{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T03:09:15Z","timestamp":1787713755550,"version":"build-2784847793"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2016,5,28]],"date-time":"2016-05-28T00:00:00Z","timestamp":1464393600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100002860","name":"China Sponsorship Council (CN)","doi-asserted-by":"publisher","award":["2008613008"],"award-info":[{"award-number":["2008613008"]}],"id":[{"id":"10.13039\/501100002860","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NL)","doi-asserted-by":"publisher","award":["612.001.107"],"award-info":[{"award-number":["612.001.107"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Funds for the Central Universities (CN)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Queueing Syst"],"published-print":{"date-parts":[[2016,10]]},"DOI":"10.1007\/s11134-016-9483-0","type":"journal-article","created":{"date-parts":[[2016,5,27]],"date-time":"2016-05-27T22:36:34Z","timestamp":1464388594000},"page":"21-48","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms"],"prefix":"10.1007","volume":"84","author":[{"given":"Yanting","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard J.","family":"Boucherie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jasper","family":"Goseling","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,5,28]]},"reference":[{"issue":"2\u20133","key":"9483_CR1","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s11134-012-9324-8","volume":"74","author":"IJBF Adan","year":"2013","unstructured":"Adan, I.J.B.F., Kapodistria, S., van Leeuwaarden, J.S.H.: Erlang arrivals joining the shorter queue. Queueing Syst. 74(2\u20133), 273\u2013302 (2013)","journal-title":"Queueing Syst."},{"issue":"4","key":"9483_CR2","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1017\/S0001867800025751","volume":"25","author":"IJBF Adan","year":"1993","unstructured":"Adan, I.J.B.F., Wessels, J., Zijm, W.H.M.: A compensation approach for two-dimensional Markov processes. Adv. Appl. Probab. 25(4), 783\u2013817 (1993)","journal-title":"Adv. Appl. Probab."},{"issue":"1","key":"9483_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/15326348508807001","volume":"1","author":"JPC Blanc","year":"1985","unstructured":"Blanc, J.P.C.: The relaxation time of two queueing systems in series. Stoch. Models 1(1), 1\u201316 (1985)","journal-title":"Stoch. Models"},{"key":"9483_CR4","unstructured":"Chen, Y., Boucherie, R.J. and Goseling, J.: Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms. arXiv preprint arXiv:1304.3316 , (2013)"},{"issue":"02","key":"9483_CR5","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1017\/S026996481400031X","volume":"29","author":"Y Chen","year":"2015","unstructured":"Chen, Y., Boucherie, R.J., Goseling, J.: The invariant measure of random walks in the quarter-plane: representation in geometric terms. Probab. Eng. Informational Sci. 29(02), 233\u2013251 (2015)","journal-title":"Probab. Eng. Informational Sci."},{"key":"9483_CR6","volume-title":"Boundary Value Problems in Queueing System Analysis","author":"JW Cohen","year":"1983","unstructured":"Cohen, J.W., Boxma, O.J.: Boundary Value Problems in Queueing System Analysis. North Holland, Amsterdam (1983)"},{"issue":"2\u20133","key":"9483_CR7","first-page":"181","volume":"74","author":"JG Dai","year":"2013","unstructured":"Dai, J.G., Miyazawa, M.: Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Syst. 74(2\u20133), 181\u2013217 (2013)","journal-title":"Queueing Syst."},{"issue":"4","key":"9483_CR8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s11134-014-9411-0","volume":"78","author":"JG Dai","year":"2014","unstructured":"Dai, J.G., Miyazawa, M., Wu, J.: A multi-dimensional SRBM: geometric views of its product form stationary distribution. Queueing Syst. 78(4), 313\u2013335 (2014)","journal-title":"Queueing Syst."},{"key":"9483_CR9","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-60001-2","volume-title":"Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications","author":"G Fayolle","year":"1999","unstructured":"Fayolle, G., Iasnogorodski, R., Malyshev, V.A.: Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications, vol. 40. Springer, Berlin (1999)"},{"issue":"4","key":"9483_CR10","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1023\/A:1016235415066","volume":"41","author":"A G\u00f3mez-Corral","year":"2002","unstructured":"G\u00f3mez-Corral, A.: A tandem queue with blocking and markovian arrival process. Queueing Syst. 41(4), 343\u2013370 (2002)","journal-title":"Queueing Syst."},{"key":"9483_CR11","unstructured":"Goseling, J., Boucherie, R.J. and van Ommeren, J.C.W.: A linear programming approach to error bounds for random walks in the quarter-plane. arXiv preprint arXiv:1409.3736 , (2014)"},{"issue":"1\u20133","key":"9483_CR12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1023\/A:1019139405059","volume":"36","author":"WK Grassmann","year":"2000","unstructured":"Grassmann, W.K., Drekic, S.: An analytical solution for a tandem queue with blocking. Queueing Syst. 36(1\u20133), 221\u2013235 (2000)","journal-title":"Queueing Syst."},{"issue":"4","key":"9483_CR13","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF01193323","volume":"10","author":"N Igaki","year":"1992","unstructured":"Igaki, N.: Exponential two server queue with $$N$$ N -policy and general vacations. Queueing Syst. 10(4), 279\u2013294 (1992)","journal-title":"Queueing Syst."},{"issue":"3\u20134","key":"9483_CR14","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/A:1019181124314","volume":"26","author":"SMR Iravani","year":"1997","unstructured":"Iravani, S.M.R., Posner, M.J.M., Buzacott, J.A.: A two-stage tandem queue attended by a moving server with holding and switching costs. Queueing Syst. 26(3\u20134), 203\u2013228 (1997)","journal-title":"Queueing Syst."},{"issue":"1","key":"9483_CR15","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.peva.2004.09.001","volume":"61","author":"V Klimenok","year":"2005","unstructured":"Klimenok, V., Breuer, L., Tsarenkov, G., Dudin, A.: The tandem queue with losses. Perform. Eval. 61(1), 17\u201340 (2005)","journal-title":"Perform. Eval."},{"key":"9483_CR16","doi-asserted-by":"crossref","unstructured":"Kobayashi, M. and Miyazawa, M.: Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In: Matrix-Analytic Methods in Stochastic Models, pp. 145\u2013185. Springer, (2013)","DOI":"10.1007\/978-1-4614-4909-6_8"},{"issue":"4","key":"9483_CR17","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/s11134-013-9381-7","volume":"77","author":"G Latouche","year":"2014","unstructured":"Latouche, G., Miyazawa, M.: Product-form characterization for a two-dimensional reflecting random walk. Queueing Syst. 77(4), 373\u2013391 (2014)","journal-title":"Queueing Syst."},{"issue":"3","key":"9483_CR18","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1080\/15326340701471042","volume":"23","author":"H Li","year":"2007","unstructured":"Li, H., Miyazawa, M., Zhao, Y.Q.: Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model. Stoch. Models 23(3), 413\u2013438 (2007)","journal-title":"Stoch. Models"},{"issue":"1\u20132","key":"9483_CR19","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1023\/B:QUES.0000021148.33178.0f","volume":"46","author":"M Miyazawa","year":"2004","unstructured":"Miyazawa, M.: A markov renewal approach to M\/G\/1 type queues with countably many background states. Queueing Syst. 46(1\u20132), 177\u2013196 (2004)","journal-title":"Queueing Syst."},{"issue":"3","key":"9483_CR20","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1287\/moor.1090.0375","volume":"34","author":"M Miyazawa","year":"2009","unstructured":"Miyazawa, M.: Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34(3), 547\u2013575 (2009)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"9483_CR21","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s11750-011-0179-7","volume":"19","author":"M Miyazawa","year":"2011","unstructured":"Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. Top 19(2), 233\u2013299 (2011)","journal-title":"Top"},{"issue":"4","key":"9483_CR22","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1017\/S0001867800013380","volume":"36","author":"M Miyazawa","year":"2004","unstructured":"Miyazawa, M., Zhao, Y.Q.: The stationary tail asymptotics in the GI\/G\/1-type queue with countably many background states. Adv. Appl. Probab. 36(4), 1231\u20131251 (2004)","journal-title":"Adv. Appl. Probab."},{"issue":"3\u20134","key":"9483_CR23","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/BF01154549","volume":"7","author":"JA Morrison","year":"1990","unstructured":"Morrison, J.A.: Two-server queue with one server idle below a threshold. Queueing Syst. 7(3\u20134), 325\u2013336 (1990)","journal-title":"Queueing Syst."},{"key":"9483_CR24","volume-title":"Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach","author":"MF Neuts","year":"1981","unstructured":"Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publications, New York (1981)"},{"issue":"2","key":"9483_CR25","first-page":"189","volume":"12","author":"T Phung-Duc","year":"2012","unstructured":"Phung-Duc, T.: An explicit solution for a tandem queue with retrials and losses. Oper. Res. 12(2), 189\u2013207 (2012)","journal-title":"Oper. Res."},{"issue":"5","key":"9483_CR26","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/S0167-6377(03)00046-4","volume":"31","author":"J Resing","year":"2003","unstructured":"Resing, J., \u00d6rmeci, L.: A tandem queueing model with coupled processors. Oper. Res. Lett. 31(5), 383\u2013389 (2003)","journal-title":"Oper. Res. Lett."},{"key":"9483_CR27","volume-title":"Stochastic Processes","author":"SM Ross","year":"1996","unstructured":"Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, New York (1996)","edition":"2"},{"issue":"1","key":"9483_CR28","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/BF02031592","volume":"49","author":"I Stavrakakis","year":"1994","unstructured":"Stavrakakis, I.: A two-server queueing system with periodic and credit-based server availabilities. Ann. Oper. Res. 49(1), 79\u2013100 (1994)","journal-title":"Ann. Oper. Res."},{"key":"9483_CR29","doi-asserted-by":"crossref","unstructured":"van Dijk, N.M.: Error bounds and comparison results: the Markov reward approach for queueing networks. In: Boucherie R.J., Van\u00a0Dijk N.M., (eds.) Queueing Networks: A Fundamental Approach, vol. 154 of International Series in Operations Research & Management Science, Springer (2011)","DOI":"10.1007\/978-1-4419-6472-4_9"},{"issue":"1","key":"9483_CR30","doi-asserted-by":"crossref","first-page":"79","DOI":"10.2307\/1427271","volume":"20","author":"NM Dijk van","year":"1988","unstructured":"van Dijk, N.M., Puterman, M.L.: Perturbation theory for Markov reward processes with applications to queueing systems. Adv. Appl. Probab. 20(1), 79\u201398 (1988)","journal-title":"Adv. Appl. Probab."},{"issue":"2\u20133","key":"9483_CR31","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1081\/STM-200056037","volume":"21","author":"ND Foreest van","year":"2005","unstructured":"van Foreest, N.D., van Ommeren, J.C.W., Mandjes, M.R.H., Scheinhardt, W.R.W.: A tandem queue with server slow-down and blocking. Stoch. Models 21(2\u20133), 695\u2013724 (2005)","journal-title":"Stoch. Models"},{"issue":"1\u20132","key":"9483_CR32","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s11134-005-1683-y","volume":"51","author":"JSH Leeuwaarden Van","year":"2005","unstructured":"Van Leeuwaarden, J.S.H., Resing, J.A.C.: A tandem queue with coupled processors: computational issues. Queueing Syst. 51(1\u20132), 29\u201352 (2005)","journal-title":"Queueing Syst."},{"key":"9483_CR33","volume-title":"Stochastic Modeling and the Theory of Queues","author":"RW Wolff","year":"1989","unstructured":"Wolff, R.W.: Stochastic Modeling and the Theory of Queues, vol. 14. Prentice Hall, Englewood Cliffs (1989)"}],"container-title":["Queueing Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-016-9483-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11134-016-9483-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-016-9483-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-016-9483-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T03:44:45Z","timestamp":1559360685000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11134-016-9483-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,28]]},"references-count":33,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2016,10]]}},"alternative-id":["9483"],"URL":"https:\/\/doi.org\/10.1007\/s11134-016-9483-0","relation":{},"ISSN":["0257-0130","1572-9443"],"issn-type":[{"value":"0257-0130","type":"print"},{"value":"1572-9443","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,5,28]]}}}