Analysis of generalized processor-sharing systems with two classes of customers and exponential services

Abstract

We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann–Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).

Numerical Key Results for Studying the Performance of TCP Traffic under Class-Based Weighted Fair Queuing System

“…Such CBWFQ system is mostly treated under a decoupling approach, where the system is approximated by two servers [10,13,14]. In Refs.…”

Power series approximations for two-class generalized processor sharing systems

“…Such integrals provide efficient algorithms for computing performance characteristics by numerical integration. This approach was taken in [8,12,15]. We choose not the pursue this approach here, as it requires a construction based on the analytic continuations of the functions K 10 and K 01 , and a detailed study of the involved conformal mappings, which is rather involved.…”

Performance Evaluation of Weighted Fair Queuing System Using Matrix Geometric Method

“…The subject of WFQ has been investigated by many authors. Models similar to our WFQ system with two classes of customers have been analyzed for Poisson arrivals and exponential service times but with infinite buffers [4] and [5]. In [4], a two class system with two queues is considered; however the WFQ system is approximated with a two server, two queue systems and a numerical solution is provided.…”

“…In [4], a two class system with two queues is considered; however the WFQ system is approximated with a two server, two queue systems and a numerical solution is provided. In [5], the same problem in [4] is considered; however an analytical solution for the system is given.…”

“…There were many solutions proposed to offer the solution for the WFQ system with two classes of customers on infinite buffer in [4] [5] and [6]. However, the class based with finite buffer is applied in a lot of computer and communication system and more realistic.…”