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

Authors:
Charles Knessl;Yongzhi Peter Yang
Affiliations:
Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA knessl@uic.edu;Department of Mathematics, University of St. Thomas, St. Paul, MN 55105-1079, USA
Venue:
Queueing Systems: Theory and Applications
Year:
2001

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Analysis of congestion periods of an m/m/∞-queue

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Abstract

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)=mc. 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.