Skip to main content
SpringerLink
Log in
Book cover

International Conference on Frontiers in Optimization: Theory and Applications

FOTA 2016: Operations Research and Optimization pp 75–90Cite as

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

  • Conference paper
  • First Online:

  • 740 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Dickson, D.C.M.: On a class of renewal risk processes. N. Am. Actuar. J. 2(3), 60–68 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dong, H., Liu, Z.: A class of Sparre Andersen risk process. Front. Math. China 5(3), 517–530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dufresne, D.: A general class of risk models. Aust. Actuar. J. 7(4), 755–791 (2011)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gerber, H.U., Shiu, E.S.: On the time value of ruin. N. Am. Actuar. J. 2(1), 48–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dickson, D., Hipp, C.: Ruin probabilities for Erlang (2) risk processes. Insur. Math. Econ. 22(3), 251–262 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dickson, D.C., Hipp, C.: On the time to ruin for Erlang (2) risk processes. Insur. Math. Econ. 29(3), 333–344 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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)

    Article  MathSciNet  Google Scholar 

  13. Asmussen, S., Rolski, T.: Computational methods in risk theory: a matrix-algorithmic approach. Insur. Math. Econ. 10(4), 259–274 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thorin, O.: Probabilities of ruin. Scand. Actuar. J. 1982(2), 65–103 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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. Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd edn. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  18. Prabhu, N.U.: On the ruin problem of collective risk theory. Ann. Math. Statist. 3, 757–764 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thampi, K.K., Jacob, M.J.: On a class of renewal queueing and risk processes. J. Risk Financ 11, 204–220 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Frostig, E.: Upper bounds on the expected time to ruin and on the expected recovery time. Adv. Appl. Probab. 36, 377–397 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kleinrock, L.: Queueing Systems Vol 1: Theory. Wiley-Interscience, New York (1975)

    Google Scholar 

  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. Chaudhry, M.L., Yang, X., Ong, B.: Computing the distribution function of the number of renewals. Am. J. Oper. Res. 3, 380–386 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar, India

    Gopinath Panda & A. D. Banik

  2. Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON, K7K 7B4, Canada

    M. L. Chaudhry

Authors
  1. Gopinath Panda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. D. Banik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. L. Chaudhry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. D. Banik .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, West Bengal, India

    Samarjit Kar

  2. Department of Computer Science and Engineering, Jadavpur University, Kolkata, West Bengal, India

    Ujjwal Maulik

  3. School of Economics and Management, Beijing University of Chemical Technology, Beijing, China

    Xiang Li

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Panda, G., Banik, A.D., Chaudhry, M.L. (2018). Computational Analysis of the GI/G/1 Risk Process Using Roots. In: Kar, S., Maulik, U., Li, X. (eds) Operations Research and Optimization. FOTA 2016. Springer Proceedings in Mathematics & Statistics, vol 225. Springer, Singapore. https://doi.org/10.1007/978-981-10-7814-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation