Mean sojourn times for phase-type discriminatory processor sharing systems

In a discriminatory processor sharing (DPS) queueing model, each job (or customer) belongs to one out of finitely many classes. The arrival processes are Poisson. Classes differ with respect to arrival rates and service time distributions. Moreover, classes have different priority levels. All jobs present are served simultaneously but the fraction of the server's capacity allocated to each one of them is proportional to their class priority parameter (while the total capacity is of course fixed). For the case of exponential service requirements, we find the mean sojourn times of various job classes, conditioning on the system's state, namely how many of each class are present. Next we derive the well-known system of linear equations solved by the unconditional values. We then switch to the much weaker assumption of phase-type service requirements, different across classes, where additionally the priority parameters may depend both on class and phase (and not only on class as assumed in existing papers). We determine mean sojourn times for a job belonging to any class, firstly conditional on the numbers of jobs from all classes present (which are shown to be affine functions) and secondly, unconditionally (as solutions of a system of linear equations).