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

Author

Listed:
  • Landy Rabehasaina

    (University Bourgogne Franche Comté)

  • Jae-Kyung Woo

    (University of New South Wales)

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 $$G/G/\infty $$ G / G / ∞ 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.

Suggested Citation

  • Landy Rabehasaina & Jae-Kyung Woo, 2018. "On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 307-350, December.
    • Handle: RePEc:spr:queues:v:90:y:2018:i:3:d:10.1007_s11134-018-9583-0
    DOI: 10.1007/s11134-018-9583-0
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    References listed on IDEAS

    as
    1. Woo, Jae-Kyung, 2016. "On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 354-363.
    2. Patch, Brendan & Nazarathy, Yoni & Taimre, Thomas, 2015. "A correction term for the covariance of renewal-reward processes with multivariate rewards," Statistics & Probability Letters, Elsevier, vol. 102(C), pages 1-7.
    3. Aliyev, Rovshan & Bayramov, Veli, 2017. "On the asymptotic behaviour of the covariance function of the rewards of a multivariate renewal–reward process," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 138-149.
    4. Woo, Jae-Kyung & Cheung, Eric C.K., 2013. "A note on discounted compound renewal sums under dependency," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 170-179.
    5. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
    6. Woollcott Smith, 1972. "Technical Note—The Infinitely-Many-Server Queue with Semi-Markovian Arrivals and Customer-Dependent Exponential Service Times," Operations Research, INFORMS, vol. 20(4), pages 907-912, August.
    7. Losidis, Sotirios & Politis, Konstadinos, 2017. "A two-sided bound for the renewal function when the interarrival distribution is IMRL," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 164-170.
    8. Brown, Mark & Solomon, Herbert, 1975. "A second-order approximation for the variance of a renewal reward process," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 301-314, July.
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