Paper Info
Title
Simple spectral representations for the M/M/1 queue
Abstract
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.
Year
DOI
Venue
1988
10.1007/BF01157854
Queueing Syst.
Keywords
Field
DocType
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
M/M/1 queue,Discrete mathematics,M/D/1 queue,M/M/c queue,M/G/1 queue,M/G/k queue,Burke's theorem,M/D/c queue,M/M/∞ queue,Mathematics
Journal
Volume
Issue
Citations 
3
4
8
PageRank 
References 
Authors
3.45
7
2
Name
Order
Citations
PageRank
J. Abate122693.40
Ward Whitt21509658.94