We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.