Computation of the Transient M/M/1 Queue cdf, pdf, and Mean with Generalized Q-Functions

Abstract
GeneralizedQ-function expressions are developed for the transient state occupancy cumulative distribution function (cdf), probability density function (pdf), and expected value for anM/M/1queue. The pdf equation is an extension of a previous result. When Parl's method is used to calculate the generalizedQ-function, the equations are computationally efficient and accurate. For aQ-function relative error of2\cdot10^{-12}, the relative error of the result is typically 10-10or better. Relative error will increase, however, for cdf and pdf values on the order of theQfunction relative error. Execution time per point on a VAX 11/750 is on the order of tens of milliseconds for the range of parameters considered.