The Fourier-series method for inverting transforms of probability distributions
Queueing Systems: Theory and Applications - Numerical computations in queues
Solving probability transform functional equations for numerical inversion
Operations Research Letters
Asymptotics for M/G/1 low-priority waiting-time tail probabilities
Queueing Systems: Theory and Applications
Relaxation Time for the Discrete D/G/1 Queue
Queueing Systems: Theory and Applications
On asymptotics of the emptiness probability in the M/GI/1 queue
Automation and Remote Control
Using singularity analysis to approximate transient characteristics in queueing systems
Probability in the Engineering and Informational Sciences
Limits and approximations for the M/G/1 LIFO waiting-time distribution
Operations Research Letters
On the numerical inversion of busy-period related transforms
Operations Research Letters
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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 asymptotic 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.