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This paper presents a method for calculating steady state probabilities of$M|E_r|c|K$ queueing systems. The infinitesimal generator matrix is used todefine all possible states in the system and their transition probabilities.While this matrix can be written down immediately for many other $M|PH|c|K$queueing systems with phase-type service times (e.g. Coxian, Hypoexponential,\ldots), it requires a more careful analysis for systems with Erlangian servicetimes. The constructed matrix may then be used to calculate steady stateprobabilities using an iterative algorithm. The resulting steady stateprobabilities can be used to calculate various performance measures, e.g. theaverage queue length. Additionally, computational issues of the implementationare discussed and an example from the field of telecommunication call-centerqueue length will be outlined to substantiate the applicability of theseefforts. In the appendix, tables of the average queueing length given aspecific number of service channels, traffic density, and system size arepresented.

An algorithm for calculating steady state probabilities of $M|E_r|c|K$ queueing systems