On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and 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/∞ 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 infinite server queue 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 provided.AMS 2000 subject classifications: Primary 60G50, 60K30, 62P05, 60K25.

Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs

“…, L ik )'s, with each L i having a Laplace transform denoted by L L i (u) = E(e −uL i ) for u ≥ 0. As in [11], we letZ(t) =Z(t; δ) := e δt Z(t; δ). The processes described in (1) are viewed as different quantities of interest in the following two areas.…”

“…respectively. We remark that the mgf defined in (5) is different from the one studied in [11] which does not consider Markovian assumption for a vector X i and joint structure with the state X Nt conditioning on the initial state X 0 . We finish this introductory section by giving some results of independent interest that will be used in the rest of the paper.…”

“…Furthermore, it is essential to focus on those risks observed at different times with each arrival adjusted by adding a random delay which lead to study the multivariate discounted Incurred But Not Reported (IBNR) claims in insurance and also the total number of customers in an infinite server queue with correlated batch arrivals. In the renewal arrival process, [13] provides explicit expressions for the joint moments of multivariate aggregate discounted IBNR claims which are recursively obtainable, and then [11] develop asymptotic approximation methods to study these joint moments and also provided some queueing theoretic applications, including the workload of the queue and infinite server queues in tandem.…”

“…The present paper considers the model which is an extension of the one given in [11]. In each batch arrival the model consists of multivariate queues (claims) which are modeled by some finite Markov chain and a renewal process is assumed for the batch arrivals.…”

Asymptotics for infinite server queues with fast/slow Markov switching and fat tailed service times

“…represents a set of k correlated queues with an innite number of servers, such that customers arrive at each time T i , with X ij customers arriving in queue j ∈ {1, ..., k}, with corresponding (same) service times L ij (as an example, the basic case where X ij = 1 for all i ∈ N and j = 1, ..., k corresponds to k customers arriving simultaneously at each instant T i ). Z j (t) can also be seen as the number of customers of class j in a (single) innite-server queue, as illustrated in [15,Figure 1]. Other innite-server queue, such as one where the customers within a batch arriving at time T i have dierent service times, may be inferred from the model (1) by choosing an appropriate value of k and Markov chain (X i ) i∈N , see [15,Section 6].…”