Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper considers a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then, this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in \(G/G/\infty \) queues with correlated batch arrivals. We study the long-term behaviour of this process as well as its moments. Asymptotic expressions and bounds for quantities of interest, and also convergence for the distribution of this process after renormalization, are studied, when interarrival times and time delays are light tailed. Next, assuming exponentially distributed delays, we derive some explicit and numerically feasible expressions for the limiting joint moments. In such a case, for an infinite server queue with a renewal arrival process, we obtain limiting results on the expectation of the workload, and the covariance of queue size and workload. Finally, some queueing theoretic applications are provided.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Aliyev, R., Bayramov, V.: On the asymptotic behaviour of the covariance function of the rewards of a multivariate renewal-reward process. Stat. Probab. Lett. 127, 138–149 (2017)

    Article  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues. Springer, Berlin (2000)

    Google Scholar 

  3. Brown, M., Ross, S.M.: Renewal Reward Processes. Issue 93 of Technical Report, Cornell University, Department of Operations Research (1969)

  4. Brown, M., Ross, S.M.: Some results for infinite server Poisson queues. J. Appl. Probab. 6, 604–611 (1969)

    Article  Google Scholar 

  5. Brown, M., Solomon, H.: A second-order approximation for the variance of a renewal reward process. Stoch. Process. Their Appl. 3(3), 301–314 (1975)

    Article  Google Scholar 

  6. Chaudhry, M.L., Templeton, J.G.C.: A First Course in Bulk Queues. Wiley, New York (1983)

    Google Scholar 

  7. Dombry, C., Rabehasaina, L.: High order expansions for renewal functions and applications to ruin theory. Ann. Appl. Probab. 27(4), 2342–2382 (2017)

    Article  Google Scholar 

  8. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1968)

    Google Scholar 

  9. Guo, L., Landriault, D., Willmot, G.E.: On the analysis of a class of loss models incorporating time dependence. Eur. Actuar. J. 3, 273–294 (2013)

    Article  Google Scholar 

  10. Haviland, E.K.: On the momentum problem for distribution functions in more than one dimension. Am. J. Math. 57(3), 562–568 (1935)

    Article  Google Scholar 

  11. Izawa, T.: The bivariate gamma distribution. Clim. Stat. 4, 9–15 (1953) (in Japanese)

  12. Karlsson, J.E.: A stochastic model for the lag in reporting of claims. J. Appl. Probab. 11, 382–387 (1974)

    Article  Google Scholar 

  13. Kayid, M.: A general family of NBU classes of life distributions. Stat. Methodol. 4, 185–195 (2007)

    Article  Google Scholar 

  14. Landriault, D., Willmot, G.E., Xu, D.: Analysis of IBNR claims in renewal insurance models. Scand. Actuar. J. 2017(7), 628–650 (2017)

    Article  Google Scholar 

  15. Léveillé, G., Adékambi, F.: Covariance of discounted compound renewal sums with a stochastic interest rate. Scand. Actuar. J. 2, 138–153 (2011)

    Article  Google Scholar 

  16. Léveillé, G., Garrido, J.: Moments of compound renewal sums with discounted claims. Insur. Math. Econ. 28(2), 217–231 (2001)

    Article  Google Scholar 

  17. Léveillé, G., Garrido, J.: Recursive moments of compound renewal sums with discounted claims. Scand. Actuar. J. 2, 98–110 (2001)

    Article  Google Scholar 

  18. Léveillé, G., Garrido, J., Wang, Y.F.: Moment generating functions of compound renewal sums with discounted claims. Scand. Actuar. J. 3, 165–184 (2010)

    Article  Google Scholar 

  19. Liu, L., Kashyap, B.R.K., Templeton, J.G.C.: On the \(GI^X/G/\infty \) system. J. Appl. Probab. (27)3, 671–683 (1990)

    Article  Google Scholar 

  20. Losidis, S., Politis, K.: A two-sided bound for the renewal function when the interarrival distribution is IMRL. Stat. Probab. Lett. 125, 164–170 (2017)

    Article  Google Scholar 

  21. Marshall, A.W., Olkin, I.: Life Distributions, Structure of Nonparametric, Semiparametric and Parametric Families. Springer, Berlin (2007)

    Google Scholar 

  22. Masuyama, H., Takine, T.: Analysis of an infinite-server queue with batch Markovian arrival streams. Queueing Syst. 42, 269–296 (2002)

    Article  Google Scholar 

  23. Neuts, M.F., Chen, S.Z.: The infinite server queue with semi-Markovian arrivals and negative exponential services. J. Appl. Probab. (9)1, 178–184 (1972)

    Article  Google Scholar 

  24. Patch, B., Nazarathy, Y., Taimre, T.: A correction term for the covariance of renewal-reward processes with multivariate rewards. Stat. Probab. Lett. 102, 1–7 (2015)

    Article  Google Scholar 

  25. Smith, W.L.: Renewal theory and its ramifications. J. R. Stat. Soc. B 20, 243–302 (1958)

    Google Scholar 

  26. Smith, W.: The infinitely many server queue with semi-Markovian arrivals and customer-dependent exponential service times. Oper. Res. (20)4, 907–912 (1972)

    Article  Google Scholar 

  27. Takács, L.: Introduction to the Theory of Queues. Oxford University Press, Oxford (1962)

    Google Scholar 

  28. Willmot, G.E.: A queueing theoretic approach to the analysis of the claims payment process. Trans. Soc. Actuar. 42, 447–497 (1990)

    Google Scholar 

  29. Willmot, G.E., Drekic, S.: On the transient analysis of the \(M^X/M/\infty \) queue. Oper. Res. Lett. 28, 137–142 (2001)

    Article  Google Scholar 

  30. Willmot, G.E., Drekic, S.: Transient analysis of some infinite server queues. Recent Adv. Stat. Methods 1, 329–338 (2002)

    Article  Google Scholar 

  31. Willmot, G.E., Drekic, S.: Time-dependent analysis of some infinite server queues with bulk Poisson arrivals. INFOR 47, 297–303 (2009)

    Google Scholar 

  32. Willmot, G.E., Woo, J.-K.: Surplus Analysis of Sparre Andersen Insurance Risk Processes. Springer Actuarial, Berlin (2017)

    Book  Google Scholar 

  33. Woo, J.-K.: On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays. Insur. Math. Econ. 70, 354–363 (2016)

    Article  Google Scholar 

  34. Woo, J.-K., Cheung, E.C.K.: A note on discounted compound renewal sums under dependency. Insur. Math. Econ. 52(2), 170–179 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referees for their careful reading and valuable comments on an earlier version of the manuscript which have led to significant improvements in the paper. This work was supported by Joint Research Scheme France/Hong Kong Procore Hubert Curien Grant No 35296 and F-HKU710/15T.

Author information

Authors and Affiliations

  1. Laboratory of Mathematics, University Bourgogne Franche Comté, 16 route de Gray, 25030, Besançon cedex, France

    Landy Rabehasaina

  2. School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney, Australia

    Jae-Kyung Woo

Authors
  1. Landy Rabehasaina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jae-Kyung Woo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Landy Rabehasaina.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rabehasaina, L., Woo, JK. On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues. Queueing Syst 90, 307–350 (2018). https://doi.org/10.1007/s11134-018-9583-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-018-9583-0

Keywords

Mathematics Subject Classification

Search

Navigation