54","author":"LM Wein","year":"2007","unstructured":"Wein, L.M., Atkinson, M.P.: The last line of defense: designing radiation detection-interdiction systems to protect cities from a nuclear terrorist attack. IEEE Trans. Nucl. Sci. 54(3), 654\u2013669 (2007)","journal-title":"IEEE Trans. Nucl. Sci."}],"container-title":["Queueing Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-020-09681-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11134-020-09681-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-020-09681-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,31]],"date-time":"2021-03-31T04:18:50Z","timestamp":1617164330000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11134-020-09681-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,25]]},"references-count":27,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["9681"],"URL":"https:\/\/doi.org\/10.1007\/s11134-020-09681-9","relation":{},"ISSN":["0257-0130","1572-9443"],"issn-type":[{"value":"0257-0130","type":"print"},{"value":"1572-9443","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,25]]},"assertion":[{"value":"16 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}