On the sojourn times for many-queue head-of-the-line Processor-sharing systems with permanent customers |
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Authors: | Andreas Brandt Manfred Brandt |
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Affiliation: | (1) Wirtschaftswissenschaftliche Fakultät, Humboldt-Universität zu Berlin, Spandauer Str. 1, D-10178 Berlin, Germany;(2) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, D-14195 Berlin, Germany |
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Abstract: | We consider a single server system consisting of e queues with different types of customers (Poisson streams) andk permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). The stability condition and a pseudo work conservation law will be given for arbitrary service time distributions; for exponential service times a pseudo conservation law for the mean sojourn tunes can be derived. In case of two queues and exponential service times, the generating function of the stationary occupancy distribution satisfies a functional equation being a Riemann-Hilbert problem which can be reduced to a Dirichlet problem for a circle. The solution yields the mean sojourn times as an elliptic integral, which can be computed numerically very efficiently. In case ofn 2 a numerical algorithm for computing the performance measures is presented, which is efficient forn 3. Since forn 4 an exact analytical or/and numerical treatment is too complex a heuristic approximation for the mean sojourn times of the different types of customers is given, which in case of a (completely) symmetric system is exact. The numerical and simulation results show that, over a wide range of parameters, the approximation works well.This work was supported by a grant from the Siemens AG. |
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Keywords: | head-of-the-line processor-sharing many queues permanent customers sojourn times pseudo conservation law Riemann-Hilbert problem Dirichlet problem |
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