A new method for finding the characteristic roots of \(E_{n }/E_{m }/1\) queues
The author derives a set of equations, one for each complex root, which can be solved by Newton's method using real arithmetic. This method simplifies the programming logic because it avoids deflation and the subsequent polishing of the roots. Using the waiting time distribution, Syski (cf. [\textit{R. Syski} and \textit{N. H. G. Morris}. Introduction to congestion theory in telephone systems. Edinburgh-London: Oliver \& Boyd (1960; Zbl 0095.32801)]) derived the distribution of the number in the system after a departure. \(E_{n }/E_{m }/1\) queues can also formulated as quasi birth-death (QBD) processes, and, in this case, the characteristic roots discussed by Syski are closely related to the eigenvalues of the QBD process. The QBD process provides information about the number in system at random times, but they are much more difficult to formulate and solve.
scientific article; zbMATH DE number 6048540