Queueing Systems

, Volume 57, Issue 1, pp 19–28 | Cite as

Processor sharing for two queues with vastly different rates

  • John A. Morrison


We consider a 2-class queueing system, operating under a generalized processor-sharing discipline, in an asymptotic regime where the arrival and service rates of the two classes are vastly different. We use regular and singular perturbation analyses in a small parameter measuring this difference in rates. It is assumed that the system is stable, and not close to instability. Three different regimes are analyzed, corresponding to an underloaded, an overloaded and a critically loaded fast queue, respectively. In the first two regimes the lowest order approximation to the joint stationary distribution of the queue lengths is derived. For a critically loaded fast queue only the mean queue lengths are investigated, and the asymptotic matching, to lowest order, with the results for an underloaded and an overloaded fast queue is established.


Asymptotics Matching Processor sharing Singular perturbations 

Mathematics Subject Classification (2000)

60K30 90B22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington (1964) Google Scholar
  2. 2.
    Altman, E., Avrachenkov, K.E., Núñez-Queija, R.: Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Probab. 36, 839–853 (2004) CrossRefGoogle Scholar
  3. 3.
    Borst, S.C., Morrison, J.A.: Bandwidth sharing with disparate flow classes and state-dependent service rates. Preprint Google Scholar
  4. 4.
    Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, New York (1971) Google Scholar
  5. 5.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954) Google Scholar
  6. 6.
    Fayolle, G., Iasnogorodski, R.: Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrsch. 47, 325–351 (1979) CrossRefGoogle Scholar
  7. 7.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 4th edn. Academic, New York (1965) Google Scholar
  8. 8.
    Guillemin, F., Pinchon, D.: Analysis of generalized processor-sharing systems with two classes of customers and exponential services. J. Appl. Probab. 41, 832–858 (2004) CrossRefGoogle Scholar
  9. 9.
    van Kessel, G., Núñez-Queija, R., Borst, S.C.: Differentiated bandwidth sharing with disparate flow sizes. In: Proc. of INFOCOM, 2005 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Consultant, Alcatel-Lucent, Bell LaboratoriesMurray HillUSA

Personalised recommendations