Analysis of queueing model with processor sharing discipline and customers impatience

Queueing systems with processor sharing represent the adequate models for sharing the resources, e.g., components of a computer or a bandwidth of communication systems. In this paper, we consider a queueing system with processor sharing discipline under quite general assumptions about the arrival and service processes. Arrivals are defined by the Markovian arrival process. The service time has a phase type distribution. Possible impatience of customers is taken into account. The number of customers, which can simultaneously obtain service, is limited. We compare two approaches for monitoring service of customers, namely, the approach counting the number of customers at each phase of service and the approach counting the phase of service of each customer and show the significant advantage of the former approach. We obtain the joint distribution of the number of customers in the system and the states of the underlying arrival and service processes as well as the loss probabilities. It is shown that the sojourn time in the system of an arbitrary customer has phase type distribution and an irreducible representation of this distribution is obtained. Numerical examples are presented. A possibility of optimal choice of the server capacity (e.g., multi-programming level) is numerically illustrated. An opportunity of increasing the speed of computations via the use of the graphics processing unit is discussed.