Queueing Systems

, Volume 3, Issue 4, pp 321–345

Simple spectral representations for the M/M/1 queue

  • Joseph Abate
  • Ward Whitt
Contributed Papers

DOI: 10.1007/BF01157854

Cite this article as:
Abate, J. & Whitt, W. Queueing Syst (1988) 3: 321. doi:10.1007/BF01157854


This paper shows that certain basic descriptions of the time-dependent behavior of the M/M/1 queue have very simple representations as mixtures of exponentials. In particular, this is true for the busy-period density, the probability that the server is busy starting at zero, the expected queue length starting at zero and the autocorrelation function of the stationary queue-length process. In each case the mixing density is a minor modification of a beta density. The last two representations also apply to regulated or reflected Brownian motion (RBM) by virtue of the heavy-traffic limit. Connections are also established to the classical spectral representations of Ledermann and Reuter (1954) and Karlin and McGregor (1958) and the associated trigonometric integral representations of Ledermann and Reuter, Vaulot (1954), Morse (1955), Riordan (1961) and Takács (1962). Overall, this paper aims to provide a more unified view of the M/M/1 transient behavior and show how several different approaches are related.


M/M/1 queue Brownian motion spectral representation mixtures of exponentials busy period autocorrelation function time-dependent mean transient behavior Chebycheff polynomials duality, the associated process 

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1988

Authors and Affiliations

  • Joseph Abate
    • 1
  • Ward Whitt
    • 2
  1. 1.AT&T Bell LaboratoriesWarrenUSA
  2. 2.AT&T Bell LaboratoriesMurray HillUSA

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