×

zbMATH — the first resource for mathematics

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. (English) Zbl 1124.60067
The main contributions of the paper are the following. First, the authors bring together the formulae for the needed kernels and first passage time distributions for fluid models from several of their earlier papers in one place. A particular feature of their approach is the clear demonstration and organization of computations in a way that once the Laplace transform matrix characterizing the busy period is determined, all the other formulae they need flow from it through solutions only of certain linear systems of equations. Then they give a complete set of transform formulae for the time dependent analysis of fluid models with a boundary at a positive level in addition to the one at zero. The authors demonstrate that their transforms can be inverted accurately. The paper contains a set of numerical examples.

MSC:
60K25 Queueing theory (aspects of probability theory)
60J25 Continuous-time Markov processes on general state spaces
60K15 Markov renewal processes, semi-Markov processes
60K37 Processes in random environments
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abate J, Whitt W. The Fourier-series method for inverting transforms of probability distributions. Queueing Syst 1992;10:5–88. · Zbl 0749.60013 · doi:10.1007/BF01158520
[2] Abate J, Whitt W. Numerical inversion of Laplace transforms of probability distributions. ORSA J Comput 1995;7(1):36–43. · Zbl 0821.65085
[3] Ahn S, Ramaswami V. Fluid flow models and queues–A connection by stochastic coupling. Stoch Models 2003;19(3):325–48. · Zbl 1021.60073 · doi:10.1081/STM-120023564
[4] Ahn S, Ramaswami V. Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch Models 2004;20(1):71–101. · Zbl 1038.60086 · doi:10.1081/STM-120028392
[5] Ahn S, Ramaswami V. Efficient algorithms for transient analysis of stochastic fluid flow models. J Appl Probab 2005;42(2):531–49. · Zbl 1085.60065 · doi:10.1239/jap/1118777186
[6] Ahn S, Ramaswami V. Transient analysis of fluid flow models via elementary level crossing arguments. Stoch Models 2006;22(1):129–47. · Zbl 1350.60095 · doi:10.1080/15326340500481788
[7] Ahn S, Jeon J, Ramaswami V. Steady state analysis of finite fluid flow models using finite QBDs. QUESTA 2005;49:223–59. · Zbl 1080.90023
[8] Akar N, Sohraby K. Infinite- and finite-buffer Markov fluid queues: a unified analysis. J Appl Probab 2004;41:557–69. · Zbl 1046.60078 · doi:10.1239/jap/1082999086
[9] Albrecher H, Boxma O. On the discounted penalty function in a Markov-dependent risk model. Insur Math Econ 2005;37(3):650–72. · Zbl 1129.91023 · doi:10.1016/j.insmatheco.2005.06.007
[10] Albrecher H, Kainhofer R. Risk theory with a nonlinear dividend barrier. Computing 2002;68:289–311. · Zbl 1076.91521 · doi:10.1007/s00607-001-1447-4
[11] Anick D, Mitra D, Sondhi MM. Stochastic theory of data handling system with multiple sources. Bell Syst Tech J 1982;61:1871–94.
[12] Asmussen S. Stationary distributions via first passage times. In: Dshalalow JH, editor. Advances in queueing: theory, methods, and open problems. Boca Raton: CRC Press; 1995. p. 79–102. · Zbl 0866.60077
[13] Asmussen S. Stationary distributions for fluid flow models with or without Brownian noise. Stoch Models 1995;11:1–20. · Zbl 0817.60086 · doi:10.1080/15326349508807330
[14] Asmussen S. Matrix-analytic models and their analysis. Scand J Stat 2000;(2):193–226. · Zbl 0959.60085 · doi:10.1111/1467-9469.00186
[15] Asmussen S, Avram F, Usabel M. Erlangian approximations for finite time ruin probabilities. ASTIN Bull 2002;32:267–81. · Zbl 1081.60028 · doi:10.2143/AST.32.2.1029
[16] Badescu A, Breuer L, Da Silva Soares A, Latouche G, Remiche MA, Stanford D. Risk processes analyzed as fluid queues. Scand Actuar J 2005;2:127–41. · Zbl 1092.91037 · doi:10.1080/03461230410000565
[17] Badescu A, Breuer L, Drekic S, Latouche G, Stanford D. The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scand Actuar J 2005;6:433–45. · Zbl 1143.91025 · doi:10.1080/03461230510009835
[18] Barlow MT, Rogers LCG, Williams D. Wiener-Hopf factorization for matrices. In: Seminaire de probabilites XIV. Lecture notes in math. vol. 784. Berlin: Springer; 1980. p. 324–31.
[19] Bean NG, O’Reilly M, Taylor PG. Hitting probabilities and hitting times for stochastic fluid flows. Stoch Proc Appl 2005;115(9):1530–56. · Zbl 1074.60078 · doi:10.1016/j.spa.2005.04.002
[20] Bühlmann H. Mathematical methods in risk theory. New York: Springer; 1970. · Zbl 0209.23302
[21] da Silva Soares A. Fluid queues–building upon the analogy with QBD processes. Doctoral Dissertation, Universite Libre de Bruxelles, Belgium; 2005.
[22] da Silva Soares A, Latouche G. Matrix-analytic methods for fluid queues with finite buffers. Perform Eval 2006;63:295–314. · doi:10.1016/j.peva.2005.02.002
[23] De Finetti B. Su un’impostazione alternativa dell teoria collectiva del rischio. Trans XV Int Congr Actuar 1957;2:433–43.
[24] Gerber HU. An introduction to mathematical risk theory. S.S. Huebner Foundation. Philadelphia: University of Pennsylvania; 1979.
[25] Gerber HU. On the probability of ruin in the presence of a linear dividend barrier. Scand Actuar J 1981;105–15. · Zbl 0455.62086
[26] Gerber HU, Shiu ESW. On the time value of ruin. North Am Actuar J 1998;2(1):48–78. · Zbl 1081.60550
[27] Gerber HU, Shiu ESW. The time value of ruin in a Sparre Andersen model. North Am Actuar J 2005;9(2):49–84. · Zbl 1085.62508
[28] Høojgard, B. Optimal dynamic premium control in non-life insurance: maximizing the dividend payouts. Scand Actuar J 2002;4:225–45. · Zbl 1039.91042 · doi:10.1080/03461230110106291
[29] Kobayashi H, Ren Q. A mathematical theory for transient analysis of communication networks. IEICE Trans Commun 1992;12:1266–76.
[30] Latouche G, Ramaswami V. Introduction to matrix analytic methods in stochastic modeling. Philadelphia: SIAM and ASA; 1999. · Zbl 0922.60001
[31] Li S, Garrido J. On a class of renewal risk models with a constant dividend barrier. Insur Math Econ 2004;35:691–701. · Zbl 1122.91345 · doi:10.1016/j.insmatheco.2004.08.004
[32] Li S, Garrido J. On a general class of renewal risk process: analysis of the Gerber-Shiu function. Adv Appl Probab 2005;37(3):836–56. · Zbl 1077.60063 · doi:10.1239/aap/1127483750
[33] Lin XS, Willmot GE, Drekic S. The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insur Math Econ 2003;33:551–66. · Zbl 1103.91369 · doi:10.1016/j.insmatheco.2003.08.004
[34] Neuts MF. Matrix-geometric solutions in stochastic models–an algorithmic approach. Baltimore: The Johns Hopkins University Press; 1981. · Zbl 0469.60002
[35] Paulsen J, Gjessing H. Optimal choice of dividend barriers for a risk process with stochastic returns on investments. Insur Math Econ 1997;20:19–44. · Zbl 0894.90048 · doi:10.1016/S0167-6687(97)00011-5
[36] Ramaswami V. Matrix analytic methods for stochastic fluid flows. In: Smith D, Key P, editors. Teletraffic engineering in a competitive world–proc. of the 16th international teletraffic congress. New York: Elsevier; 1999. p. 1019–30.
[37] Ramaswami V. Passage times in fluid models with application to risk processes. Methodol Comput Appl Prob 2006;8:497–515. · Zbl 1110.60067 · doi:10.1007/s11009-006-0426-9
[38] Ramaswami V, Poole D, Ahn S, Byers S, Kaplan AE. Ensuring access to emergency services in the presence of long Internet dial up calls. Interfaces 2005;35(5):411–22. · doi:10.1287/inte.1050.0155
[39] Remiche MA. Compliance of the token-bucket model with Markovian traffic. Stoch Models 2005;21:615–30. · Zbl 1082.60088 · doi:10.1081/STM-200057884
[40] Rogers LCG. Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann Appl Probab 1994;4(2):390–413. · Zbl 0806.60052 · doi:10.1214/aoap/1177005065
[41] Scheinhardt W. Markov-modulated and feedback fluid queues. Thesis, University of Twente, Enscheide, The Netherlands; 1998.
[42] Segerdahl C. On some distributions in time connected with the collective theory of risk. Scand Actuar J 1970;167–92. · Zbl 0229.60063
[43] Sericola B. Transient analysis of stochastic fluid models. Perform Eval 1998;32:245–63. · doi:10.1016/S0166-5316(98)00004-2
[44] Thorin O, Wikstad N. Numerical evaluation of ruin probabilities for a finite period. Astin Bull 1973;7:137–53.
[45] Tijms HC. Stochastic modelling and analysis: a computational approach. New York: Wiley; 1986.
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.