{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T09:13:05Z","timestamp":1744189985201,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T00:00:00Z","timestamp":1661817600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T00:00:00Z","timestamp":1661817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Queueing Syst"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s11134-022-09860-w","type":"journal-article","created":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T03:27:16Z","timestamp":1661830036000},"page":"227-267","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process"],"prefix":"10.1007","volume":"102","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7879-6219","authenticated-orcid":false,"given":"Toshihisa","family":"Ozawa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,30]]},"reference":[{"key":"9860_CR1","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198527688.001.0001","volume-title":"Numerical Solution of Structured Markov Chains","author":"DA Bini","year":"2005","unstructured":"Bini, D.A., Latouche, G., Meini, B.: Numerical Solution of Structured Markov Chains. Oxford University Press, Oxford (2005)"},{"key":"9860_CR2","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1070\/RM2001v056n05ABEH000398","volume":"56","author":"AA Borovkov","year":"2001","unstructured":"Borovkov, A.A., Mogul\u2019ski\u012d, A.A.: Large deviations for Markov chains in the positive quadrant. Russ. Math. Surv. 56, 803\u2013916 (2001)","journal-title":"Russ. Math. Surv."},{"issue":"1","key":"9860_CR3","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1287\/10-SSY022","volume":"1","author":"JG Dai","year":"2011","unstructured":"Dai, J.G., Miyazawa, M.: Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Syst. 1(1), 146\u2013208 (2011)","journal-title":"Stoch. Syst."},{"key":"9860_CR4","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s11134-012-9339-1","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, 181\u2013217 (2013)","journal-title":"Queueing Syst."},{"issue":"6","key":"9860_CR5","doi-asserted-by":"publisher","first-page":"2991","DOI":"10.1214\/21-AAP1681","volume":"31","author":"PA Ernst","year":"2021","unstructured":"Ernst, P.A., Franceschi, S.: Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary. Ann. Appl. Probab. 31(6), 2991\u20133016 (2021)","journal-title":"Ann. Appl. Probab."},{"key":"9860_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511984020","volume-title":"Topics in the Constructive Theory of Countable Markov Chains","author":"G Fayolle","year":"1995","unstructured":"Fayolle, G., Malyshev, V.A., Menshikov, M.V.: Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge (1995)"},{"key":"9860_CR7","doi-asserted-by":"publisher","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.: Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications. Springer, Berlin (1999)"},{"key":"9860_CR8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655","volume-title":"Analytic Combinatorics","author":"P Flajolet","year":"2009","unstructured":"Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)"},{"issue":"1B","key":"9860_CR9","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1214\/105051604000000666","volume":"15","author":"RD Foley","year":"2005","unstructured":"Foley, R.D., McDonald, D.R.: Large deviations of a modified Jackson network: stability and rough asymptotics. Ann. Appl. Probab. 15(1B), 519\u2013541 (2005)","journal-title":"Ann. Appl. Probab."},{"issue":"1","key":"9860_CR10","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1287\/16-SSY218","volume":"7","author":"S Franceschi","year":"2017","unstructured":"Franceschi, S., Kourkova, I.: Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. Stoch. Syst. 7(1), 32\u201394 (2017)","journal-title":"Stoch. Syst."},{"key":"9860_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"RA Horn","year":"1991","unstructured":"Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"9860_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6200-8","volume-title":"Markov Chain Models - Rarity and Exponentiality","author":"J Keilson","year":"1979","unstructured":"Keilson, J.: Markov Chain Models - Rarity and Exponentiality. Springer, New York (1979)"},{"key":"9860_CR13","doi-asserted-by":"crossref","unstructured":"Kobayashi, M., Miyazawa, M.: Revisit to the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. Matrix-Analytic Methods in Stochastic Models, 145\u2013185 (2013)","DOI":"10.1007\/978-1-4614-4909-6_8"},{"key":"9860_CR14","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719734","volume-title":"Introduction to Matrix Analytic Methods in Stochastic Modeling","author":"G Latouche","year":"1999","unstructured":"Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999)"},{"issue":"1","key":"9860_CR15","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1287\/12-SSY082","volume":"3","author":"Z Liang","year":"2013","unstructured":"Liang, Z., Hasenbein, J.J.: Optimal paths in large deviations of symmetric reflected Brownian motion in the octant. Stoch. Syst. 3(1), 187\u2013229 (2013)","journal-title":"Stoch. Syst."},{"issue":"1","key":"9860_CR16","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/BF00967270","volume":"14","author":"VA Malyshev","year":"1973","unstructured":"Malyshev, V.A.: Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks. Sib. Math. J. 14(1), 109\u2013118 (1973)","journal-title":"Sib. Math. J."},{"key":"9860_CR17","volume-title":"Theory of Functions of a Complex Variable","author":"AI Markushevich","year":"2005","unstructured":"Markushevich, A.I.: Theory of Functions of a Complex Variable. AMS Chelsea Publishing, Providence (2005)"},{"issue":"3","key":"9860_CR18","doi-asserted-by":"publisher","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":"9860_CR19","doi-asserted-by":"publisher","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"},{"key":"9860_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11134-015-9454-x","volume":"81","author":"M Miyazawa","year":"2015","unstructured":"Miyazawa, M.: Superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov modulated two dimensional reflecting process. Queueing Syst. 81, 1\u201348 (2015)","journal-title":"Queueing Syst."},{"key":"9860_CR21","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1080\/15326349.2020.1832525","volume":"37","author":"M Miyazawa","year":"2021","unstructured":"Miyazawa, M.: Markov modulated fluid network process: tail asymptotics of the stationary distribution. Stoch. Model. 37, 127\u2013167 (2021)","journal-title":"Stoch. Model."},{"key":"9860_CR22","volume-title":"Matrix-Geometric Solutions in Stochastic Models","author":"MF Neuts","year":"1994","unstructured":"Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. Dover Publications, New York (1994)"},{"key":"9860_CR23","volume-title":"Structured Stochastic Matrices of M\/G\/1 Type and their Applications","author":"MF Neuts","year":"1989","unstructured":"Neuts, M.F.: Structured Stochastic Matrices of M\/G\/1 Type and their Applications. Marcel Dekker, New York (1989)"},{"issue":"2","key":"9860_CR24","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1214\/aop\/1176992159","volume":"15","author":"P Ney","year":"1987","unstructured":"Ney, P., Nummelin, E.: Markov additive processes I. Eigenvalue properties and limit theorems. Ann. Probab. 15(2), 561\u2013592 (1987)","journal-title":"Ann. Probab."},{"key":"9860_CR25","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511526237","volume-title":"General Irreducible Markov Chains and Non-negative Operators","author":"E Nummelin","year":"1984","unstructured":"Nummelin, E.: General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, Cambridge (1984)"},{"key":"9860_CR26","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s11134-012-9323-9","volume":"74","author":"T Ozawa","year":"2013","unstructured":"Ozawa, T.: Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process. Queueing Syst. 74, 109\u2013149 (2013)","journal-title":"Queueing Syst."},{"key":"9860_CR27","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s11134-018-9586-x","volume":"90","author":"T Ozawa","year":"2018","unstructured":"Ozawa, T., Kobayashi, M.: Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process. Queueing Syst. 90, 351\u2013403 (2018)","journal-title":"Queueing Syst."},{"key":"9860_CR28","unstructured":"Ozawa, T., Kobayashi, M.: Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs, arXiv:1805.04802 (2018)"},{"key":"9860_CR29","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.peva.2018.11.004","volume":"130","author":"T Ozawa","year":"2019","unstructured":"Ozawa, T.: Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models. Perform. Eval. 130, 101\u2013118 (2019)","journal-title":"Perform. Eval."},{"key":"9860_CR30","doi-asserted-by":"crossref","unstructured":"Ozawa, T.: Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk (2020). arXiv:2001.00700","DOI":"10.1007\/s11134-020-09673-9"},{"key":"9860_CR31","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s11134-020-09673-9","volume":"97","author":"T Ozawa","year":"2021","unstructured":"Ozawa, T.: Asymptotic properties of the occupation measure in a multidimensional skip-free Markov modulated random walk. Queueing Syst. 97, 125\u2013161 (2021)","journal-title":"Queueing Syst."}],"container-title":["Queueing Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-022-09860-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11134-022-09860-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-022-09860-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,5]],"date-time":"2022-10-05T10:13:54Z","timestamp":1664964834000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11134-022-09860-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,30]]},"references-count":31,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["9860"],"URL":"https:\/\/doi.org\/10.1007\/s11134-022-09860-w","relation":{},"ISSN":["0257-0130","1572-9443"],"issn-type":[{"type":"print","value":"0257-0130"},{"type":"electronic","value":"1572-9443"}],"subject":[],"published":{"date-parts":[[2022,8,30]]},"assertion":[{"value":"21 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 August 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 August 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}