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A QUEUEING MODEL WITH RANDOMIZED DEPLETION OF INVENTORY

Published online by Cambridge University Press:  13 September 2016

Hansjörg Albrecher ,
Onno Boxma ,
Rim Essifi  and
Richard Kuijstermans
Hansjörg Albrecher
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne and Swiss Financial Institute, UNIL-Dorigny, CH-1015 Lausanne, Switzerland
Onno Boxma
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: o.j.boxma@tue.nl
Rim Essifi
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: o.j.boxma@tue.nl
Richard Kuijstermans
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: o.j.boxma@tue.nl
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Abstract

In this paper, we study an M/M/1 queue, where the server continues to work during idle periods and builds up inventory. This inventory is used for new arriving service requirements, but it is completely emptied at random epochs of a non-homogeneous Poisson process, whose rate depends on the current level of the acquired inventory. For several shapes of depletion rates, we derive differential equations for the stationary density of the workload and the inventory level and solve them explicitly. Finally, numerical illustrations are given for some particular examples, and the effects of this depletion mechanism are discussed.

Keywords

applied probabilityinventory theoryqueueing theorystochastic modelling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 1 , January 2017 , pp. 43 - 59
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Abramowitz, M. & Stegun, I.A. (1964). Handbook of mathematical functions with formulas, graphs and mathematical tables. New York, NY: Dover Publications.Google Scholar
2. Albrecher, H. & Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bulletin 43: 213–243.Google Scholar
3. Albrecher, H., Cheung, E.C.K., & Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scandinavian Actuarial Journal 6: 424–452.CrossRefGoogle Scholar
4. Albrecher, H., Gerber, H.U., & Shiu, E.S.W. (2011). The optimal dividend barrier in the Gamma–Omega model. European Actuarial Journal 1: 43–55.Google Scholar
5. Asmussen, S. & Albrecher, H. (2010). Ruin probabilities. Hackensack, NJ: World Scientific.Google Scholar
6. Berman, O. & Kim, E. (2001). Dynamic order replenishment policy in internet-based supply chains. Mathematical Methods of Operations Research 53: 371–390.Google Scholar
7. Berman, O., Parlar, M., Perry, D., & Posner, M.J.M. (2005). Production/clearing models under continuous and sporadic review. Methodology and Computing in Applied Probability 7: 203–224.Google Scholar
8. Boxma, O.J., Essifi, R., & Janssen, A.J.E.M. (2015). A queueing/inventory and an insurance risk model. To appear in Journal of Applied Probability Google Scholar
9. Doshi, B.T. (1992). Level crossing analysis of queues . In Bhat, U.N. & Basawa, I.V. (eds.), Queueing and related models, Oxford: Clarendon Press, pp. 3–33 Google Scholar
10. Mohebbi, E. & Posner, M.J.M. (1998). A continuous-review inventory system with lost sales and variable lead time. Naval Research Logistics 45: 259–278.Google Scholar
11. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., & Clark, C.W. (2010). NIST handbook of mathematical functions. NIST, New York, NY: Cambridge University Press.Google Scholar
12. Perry, D., Stadje, W., & Zacks, S. (2005). Sporadic and continuous clearing policies for a production/inventory system under an M/G demand process. Mathematics of Operations Research 30: 354–368.Google Scholar
13. Polyanin, A.D., & Zaitsev, V.F. (2003). Handbook of exact solutions for ordinary differential equations, 2nd ed. Boca Raton, FL: CRC Press.Google Scholar
14. Saffari, M., Asmussen, S., & Haji, R. (2013). The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Systems 75: 65–77.Google Scholar
15. Sahin, I. (1990). Regenerative inventory systems, operating characteristics and optimization, New York, NY: Springer.Google Scholar
16. Schwarz, M. & Daduna, H. (2006). Queueing systems with inventory management with random lead times and with backordering. Mathematical Methods of Operations Research 64: 383–414.CrossRefGoogle Scholar
17. Schwarz, M., Sauer, C., Daduna, H., Kulik, R., & Szekli, R. (2006). M/M/1 queueing systems with inventory. Queueing Systems 54: 55–78.Google Scholar