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 characterized by the Kendall functional equation for its Laplace transform. The Kendall functional equation can be solved iteratively to obtain transform values to use in numerical inversion algorithms. However, we show that the busy-period density can also be numerically inverted directly, without iterating a functional equation, exploiting a contour integral representation due to Cox and Smith (1961). The contour integral representation was originally proposed as a basis for asymptotic approximations. We derive heavy-traffic expansions for the aysmptotic parameters 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 integral representations for the probability of emptiness, which is directly related to the waiting-time distribution with the LIFO discipline. The asymptotics and the numerical inversion reveal the striking difference between the waiting-time distributions for the FIFO and LIFO disciplines.