Abstract
We consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach. 22 (1975) 248–260.
D. Bertsekas and R. Gallager, Data Networks (Prentice Hall, New York, 1987).
D. Bertsimas, I.C. Paschalidis and J.N. Tsitsiklis, Large deviation analysis of the generalized processor sharing policy, Queueing Systems 32 (1999) 319–349.
P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
T. Bonald and L. Massoulié, Impact of fairness on Internet performance, in: Proc. of ACM SIGMETRICS, 2001.
T. Bonald and A. Proutière, Insensitivity in processor-sharing networks, Perform. Eval. 49 (2002) 193–209.
T. Bonald and A. Proutière, Insensitive bandwidth sharing in data networks, Queueing Systems 44(1) (2003) 69–100.
S. Borst, O.J. Boxma and P. Jelenkovic, Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows, Queueing Systems 43(4) (2003) 273–306.
G.L. Choudhury, A. Mandelbaum, M.I. Reiman and W. Whitt, Fluid and diffusion limits for queues in slowly changing environments, Stochastic Models 13 (1997) 121–146.
J.W. Cohen, The multiple phase service network with generalized processor sharing, Acta Inform. 12 (1979) 245–284.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Jones & Barlett, Boston, MA, 1993).
G. Fayolle, I. Mitrani and R. lasnogorodski, Sharing a processor among many classes, J. Assoc. Comput. Mach. 27 (1980) 519–532.
P.R. Jelenkovic, The effect of multiple time scales and subexponentiality on the behavior of a broadband network multiplexer, Ph.D. Thesis, Columbia University, New York (October 1996).
F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).
F.P. Kelly, A. Maulloo and D. Tan, Rate control for communication networks: Shadow prices, proportional fairness and stability, J. Oper. Res. Soc. 49 (1998) 237–252.
F.P. Kelly and R.J. Williams, Fluid models for a network operating under a fair bandwidth-sharing policy, to appear in Ann. Appl. Probab.
L. Kleinrock, Queueing Systems, Vol. 2 (Wiley, New York, 1975).
H.J. Kushner, Heavy-Traffic of Controlled Queueing and Commmunication Networks (Springer, Berlin, 2001).
L. Massoulié, Large deviation estimates for polling and weighted fair queueing systems, Adv. Perform. Anal. (1999).
L. Massoulié and J.W. Roberts, Bandwidth sharing and admission control for elastic traffic, Telecom. Systems 15 (2000) 185–201.
L. Massoulié and J.W. Roberts, Bandwidth sharing: objectives and algorithms, IEEE/ACM Trans. Networking 10(3) (2002) 320–328.
R. Núñez Queija, Processor-sharing models for integrated-services networks, Ph.D. Thesis, Eindhoven University of Technology (1999).
R.F. Serfozo, Introduction to Stochastic Networks (Springer, Berlin, 1999).
W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues (Springer, Berlin, 2002).
G.G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach (Springer, Berlin, 1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bonald, T., Proutière, A. On Stochastic Bounds for Monotonic Processor Sharing Networks. Queueing Systems 47, 81–106 (2004). https://doi.org/10.1023/B:QUES.0000032802.41986.c6
Issue Date:
DOI: https://doi.org/10.1023/B:QUES.0000032802.41986.c6