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A queueing/inventory and an insurance risk model

Abstract

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.

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Corresponding author

* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
** Email address: r.essifi@tue.nl

References

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[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.
[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.
[3] Albrecher, H. and Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bull. 43, 213243.
[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, 4355.
[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.
[6] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
[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, 203224.
[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.
[9] Brill, P. H. (2008). Level Crossing Methods in Stochastic Models. Springer, New York.
[10] Cohen, J. W. (1975). The Wiener–Hopf technique in applied probability. In Perspectives in Probability and Statistics, Applied Probability Trust, Sheffield, pp.145156.
[11] Cohen, J. W. (YEAR). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.
[12] Kashyap, B. R. K. (1966). The double-ended queue with bulk service and limited waiting space. Operat. Res. 14, 822834.
[13] Liu, X., Gong, Q. and Kulkarni, V. G. (2015). Diffusion models for double-ended queues with renewal arrival processes. Stoch. Systems 5, 161.
[14] Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). The NIST Handbook of Mathematical Functions. Cambridge University Press.
[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, 354368.
[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.
[17] Srivastava, H. M. and Kashyap, B. R. K. (1982). Special Functions in Queuing Theory and Related Stochastic Processes. Academic Press, New York.
[18] Titchmarsh, E. C. (1939). Theory of Functions, 2nd edn. Oxford University Press.
[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, 736752.
[20] Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.

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