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Random Fluid Limit of an Overloaded Polling Model

Published online by Cambridge University Press:  22 February 2016

Maria Remerova*
Affiliation:
CWI
Sergey Foss*
Affiliation:
Heriot-Watt University and Sobolev Institute of Mathematics
Bert Zwart*
Affiliation:
CWI, EURANDOM, VU University Amsterdam and Georgia Institute of Technology
*
Current address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗ Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK.
Current address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
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Abstract

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In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

Type
Research Article
Copyright
© Applied Probability Trust 

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