On the Sojourn Times for Many-Queue Head-of-the-Line Processor-Sharing Systems with Permanent Customers
Please always quote using this URN: urn:nbn:de:0297-zib-2005
- We consider a single server system consisting of $n$ queues with different types of customers (Poisson streams) and $k$ 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 times can be derived. In case of two queues and exponential service times, the generating function of the stationary 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 $n\ge 2$ a numerical algorithm for computing the performance measures is presented, which is efficient for $n=2,3$. Since for $n\ge 4$ 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 (complete) symmetric system is exact. The numerical and simulation results show that, over a wide range of parameters, the approximation works well.
Author: | Andreas Brandt, Manfred Brandt |
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Document Type: | ZIB-Report |
Date of first Publication: | 1995/11/30 |
Series (Serial Number): | ZIB-Report (SC-95-34) |
ZIB-Reportnumber: | SC-95-34 |
Published in: | Appeared in: Mathematical Methods of Operations Research 47 (1998) 181-220 |