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

Abstract

We consider a general k dimensional discounted infinite server queues process (alternatively, an Incurred But Not Reported (IBNR) claim process) where the multivariate inputs (claims) are given by a k dimensional finite state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study a semi-Markov modulated infinite queues process where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and on the switching times.AMS 2000 subject classifications: Primary 60G50, 60K30, 62P05, 60K25.

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

“…is a particular case of the process in (2) with discount factor a = 0, the LT ψ(s, t) dened in ( 5) is characterized by [16,Proposition 4], which we rewrite here: Proposition 1. When {N t , t ≥ 0} is a Poisson process with intensity λ > 0, then ψ(s, t) is the unique solution to the rst order linear (matrix) dierential equation…”

“…The proof of Corollary 4 is included in Subsections 3.2, for the convergence ( 14), and 3.3, for the convergence (16). As a concluding remark, we will discuss in Section 4 some computational aspect for the limiting distributions mentioned in those dierent regimes in Theorem 2 in the particular case when α is a rational number lying in (0, 1).…”

“…The present paper follows [16], which studies the transient or limiting distribution of a discounted version of Z(t) of the form…”

“…The main dierence with [16] is that the latter has more general assumptions on the interarrival and service distributions, whereas we focus here on Poisson arrivals. Even though the assumptions are more restrictive than in [16], the goal here is dierent in that we are trying to exhibit dierent behaviours for the limiting models when the arrival rate is increased and the service times are decreased by suitable factors, whereas [16] is more focused on analytical stochastic properties such as the moments of Z(t) or its limiting distribution as t → ∞. The discounting factor a ≥ 0 in (2) is important in situations e.g.…”

“…such that customers arrive on the switching times of the Markov chain, with service times depending on the state of the background process (see [12], [11,Model II]). Then [16,Section 6] explains how this process {Z(t), t ≥ 0} can be embedded in a process {Z(t), t ≥ 0} dened by (1) with an appropriate choice of k, K in function of κ, as well as of the Markov chain (X i ) i∈N and the sequence (L ij ) i∈N,j=1,...,k of service times. Thus, studying a general process {Z(t), t ≥ 0} in (1) allows to study a broad class of innite server queue models in a similar Markov modulated context.…”