On the influence of high priority customers on a generalized processor sharing queue

In this paper, we study a hybrid scheduling mechanism in discrete-time. This mechanism combines the well-known Generalized Processor Sharing (GPS) scheduling with strict priority. We assume three customer classes with one class having strict priority over the other classes, whereby each customer requires a single slot of service. The latter share the remaining bandwith according to GPS. This kind of scheduling is used in practice for the scheduling of jobs on a processor and in Quality of Service modules of telecommunication network devices. First, we derive a functional equation of the joint probability generating function of the queue contents. To explicitly solve the functional equation, we introduce a power series in the weight parameter of GPS. Subsequently, an iterative procedure is presented to calculate consecutive coefficients of the power series. Lastly, the approximation resulting from a truncation of the power series is verified with simulation results. We also propose rational approximations. We argue that the approximation performs well and is extremely suited to study these systems and their sensitivity in their parameters (scheduling weights, arrival rates, loads ...). This method provides a fast way to observe the behaviour of such type of systems avoiding time-consuming simulations.In this paper, we study a hybrid scheduling mechanism in discrete-time. This mechanism combines the well-known Generalized Processor Sharing (GPS) scheduling with strict priority. We assume three customer classes with one class having strict priority over the other classes, whereby each customer requires a single slot of service. The latter share the remaining bandwith according to GPS. This kind of scheduling is used in practice for the scheduling of jobs on a processor and in Quality of Service modules of telecommunication network devices. First, we derive a functional equation of the joint probability generating function of the queue contents. To explicitly solve the functional equation, we introduce a power series in the weight parameter of GPS. Subsequently, an iterative procedure is presented to calculate consecutive coefficients of the power series. Lastly, the approximation resulting from a truncation of the power series is verified with simulation results. We also propose rational approximations. We argue that the approximation performs well and is extremely suited to study these systems and their sensitivity in their parameters (scheduling weights, arrival rates, loads ...). This method provides a fast way to observe the behaviour of such type of systems avoiding time-consuming simulations.P