Queueing Systems

, Volume 90, Issue 3–4, pp 307–350 | Cite as

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

  • Landy RabehasainaEmail author
  • Jae-Kyung Woo


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.


Renewal-reward process Multivariate discounted rewards Incurred But Not Reported (IBNR) claims Infinite server queues Workload Convergence in distribution 

Mathematics Subject Classification

Primary 60G50 60K30 62P05 60K25 



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.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of MathematicsUniversity Bourgogne Franche ComtéBesançon cedexFrance
  2. 2.School of Risk and Actuarial Studies, Australian School of BusinessUniversity of New South WalesSydneyAustralia

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