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Optimal routeing in two-queue polling systems

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  16 November 2018

I. J. B. F. Adan ,
V. G. Kulkarni ,
N. Lee  and
E. Lefeber
I. J. B. F. Adan*
Affiliation:
Eindhoven University of Technology
V. G. Kulkarni*
Affiliation:
University of North Carolina
N. Lee*
Affiliation:
University of North Carolina
E. Lefeber*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands. Email address: i.adan@tue.nl
** Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
** Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
*** Postal address: Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands.
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Abstract

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We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs andcharacterize individually and sociallyoptimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

Keywords

Customer routeingdynamic programmingfluid queuelinear quadratic regulatorNash equilibriumpolling systemRicatti equationindividual optimum, social optimum

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 944 - 967
Copyright
Copyright © Applied Probability Trust 2018 

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