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

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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.

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References

  1. Artzrouni, M.: On the convergence of infinite products of matrices. Linear Algebra Appl. 74, 11–21 (1986)

    Article  Google Scholar 

  2. Athreya, A.B., Ramamurthy, K.: Feller’s renewal theorem for systems of renewal equations. J. Indian Inst. Sci. 58, 437–459 (1976)

    Google Scholar 

  3. Blom, J., De Turck, K., Mandjes, M.: Refined large deviations asymptotics for Markov-modulated infinite-server systems. Eur. J. Oper. Res. 259, 1036–1044 (2017)

    Article  Google Scholar 

  4. Blom, J., Kella, O., Mandjes, M., Thorsdottir, H.: Markov-modulated infinite-server queues with general service times. Queueing Syst. 76, 403–424 (2014)

    Article  Google Scholar 

  5. D’Auria, B.: \(M/M/\infty \) queues in semi-Markovian random environment. Queueing Syst. 58, 221–237 (2008)

    Article  Google Scholar 

  6. Francq, C., Gautier, A.: Estimation of time-varying ARMA models with Markovian changes in regime. Stat. Prob. Lett. 70, 243–251 (2004)

    Article  Google Scholar 

  7. Fralix, B.H., Adan, I.J.B.F.: An infinite-server queue influenced by a semi-Markovian environment. Queueing Syst. 61, 65–84 (2009)

    Article  Google Scholar 

  8. Karlsson, J.-E.: A stochastic model for time lag in reporting of claims. J. Appl. Prob. 11(2), 382–387 (1974)

    Article  Google Scholar 

  9. Mandjes, M., De Turck, K.: Markov-modulated infinite-server queues driven by a common background process. Stoch. Models 32(2), 206–232 (2016)

    Article  Google Scholar 

  10. Masuyama, H., Takine, T.: Analysis of an infinite-server queue with batch Markovian arrival streams. Queueing Syst. 42(3), 269–296 (2002)

    Article  Google Scholar 

  11. O’Cinneide, C., Purdue, P.: The \(M/M/\infty \) in a random environment. J. Appl. Prob. 23, 175–184 (1986)

    Google Scholar 

  12. Rabehasaina, L., Woo, J.-K.: On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR process and infinite server queues. Queueing Syst. 90(3), 307–350 (2018)

    Article  Google Scholar 

  13. Takács, L.: Introduction to the Theory of Queues. Oxford University Press, Oxford (1962)

    Google Scholar 

  14. Willmot, G.E.: A queueing theoretic approach to the analysis of the claims payment process. Trans. Soc. Actuar. 42, 447–497 (1990)

    Google Scholar 

  15. Willmot, G.E., Drekic, S.: On the transient analysis of the \(M^X/M/\infty \) queue. Oper. Res. Lett. 28, 137–142 (2001)

    Article  Google Scholar 

  16. Woo, J.-K.: On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays. Insur. Math. Econ. 70, 354–363 (2016)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous reviewers for their helpful comments and suggestions. This work was supported by Joint Research Scheme France/Hong Kong Procore Hubert Curien Grant Nos. 35296, F-HKU710/15T, and the UNSW Business School 2018 International Research Collaboration Travel Funds. Also, Jae-Kyung Woo gratefully acknowledges the support from the 2019 Business School Research Grant.

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Authors and Affiliations

  1. Laboratory of Mathematics, University Bourgogne Franche Comté, 16 Route de Gray, 25030, Besançon Cedex, France

    Landy Rabehasaina

  2. School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney, Australia

    Jae-Kyung Woo

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  1. Landy Rabehasaina
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  2. Jae-Kyung Woo
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Correspondence to Landy Rabehasaina.

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Rabehasaina, L., Woo, JK. Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs. Queueing Syst 94, 393–420 (2020). https://doi.org/10.1007/s11134-020-09646-y

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  • DOI: https://doi.org/10.1007/s11134-020-09646-y

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