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Polling systems with periodic server routeing in heavy traffic: distribution of the delay

Part of: Operations research and management science Computer system organization Special processes

Published online by Cambridge University Press:  14 July 2016

Tava Lennon Olsen  and
R. D. van der Mei
Tava Lennon Olsen*
Affiliation:
Washington University in St. Louis
R. D. van der Mei*
Affiliation:
Vrije Universiteit, Amsterdam, and TNO Telecom
*
∗ Postal address: John M. Olin School of Business, Washington University in St. Louis, Campus Box 1133, St. Louis, MO 63130—4899, USA. Email address: [email protected]
∗∗ Postal address: Vrije Universiteit, Faculty of Exact Sciences, 1081HV Amsterdam, Netherlands.
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Abstract

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

Keywords

Polling systempolling tableperiodic routeingdelaywaiting timeheavy trafficdistributional approximationswitch-over timesetup time

MSC classification

Primary: 68M20: Performance evaluation; queueing; scheduling 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 305 - 326
Copyright
Copyright © Applied Probability Trust 2003 

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