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

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 mult...

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Bibliographic Details
Published inQueueing systems Vol. 94; no. 3-4; pp. 393 - 420
Main Authors Rabehasaina, Landy, Woo, Jae-Kyung
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2020
Springer Nature B.V
Springer Verlag
Subjects
Business and Management
Computer Communication Networks
Control
Integral equations
Markov analysis
Markov chains
Mathematics
Multivariate analysis
Operations Research/Decision Theory
Probability
Probability Theory and Stochastic Processes
Queues
Queuing theory
Supply Chain Management
Systems Theory
IBNR process
60K30
60G50
62P05
60K25
Markov modulation
Semi-Markovian multivariate discounted inputs
Infinite server queues
and phrases: Semi-Markovian multivariate discounted inputs
Queueing
Innite server queues
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Summary: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.
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ISSN:0257-0130
1572-9443
DOI:10.1007/s11134-020-09646-y