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

Landy Rabehasaina () and Jae-Kyung Woo ()
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Landy Rabehasaina: University Bourgogne Franche Comté
Jae-Kyung Woo: University of New South Wales

Queueing Systems: Theory and Applications, 2020, vol. 94, issue 3, No 8, 393-420

Abstract: Abstract We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported 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 semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

Keywords: Semi-Markovian multivariate discounted inputs; Infinite server queues; IBNR process; Markov modulation; 60G50; 60K25; 60K30; 62P05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s11134-020-09646-y

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