Queueing Systems

, Volume 2, Issue 1, pp 41–65

Transient behavior of the M/M/l queue: Starting at the origin

  • Joseph Abate
  • Ward Whitt
Contributed Paper

DOI: 10.1007/BF01182933

Cite this article as:
Abate, J. & Whitt, W. Queueing Syst (1987) 2: 41. doi:10.1007/BF01182933


This paper presents some new perspectives on the time-dependent behavior of the M/M/1 queue. The factorial moments of the queue length as functions of time when the queue starts empty have interesting structure, which facilitates developing approximations. Simple exponential and hyperexponential approximations for the first two moment functions help show how the queue approaches steady state as time evolves. These formulas also help determine if steady-state descriptions are reasonable when the arrival and service rates are nearly constant over some interval but the process does not start in steady state.


Transient behaviorapproach to steady staterelaxation timesbirth- and-death processqueuesBrownian motionfirst passage timesbusy periodcomplete monotonicity

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1987

Authors and Affiliations

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