Queueing Systems

, Volume 39, Issue 2, pp 213–256

Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue


  • Charles Knessl
    • Department of Mathematics, Statistics and Computer Science (M/C 249)University of Illinois at Chicago
  • Yongzhi Peter Yang
    • Department of MathematicsUniversity of St. Thomas

DOI: 10.1023/A:1012752719211

Cite this article as:
Knessl, C. & Yang, Y.P. Queueing Systems (2001) 39: 213. doi:10.1023/A:1012752719211


We consider the M/M/∞ queue with arrival rate λ, service rate μ and traffic intensity ρ=λ/μ. We analyze the first passage distribution of the time the number of customers N(t) reaches the level c, starting from N(0)=m>c. If m=c+1 we refer to this time period as the congestion period above the level c. We give detailed asymptotic expansions for the distribution of this first passage time for ρ→∞, various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.

M/M/∞ queuebusy periodasymptotics

Copyright information

© Kluwer Academic Publishers 2001