Computational Analysis of the GI/G/1 Risk Process Using Roots

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 225)

Abstract

In this paper, we analyze an insurance risk model wherein the arrival of claims and their sizes occur as renewal processes. Using the duality relation in queueing theory and roots method, we derive closed-form expressions for the ultimate ruin probability, the distribution of the deficit at the time of ruin, and the expected time to ruin in terms of the roots of the characteristic equation. Finally, some numerical computations are portrayed with the help of tables.

Keywords

Risk processes Ruin probability Duality Padé approximation Time to ruin GI/G/1 queue Deficit at the time of ruin 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

The third author’s research work was partially supported by NSERC, Canada.

References

  1. 1.
    Dickson, D.C.M.: On a class of renewal risk processes. N. Am. Actuar. J. 2(3), 60–68 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dong, H., Liu, Z.: A class of Sparre Andersen risk process. Front. Math. China 5(3), 517–530 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dufresne, D.: A general class of risk models. Aust. Actuar. J. 7(4), 755–791 (2011)Google Scholar
  4. 4.
    Gerber, H.U., Shiu, E.S.: The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insur. Math. Econ. 21(2), 129–137 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gerber, H.U., Shiu, E.S.: On the time value of ruin. N. Am. Actuar. J. 2(1), 48–72 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dickson, D., Hipp, C.: Ruin probabilities for Erlang (2) risk processes. Insur. Math. Econ. 22(3), 251–262 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dickson, D.C., Hipp, C.: On the time to ruin for Erlang (2) risk processes. Insur. Math. Econ. 29(3), 333–344 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Landriault, D., Willmot, G.: On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. Insur. Math. Econ. 42(2), 600–608 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gerber, H.U., Shiu, E.S.: The time value of ruin in a Sparre Andersen model. N. Am. Actuar. J. 9(2), 49–69 (2005)Google Scholar
  10. 10.
    Li, S., Garrido, J., et al.: On a general class of renewal risk process: analysis of the Gerber-Shiu function. Adv. Appl. Probab. 37(3), 836–856 (2005)Google Scholar
  11. 11.
    Dickson, D., Drekic, S.: The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. Insur. Math. Econ. 34(1), 97–107 (2004)Google Scholar
  12. 12.
    Rodríguez-Martínez, E.V., Cardoso, R.M., Dos Reis, A.D.E.: Some advances on the erlang (n) dual risk model. Astin Bull. 45(01), 127–150 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Asmussen, S., Rolski, T.: Computational methods in risk theory: a matrix-algorithmic approach. Insur. Math. Econ. 10(4), 259–274 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Avram, F., Usabel, M.: Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times. Astin Bull. 34, 315–332 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Thorin, O.: Probabilities of ruin. Scand. Actuar. J. 1982(2), 65–103 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Panda, G., Banik, A.D., Chaudhry, M.L.: Inverting the transforms arising in the \(GI/M/1\) risk process using roots. In: Mathematics and Computing 2013, pp. 297–312. Springer (2014)Google Scholar
  17. 17.
    Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd edn. World Scientific, Singapore (2010)CrossRefMATHGoogle Scholar
  18. 18.
    Prabhu, N.U.: On the ruin problem of collective risk theory. Ann. Math. Statist. 3, 757–764 (1961)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Thampi, K.K., Jacob, M.J.: On a class of renewal queueing and risk processes. J. Risk Financ 11, 204–220 (2010)CrossRefMATHGoogle Scholar
  20. 20.
    Drekic, S., Dickson, D.C., Stanford, D.A., Willmot, G.E.: On the distribution of the deficit at ruin when claims are phase-type. Scand. Actuar. J. 2004(2), 105–120 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Frostig, E.: Upper bounds on the expected time to ruin and on the expected recovery time. Adv. Appl. Probab. 36, 377–397 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chaudhry, M.L., Agarwal, M., Templeton, J.G.: Exact and approximate numerical solutions of steady-state distributions arising in the queue \(GI/G/1\). Queueing Syst. 10(1–2), 105–152 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kleinrock, L.: Queueing Systems Vol 1: Theory. Wiley-Interscience, New York (1975)Google Scholar
  24. 24.
    Komota, Y., Nogami, S., Hoshiko, Y.: Analysis of the GI/G/1 queue by the supplementary variables approach. Electron. Commun. Jpn (Part I: Commun.) 66(5), 10–19 (1983)Google Scholar
  25. 25.
    Chaudhry, M.L., Yang, X., Ong, B.: Computing the distribution function of the number of renewals. Am. J. Oper. Res. 3, 380–386 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology BhubaneswarBhubaneswarIndia
  2. 2.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada

Personalised recommendations