{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T13:39:06Z","timestamp":1771335546661,"version":"3.50.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2021,3,8]],"date-time":"2021-03-08T00:00:00Z","timestamp":1615161600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,8]],"date-time":"2021-03-08T00:00:00Z","timestamp":1615161600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100011958","name":"Danmarks Frie Forskningsfond","doi-asserted-by":"publisher","award":["8049-00021B"],"award-info":[{"award-number":["8049-00021B"]}],"id":[{"id":"10.13039\/501100011958","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Queueing Syst"],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s11134-021-09696-w","type":"journal-article","created":{"date-parts":[[2021,3,8]],"date-time":"2021-03-08T02:02:17Z","timestamp":1615168937000},"page":"3-19","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On scale functions for L\u00e9vy processes with negative phase-type jumps"],"prefix":"10.1007","volume":"98","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3664-8365","authenticated-orcid":false,"given":"Jevgenijs","family":"Ivanovs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,8]]},"reference":[{"issue":"1","key":"9696_CR1","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1080\/15326349508807330","volume":"11","author":"S Asmussen","year":"1995","unstructured":"Asmussen, S.: Stationary distributions for fluid flow models with or without Brownian noise. Commun. Stat. Stochastic Models 11(1), 21\u201349 (1995). https:\/\/doi.org\/10.1080\/15326349508807330","journal-title":"Commun. Stat. Stochastic Models"},{"key":"9696_CR2","series-title":"Stochastic Modelling and Applied Probability","volume-title":"Applied Probability and Queues. Applications of Mathematics","author":"S Asmussen","year":"2003","unstructured":"Asmussen, S.: Applied Probability and Queues. Applications of Mathematics. Stochastic Modelling and Applied Probability, vol. 51, 2nd edn. Springer, New York (2003)","edition":"2"},{"issue":"4","key":"9696_CR3","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1080\/15326349.2014.958424","volume":"30","author":"S Asmussen","year":"2014","unstructured":"Asmussen, S.: L\u00e9vy processes, phase-type distributions, and martingales. Stoch. Models 30(4), 443\u2013468 (2014). https:\/\/doi.org\/10.1080\/15326349.2014.958424","journal-title":"Stoch. Models"},{"key":"9696_CR4","series-title":"Advanced Series on Statistical Science and Applied Probability","doi-asserted-by":"publisher","DOI":"10.1142\/9789814282536","volume-title":"Ruin Probabilities","author":"S Asmussen","year":"2010","unstructured":"Asmussen, S., Albrecher, H.: Ruin Probabilities. Advanced Series on Statistical Science and Applied Probability, vol. 14, 2nd edn. World Scientific Publishing Co. Pte. Ltd., Hackensack (2010). https:\/\/doi.org\/10.1142\/9789814282536","edition":"2"},{"issue":"1","key":"9696_CR5","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.spa.2003.07.005","volume":"109","author":"S Asmussen","year":"2004","unstructured":"Asmussen, S., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type L\u00e9vy models. Stoch. Process. Appl. 109(1), 79\u2013111 (2004). https:\/\/doi.org\/10.1016\/j.spa.2003.07.005","journal-title":"Stoch. Process. Appl."},{"issue":"1","key":"9696_CR6","doi-asserted-by":"publisher","first-page":"17","DOI":"10.3390\/risks7010017","volume":"7","author":"S Asmussen","year":"2019","unstructured":"Asmussen, S., Laub, P.J., Yang, H.: Phase-type models in life insurance: fitting and valuation of equity-linked benefits. Risks 7(1), 17 (2019)","journal-title":"Risks"},{"key":"9696_CR7","doi-asserted-by":"publisher","first-page":"454","DOI":"10.1051\/ps\/2019022","volume":"24","author":"F Avram","year":"2020","unstructured":"Avram, F., Grahovac, D., Vardar-Acar, C.: The $$W$$, $$Z$$ scale functions kit for first passage problems of spectrally negative L\u00e9vy processes, and applications to control problems. ESAIM Probab. Stat. 24, 454\u2013525 (2020). https:\/\/doi.org\/10.1051\/ps\/2019022","journal-title":"ESAIM Probab. Stat."},{"key":"9696_CR8","volume-title":"L\u00e9vy Processes. Cambridge Tracts in Mathematics","author":"J Bertoin","year":"1996","unstructured":"Bertoin, J.: L\u00e9vy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)"},{"issue":"3\u20134","key":"9696_CR9","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF02207694","volume":"13","author":"DA Bini","year":"1996","unstructured":"Bini, D.A.: Numerical computation of polynomial zeros by means of Aberth\u2019s method. Numer. Algorithms 13(3\u20134), 179\u2013200 (1996). https:\/\/doi.org\/10.1007\/BF02207694","journal-title":"Numer. Algorithms"},{"key":"9696_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-7049-0","volume-title":"Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling","author":"M Bladt","year":"2017","unstructured":"Bladt, M., Nielsen, B.F.: Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling, vol. 81. Springer, New York (2017). https:\/\/doi.org\/10.1007\/978-1-4939-7049-0"},{"key":"9696_CR11","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1080\/03461238.2013.865257","volume":"7","author":"M Bladt","year":"2015","unstructured":"Bladt, M., Nielsen, B.F., Samorodnitsky, G.: Calculation of ruin probabilities for a dense class of heavy tailed distributions. Scand. Actuar. J. 7, 573\u2013591 (2015). https:\/\/doi.org\/10.1080\/03461238.2013.865257","journal-title":"Scand. Actuar. J."},{"issue":"4","key":"9696_CR12","doi-asserted-by":"publisher","first-page":"1048","DOI":"10.1239\/jap\/1294170518","volume":"47","author":"B D\u2019Auria","year":"2010","unstructured":"D\u2019Auria, B., Ivanovs, J., Kella, O., Mandjes, M.: First passage of a Markov additive process and generalized Jordan chains. J. Appl. Probab. 47(4), 1048\u20131057 (2010). https:\/\/doi.org\/10.1239\/jap\/1294170518","journal-title":"J. Appl. Probab."},{"key":"9696_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20693-6","volume-title":"Queues and L\u00e9vy Fluctuation Theory","author":"K D\u0229bicki","year":"2015","unstructured":"D\u0229bicki, K., Mandjes, M.: Queues and L\u00e9vy Fluctuation Theory. Springer, Berlin (2015)"},{"key":"9696_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2013.12.044","volume":"264","author":"M Egami","year":"2014","unstructured":"Egami, M., Yamazaki, K.: Phase-type fitting of scale functions for spectrally negative L\u00e9vy processes. J. Comput. Appl. Math. 264, 1\u201322 (2014). https:\/\/doi.org\/10.1016\/j.cam.2013.12.044","journal-title":"J. Comput. Appl. Math."},{"key":"9696_CR15","doi-asserted-by":"publisher","unstructured":"Hubalek, F., Kyprianou, E.: Old and new examples of scale functions for spectrally negative L\u00e9vy processes. In: Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol.\u00a063, pp. 119\u2013145. Birkh\u00e4user\/Springer, Basel (2011). https:\/\/doi.org\/10.1007\/978-3-0348-0021-1_8","DOI":"10.1007\/978-3-0348-0021-1_8"},{"issue":"4","key":"9696_CR16","doi-asserted-by":"publisher","first-page":"1034","DOI":"10.1239\/jap\/1294170517","volume":"47","author":"J Ivanovs","year":"2010","unstructured":"Ivanovs, J.: Markov-modulated Brownian motion with two reflecting barriers. J. Appl. Probab. 47(4), 1034\u20131047 (2010). https:\/\/doi.org\/10.1239\/jap\/1294170517","journal-title":"J. Appl. Probab."},{"key":"9696_CR17","unstructured":"Ivanovs, J.: One-sided Markov additive processes and related exit problems, PhD thesis. BOXPress (2011)"},{"issue":"9","key":"9696_CR18","doi-asserted-by":"publisher","first-page":"3342","DOI":"10.1016\/j.spa.2012.05.016","volume":"122","author":"J Ivanovs","year":"2012","unstructured":"Ivanovs, J., Palmowski, Z.: Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122(9), 3342\u20133360 (2012). https:\/\/doi.org\/10.1016\/j.spa.2012.05.016","journal-title":"Stoch. Process. Appl."},{"issue":"3","key":"9696_CR19","doi-asserted-by":"publisher","first-page":"1101","DOI":"10.1214\/11-AAP787","volume":"22","author":"A Kuznetsov","year":"2012","unstructured":"Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic L\u00e9vy processes and their fluctuation identities. Ann. Appl. Probab. 22(3), 1101\u20131135 (2012). https:\/\/doi.org\/10.1214\/11-AAP787","journal-title":"Ann. Appl. Probab."},{"key":"9696_CR20","doi-asserted-by":"publisher","unstructured":"Kuznetsov, A., Kyprianou, A.E., Rivero, V.: The theory of scale functions for spectrally negative L\u00e9vy processes. In: L\u00e9vy Matters II. Lecture Notes in Mathematics, vol. 2061, pp. 97\u2013186. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-31407-0_2","DOI":"10.1007\/978-3-642-31407-0_2"},{"key":"9696_CR21","volume-title":"Introductory Lectures on Fluctuations of L\u00e9vy Processes with Applications","author":"AE Kyprianou","year":"2006","unstructured":"Kyprianou, A.E.: Introductory Lectures on Fluctuations of L\u00e9vy Processes with Applications. Springer, Berlin (2006)"},{"key":"9696_CR22","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1080\/03461238.2019.1663444","volume":"4","author":"D Landriault","year":"2020","unstructured":"Landriault, D., Willmot, G.E.: On series expansions for scale functions and other ruin-related quantities. Scand. Actuar. J. 4, 292\u2013306 (2020). https:\/\/doi.org\/10.1080\/03461238.2019.1663444","journal-title":"Scand. Actuar. J."},{"key":"9696_CR23","doi-asserted-by":"publisher","unstructured":"Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, American Statistical Association, Alexandria (1999). https:\/\/doi.org\/10.1137\/1.9780898719734","DOI":"10.1137\/1.9780898719734"},{"key":"9696_CR24","unstructured":"Mordecki, E.: The distribution of the maximum of a L\u00e9vy processes with positive jumps of phase-type. In: Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), vol.\u00a08, pp. 309\u2013316 (2002)"},{"key":"9696_CR25","doi-asserted-by":"publisher","unstructured":"Pistorius, M.R.: A potential-theoretical review of some exit problems of spectrally negative L\u00e9vy processes. In: S\u00e9minaire de Probabilit\u00e9s XXXVIII. Lecture Notes in Mathematics, vol. 1857, pp. 30\u201341. Springer, Berlin (2005). https:\/\/doi.org\/10.1007\/978-3-540-31449-3_4","DOI":"10.1007\/978-3-540-31449-3_4"},{"issue":"3","key":"9696_CR26","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/BF00939805","volume":"63","author":"F Potra","year":"1989","unstructured":"Potra, F.: On q-order and r-order of convergence. J. Optim. Theory Appl. 63(3), 415\u2013431 (1989)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Queueing Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-021-09696-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11134-021-09696-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11134-021-09696-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,20]],"date-time":"2021-05-20T08:29:09Z","timestamp":1621499349000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11134-021-09696-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,8]]},"references-count":26,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["9696"],"URL":"https:\/\/doi.org\/10.1007\/s11134-021-09696-w","relation":{},"ISSN":["0257-0130","1572-9443"],"issn-type":[{"value":"0257-0130","type":"print"},{"value":"1572-9443","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,8]]},"assertion":[{"value":"10 February 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 March 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}