On the stationary waiting-time distribution in the GI/G/1 queue, I: Transform methods and almost-phase-type distributions

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Abstract
This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying theGI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for theGI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.The algorithm has been coded and examples of numerical results are included.