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Convergence of tandem Brownian queues

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  21 June 2016

Sergio I. López
Sergio I. López*
Affiliation:
Universidad Nacional Autónoma de México
*
* Postal address: Departamento de Matemáticas, Universidad Nacional Autónoma de México, Av. Universidad No 3000, C.U., Distrito Federal, 04510, Mexico. Email address: silo@ciencias.unam.mx
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Abstract

It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

Keywords

Brownian queuetandem queuesBurke's theorem

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B15: Network models, stochastic
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 585 - 592
Copyright
Copyright © Applied Probability Trust 2016 

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