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This paper considers a particular renewal-reward process with multivariatediscounted rewards (inputs) where the arrival epochs are adjusted by addingsome random delays. Then this accumulated reward can be regarded asmultivariate discounted Incurred But Not Reported (IBNR) claims in actuarialscience and some important quantities studied in queueing theory such as thenumber of customers in G/G/infinity queues with correlated batch arrivals. Westudy the long term behavior of this process as well as its moments. Asymptoticexpressions and bounds for the quantities of our interest, and also convergenceresult for the distribution of this process after renormalization, are studied,when interarrival times and time delays are light tailed. Next, assumingexponentially distributed delays, we derive some explicit and numericallyfeasible expressions for the limiting joint moments. In such case, for aninnite server queues with renewal arrival process, we obtain limiting resultson the expectation of the workload, and the covariance of queue size andworkload. Finally, some queueing theoretic applications are providedMSCclassification: 60G50, 60K30, 62P05, 60K25

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