Complex functions in queueing theory

Pollaczek's queueing-oriented research dates mainly from the 2nd quarter of the present century. His analysis of the GI/G/1 queueing model was and still is significant from a methodological as well as an applicability point of view. Techniques from the theory of Complex Functions constitute the core of his analysis. The advances in Probability Theory and in the Theory of Boundary Value Problems for analytic functions, which became known in the third quarter of our century have provided a wider context for the analysis of the GI/G/1 queueing model. The present study exposes the analysis out from these backgrounds. For the time dependent as well as the stationary case this approach leads to the formulation of Riemann Boundary Value Problems. The analysis of it for the stationary case is exposed and its solution is derived under slightly weaker conditions than originally considered by Pollaczek.