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Analysis of generalized processor-sharing systems with two classes of customers and exponential services. (English) Zbl 1065.60132
Summary: We derive 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).

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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