Stochastic Properties of Waiting Lines

Abstract

The stochastic properties of waiting lines may be analyzed by a two-stage process: first solving the time-dependent equations for the state probabilities and then utilising these transient solutions to obtain the auto-correlation function for queue length and the root-mean-square frequency spectrum of its fluctuations from mean length. The procedure is worked out in detail for the one-channel, exponential service facility with Poisson arrivals, and the basic solutions for the m -channel exponential service case are given. The analysis indicates that the transient behavior of the queue length n ( t ) may be measured by a “relaxation time,” the mean time any deviation of n ( t ) away from its mean value L takes to return (1/ e ) of the way back to L . This relaxation time increases as (1 − ρ) −2 as the utilization factor ρ approaches unity, whereas the mean length L increases as (1 − ρ) −1 . In other words, as saturation of the facility is approached, the mean length of line increases; but, what is often more detrimental, the length of time for the line to return to average, once it diverges from average, increases even more markedly. Operations Research , ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.