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 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 occupancy distribution satisfies a functional equation being a Riemanu-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 of N (formula) 2 a numerical algorithm for computing the performance measures is presented, which is efficient for n (formula) 3. Since for n (formula) 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