The Analysis of the Time-dependent Behavior of Queues: the Application of Wiener-hopf Techniques

This book studies the time-dependent behavior of queuing systems which has non-empty initial work load. The technique used to analyze the behavior of the queueing systems studied in this book is based on the Wiener-Hopf factorization. The first major step in the analysis is the derivation of the (system of) transformed Wiener-Hopf equation(s). Wiener-Hopf factorization is then applied to its symbol. Since the queueing systems we consider have a non-zero initial working load, the Wiener-Hopf factorization should be followed by a decomposition on a certain (matrix) function. The Wiener-Hopf factorization and the decomposition yields a (formal) solution of the (system of) equation(s). If the stability condition is fulfilled, then the steady-state distributions of interest can be determined by applying Abel's limit theorem to the solution of the (system of) equation(s). In this book it is shown how the time-dependent distributions converge to the steady-state distribution.