CALCULATING THE M G/1 BUSY-PERIOD DENSITY AND LIFO WAITING-TIME DISTRIBUTION BY DIRECT NUMERICAL TRANSFORM INVERSION/

It is well known that the M/G/1 busy-period density can be characteriz ed by the Kendall functional equation for its Laplace transform. The K endall functional equation can be solved iteratively to obtain transfo rm values to use in numerical inversion algorithms. However, we show t hat the busy-period density can also be numerically inverted directly, without iterating a functional equation, exploiting a contour integra l representation due to Cox and Smith (1961). The contour integral rep resentation was originally proposed as a basis for asymptotic approxim ations. We derive heavy-traffic expansions for the aysmptotic paramete rs appearing there. We also use the integral representation to derive explicit series representations of the busy-period density for serval service-time distributions. In addition, we discuss related contour in tegral representations for the probability of emptiness, which is dire ctly related to the waiting-time distribution with the LIFO discipline . The asymptotics and the numerical inversion reveal the striking diff erence between the waiting-time distributions for the FIFO and LIFO di sciplines.