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Queues and Risk Models with Simultaneous Arrivals

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  22 February 2016

E. S. Badila ,
O. J. Boxma ,
J. A. C. Resing  and
E. M. M. Winands
E. S. Badila*
Affiliation:
Eindhoven University of Technology
O. J. Boxma*
Affiliation:
Eindhoven University of Technology
J. A. C. Resing*
Affiliation:
Eindhoven University of Technology
E. M. M. Winands*
Affiliation:
University of Amsterdam
*
∗ 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.
∗∗ Postal address: Korteweg de Vries Instituut voor Wiskunde, University of Amsterdam, PO Box 94248, 1090 GE Amsterdam, The Netherlands.
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Abstract

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We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

Keywords

Queue with simultaneous arrivalworkloadstochastic decompositiondualitymultivariate risk model

MSC classification

Primary: 60K25: Queueing theory
Secondary: 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 812 - 831
Copyright
© Applied Probability Trust 

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