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

TL;DR

This paper analyzes a multivariate renewal-reward process with random delays and discounting, providing asymptotic behavior, moments, and applications to IBNR claims and infinite server queues, especially under light-tailed distributions.

Contribution

It introduces a novel multivariate renewal-reward process with delayed arrivals, deriving explicit limiting moments and applying results to queueing systems and actuarial models.

Findings

Asymptotic bounds for the process are established.

Explicit joint moments are derived under exponential delays.

Results on workload and queue size covariance in infinite server queues.

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 (IBNR) claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in G/G/infinity queues with correlated batch arrivals. We study the long term behavior of this process as well as its moments. Asymptotic expressions and bounds for the quantities of our interest, and also convergence result 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 case, for an innite server queues with 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 providedMSC classification: 60G50, 60K30, 62P05, 60K25