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 Takacs (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.