Queueing Systems

, Volume 39, Issue 4, pp 377–397 | Cite as

Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory

  • Fabrice Guillemin
  • Jacqueline Boyer
Article

Abstract

We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.

M/M/1 queue processor sharing spectral theory orthogonal polynomials queue asymptotics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55 (National Bureau of Standards, 1972).Google Scholar
  2. [2]
    S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).Google Scholar
  3. [3]
    M.S. Birman and M.Z. Solomjak, Spectral Theory of Self–Adjoint Operators in Hilbert Space (Reidel/Kluwer Academic, 1982).Google Scholar
  4. [4]
    E.G. Coffman, R.R. Muntz, and H. Trotter, Waiting time distributions for processor–sharing systems, J. ACM 17 (1970) 123–130.Google Scholar
  5. [5]
    R. Dautray and J.L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques (Masson, Paris, 1985).Google Scholar
  6. [6]
    N. Dunford and J.T. Schwartz, Linear Operators II: Spectral Theory (Interscience, New York, 1963).Google Scholar
  7. [7]
    A. Erdélyi, Higher Transcendal Functions (McGraw–Hill, New York, 1953).Google Scholar
  8. [8]
    P. Flajolet and F. Guillemin, The formal theory of birth and death processes, lattice path combinatorics and continued fractions, Adv. in Appl. Probab. 32 (2000) 750–778.Google Scholar
  9. [9]
    M. Ismail and D. Kelker, Special functions, Stieltjes transforms, and infinite divisibility, SIAM J. Math. Anal. 10(5) (1979) 884–901.Google Scholar
  10. [10]
    J. Morrison, Response time for a processor–sharing system, SIAM J. Appl. Math. 45(1) (1985) 152–167.Google Scholar
  11. [11]
    F. Pollaczek, Sur une généralisation des polynômes de Jacobi, C. R. Acad. Sci. Paris 35 (1950) 1–54.Google Scholar
  12. [12]
    J. Roberts and L. Massoulié, Bandwidth sharing and admission control for elastic traffic, in: Proc. Infocom'99 (1998).Google Scholar
  13. [13]
    W. Rudin, Functional Analysis (McGraw–Hill, New York, 1973).Google Scholar
  14. [14]
    B. Sengupta and D.L. Jagerman, A conditional response time of the M/M/1 processor–sharing queue, AT&T Techn. J. 64(2) (1985).Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Fabrice Guillemin
    • 1
  • Jacqueline Boyer
    • 1
  1. 1.France Telecom R&DLannionFrance

Personalised recommendations