×

zbMATH — the first resource for mathematics

Passage times in fluid models with application to risk processes. (English) Zbl 1110.60067
The author derives a variety of passage time distributions in the canonical Markov modulated fluid flow model in terms of its busy period distribution and that of its reflection about the time axis. The use of these distributions is illustrated in the context of a general insurance risk model with Markovian arrival of claims and phase type distributed claim sizes. The paper leans heavily on results previously derived by S. Ahn and the author [Stoch. Models 20, No. 1, 71–101 (2004; Zbl 1038.60086); J. Appl. Probab. 42, No. 2, 531–549 (2005; Zbl 1085.60065); Stoch. Models 22, No. 1, 129–147 (2006)].

MSC:
60J25 Continuous-time Markov processes on general state spaces
60K15 Markov renewal processes, semi-Markov processes
60K25 Queueing theory (aspects of probability theory)
60K37 Processes in random environments
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Ahn, and V. Ramaswami, ”Fluid flow models & queues–A connection by stochastic coupling,” Stochastic Models vol. 19(3) pp. 325–348, 2003. · Zbl 1021.60073 · doi:10.1081/STM-120023564
[2] S. Ahn, and V. Ramaswami, ”Transient analysis of fluid flow models via stochastic coupling to a queue,” Stochastic Models vol. 20(1) pp. 71–101, 2004. · Zbl 1038.60086 · doi:10.1081/STM-120028392
[3] S. Ahn, and V. Ramaswami, ”Efficient algorithms for transient analysis of stochastic fluid flow models,” Journal of Applied Probability vol. 42(2) pp. 531–549, 2005. · Zbl 1085.60065 · doi:10.1239/jap/1118777186
[4] S. Ahn, and V. Ramaswami, ”Transient analysis of fluid models via elementary level crossing arguments,” Stochastic Models vol. 22(1) pp. 129–147, 2006. · Zbl 1350.60095 · doi:10.1080/15326340500481788
[5] S. Ahn, J. Jeon, and V. Ramaswami, ”Steady state analysis of finite fluid flow models using finite QBDs,” QUESTA vol. 49 pp. 223–259, 2005a. · Zbl 1080.90023
[6] S. Ahn, A. L. Badescu, and V. Ramaswami, ”Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier,” 2005b (in review). · Zbl 1124.60067
[7] S. Asmussen, ”Busy period analysis, rare events and transient behavior in fluid flow models,” Journal of Applied Mathematics and Stochastic Analysis vol. 7(3) pp. 269–299, 1994. · Zbl 0826.60086 · doi:10.1155/S1048953394000262
[8] S. Asmussen, Applied Probability & Queues, 2nd Edition, Wiley: New York, 2004. · Zbl 1029.60001
[9] S. Asmussen, and G. Koole, ”Marked point processes as limits of Markovian arrival streams,” Journal of Applied Probability vol. 30 pp. 365–372, 1993. · Zbl 0778.60035 · doi:10.2307/3214845
[10] A. Badescu, L. Breuer, A. Soares, G. Latouche, M. A. Remich, and D. Stanford, ”Risk processes analyzed as fluid queues,” Scandinavian Actuarial Journal vol. 2005(2) pp. 103–126, 2005. · Zbl 1092.91036 · doi:10.1080/03461230510006946
[11] A. Graham, Kronecker Products and Matrix Calculus with Applications, Wiley: New York, 1981. · Zbl 0497.26005
[12] G. Latouche, and V. Ramaswami, ”A logarithmic reduction algorithm for Quasi-Birth-and-Death processes,” Journal of Applied Probability vol. 30 pp. 650–674, 1993. · Zbl 0789.60055 · doi:10.2307/3214773
[13] G. Latouche, and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM & ASA: Philadelphia, 1999. · Zbl 0922.60001
[14] M. F. Neuts, ”A versatile Markovian point process,” Journal of Applied Probability vol. 16 pp. 764–779, 1979. · Zbl 0422.60043 · doi:10.2307/3213143
[15] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models–An Algorithmic Approach, The Johns Hopkins University Press: Baltimore, Maryland, 1981. · Zbl 0469.60002
[16] M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker: New York, 1989.
[17] V. Ramaswami, ”Matrix analytic methods for stochastic fluid flows.” In D. Smith, and P. Key (eds.), Teletraffic Engineering in a Competitive World–Proceedings of the 16th International Teletraffic Congress, pp. 1019–1030, Elsevier: New York, 1999.
[18] W. Scheinhardt, ”Markov-modulated and feedback fluid queues,” Thesis, University of Twente, Enscheide, The Netherlands, 1998.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.