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A queueing/inventory and an insurance risk model

Part of: General theory of linear operators Operations research and management science Special processes Mathematical economics

Published online by Cambridge University Press:  11 January 2017

Onno Boxma ,
Rim Essifi  and
Augustus J. E. M. Janssen
Onno Boxma*
Affiliation:
Eindhoven University of Technology
Rim Essifi*
Affiliation:
Eindhoven University of Technology
Augustus J. E. M. Janssen*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

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We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = a x. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

Keywords

M/G/1 queueCramér–Lundberg insurance risk modelworkloadinventoryruin probabilityWiener–Hopf technique

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service 91B30: Risk theory, insurance
Secondary: 47A68: Factorization theory (including Wiener-Hopf and spectral factorizations)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1139 - 1160
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, DC.Google Scholar
[2] Albrecher, H. and Ivanovs, J. (2015). Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Preprint. Available at http://arxiv.org/abs/1507.03848.Google Scholar
[3] Albrecher, H. and Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bull. 43, 213–243.Google Scholar
[4] Albrecher, H., Gerber, H. U. and Shiu, E. S. W. (2011). The optimal dividend barrier in the gamma–omega model. Europ. Actuarial J. 1, 43–55.Google Scholar
[5] Albrecher, H., Boxma, O. J., Essifi, R. and Kuijstermans, R. A. C. (2016). A queueing model with randomized depletion of inventory. To appear in em Probab. Eng. Inform. Sc. Google Scholar
[6] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.Google Scholar
[7] Berman, O., Parlar, M., Perry, D. and Posner, M. J. (2005). Production/clearing models under continuous and sporadic reviews. Methodol. Comput. Appl. Prob. 7, 203–224.Google Scholar
[8] Boxma, O., Essifi, R. and Janssen, A. J. E. M. (2015). A queueing/inventory and an insurance risk model. Preprint. Available at http://arxiv.org/abs/1510.07610.Google Scholar
[9] Brill, P. H. (2008). Level Crossing Methods in Stochastic Models. Springer, New York.Google Scholar
[10] Cohen, J. W. (1975). The Wiener–Hopf technique in applied probability. In Perspectives in Probability and Statistics, Applied Probability Trust, Sheffield, pp.145–156.Google Scholar
[11] Cohen, J. W. (YEAR). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
[12] Kashyap, B. R. K. (1966). The double-ended queue with bulk service and limited waiting space. Operat. Res. 14, 822–834.CrossRefGoogle Scholar
[13] Liu, X., Gong, Q. and Kulkarni, V. G. (2015). Diffusion models for double-ended queues with renewal arrival processes. Stoch. Systems 5, 1–61.Google Scholar
[14] Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). The NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
[15] Perry, D., Stadje, W. and Zacks, S. (2005). Sporadic and continuous clearing policies for a production/inventory system under an M/G demand process. Math. Operat. Res. 30, 354–368.Google Scholar
[16] Polyanin, A. D. and Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[17] Srivastava, H. M. and Kashyap, B. R. K. (1982). Special Functions in Queuing Theory and Related Stochastic Processes. Academic Press, New York.Google Scholar
[18] Titchmarsh, E. C. (1939). Theory of Functions, 2nd edn. Oxford University Press.Google Scholar
[19] Welch, P. D. (1964). On a generalized M/G/1 queuing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.Google Scholar
[20] Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar