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Convergence of tandem Brownian queues

Part of: Operations research and management science Special processes

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: [email protected]
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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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References

[1]Anantharam, V. (1993).Uniqueness of stationary ergodic fixed point for a ∙/M/K node.Ann. Appl. Prob. 3, 154–172. (Correction: 4 (1994), 607.) CrossRefGoogle Scholar
[2]Brémaud, P. (1981).Point Processes and Queues: Martingale Dynamics.Springer, New York.Google Scholar
[3]Burke, P. J. (1956).The output of a queuing system.Operat. Res. 4, 699–704.Google Scholar
[4]Draief, M., Mairesse, J. and O'Connell, N. (2003).Joint Burke's theorem and RSK representation for a queue and a store. In Discrete Random Walks, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp.69–82.Google Scholar
[5]Ferrari, P. A. and Fontes, L. R. G. (1994).The net output process of a system with infinitely many queues.Ann. Appl. Prob. 4, 1129–1144.Google Scholar
[6]Harrison, J. M. (1985).Brownian Motion and Stochastic Flow Systems.John Wiley, New York.Google Scholar
[7]Harrison, J. M. and Williams, R. J. (1990).On the quasireversibility of a multiclass Brownian service station.Ann. Prob. 18, 1249–1268.Google Scholar
[8]Kallenberg, O. (2002).Foundations of Modern Probability, 2nd edn.Springer, New York.Google Scholar
[9]Konstantopoulos, T. and Anantharam, V. (1995).Optimal flow control schemes that regulate the burstiness of traffic.IEEE/ACM Trans. Networking 3, 423–432.CrossRefGoogle Scholar
[10]Lieshout, P. and Mandjes, M. (2007).Tandem Brownian queues.Math. Meth. Operat. Res. 66, 275–298.CrossRefGoogle Scholar
[11]Lieshout, P. and Mandjes, M. (2008).Asymptotic analysis of Lévy-driven tandem queues.Queueing Systems 60, 203–226.Google Scholar
[12]Martin, J. B. and Prabhakar, B. (2010).Fixed points for multi-class queues. Preprint. Available at http://arxiv.org/abs/1003.3024v1.Google Scholar
[13]Mountford, T. and Prabhakar, B. (1995).On the weak convergence of departures from an infinite series of ∙/M/1 queues.Ann. Appl. Prob. 5, 121–127.Google Scholar
[14]O'Connell, N. and Yor, M. (2001).Brownian analogues of Burke's theorem.Stoch. Process. Appl. 96, 285–304.Google Scholar
[15]Salminen, P. and Norros, I. (2001).On busy periods of the unbounded Brownian storage.Queueing Systems 39, 317–333.Google Scholar
[16]Whitt, W. (2002).Stochastic-Process Limits.Springer, New York.Google Scholar