On the Influence of High Priority Customers on a Generalized Processor Sharing Queue

Abstract

Abstract. 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.

Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

“…1. We extend the class of stochastic processes in which PSA method (see [43,44,45,46]) can be applied, to the case of Markov-modulated reflected random walks in the quarter plane with a sort of coupling (see also [9,10] for some initial work on this direction). We obtain power series expansions of the pgf of the joint stationary distribution for any state of the phase process.…”

Stationary analysis of a tandem queue with coupled processors subject to global breakdowns

“…In the latter one, power series expansions of steady-state probabilities as functions of a certain parameter of the system (usually the load) were derived. Recently, the authors in [39] studied generalized processor sharing queues and introduced an alternative method, by providing power series expansions of the generating function; see also [36,35,15].…”

“…Fundamental contribution Based on the generating function approach, we apply two different methods to investigate the stationary behaviour of the underline Markov modulated random walk in the quarter plane. First, we apply the power series approximation method, initially introduced in [39] (see also [11,35,36]), for two-parallel generalized processor sharing queues, described by a typical random walk in the quarter plane (RWQP). In this work we show that this method is still valid for related Markov modulated RWQP, and thus, extend the class of models that can be applied.…”