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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

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 

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