This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying the GI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for the GI/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.

Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation for generators is equivalent to Fristedt's Lévy measure factorisation and to process decomposition.

The joint distributions of the exit time and exit value of a spectrally positive process, from semi-infinite and finite intervals, are derived in the form of Fourier-Laplace transforms. Also the probability that such a process makes its first exit from a finite interval via the lower end point is obtained explicitly.