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

TL;DR

This paper develops a comprehensive mathematical framework for analyzing multidimensional infinite server queues with semi-Markovian inputs, deriving integral equations, moments, and explicit expressions for special cases, with applications to claim processes.

Contribution

It introduces a multidimensional integral equation for the moment generating function of semi-Markovian queue inputs and provides explicit transient moment expressions for specific distributions.

Findings

Derived a multidimensional integral equation for the joint moment generating function.

Obtained asymptotic results for first and second moments of the process.

Provided explicit transient moment expressions for exponential and deterministic interarrival times.

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 server queues 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 classications: Primary 60G50, 60K30, 62P05, 60K25.