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

Generalized
Q
-function expressions are developed for the transient state occupancy cumulative distribution function (cdf), probability density function (pdf), and expected value for an
M/M/1
queue. The pdf equation is an extension of a previous result. When Parl's method is used to calculate the generalized
Q
-function, the equations are computationally efficient and accurate. For a
Q
-function relative error of
2\cdot10^{-12}
, the relative error of the result is typically 10
^-10
or better. Relative error will increase, however, for cdf and pdf values on the order of the
Q
function 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.