On the sojourn times for many-queue head-of-the-line Processor-sharing systems with permanent customers

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.

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