TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the Sojourn Times for Many-Queue Head-of-the-Line Processor-Sharing Systems with Permanent Customers N2 - 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. T3 - ZIB-Report - SC-95-34 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2005 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - A sample path relation for the sojourn times in G/G/1-PS systems and its applications N2 - For the general G/G/1 processor sharing (PS) system a sample path result for the sojourn times in a busy period is proved, which yields a relation between the sojourn times under PS and FCFS discipline. In particular, the result provides a formula for the mean sojourn time in G/D/1-PS in terms of the mean sojourn time in the corresponding G/D/1-FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of the formula provide the basis for a two-moment approximation of the mean sojourn time in G/GI/1-PS in terms of a related G/D/1-FCFS. T3 - ZIB-Report - 03-18 KW - G/G/1 KW - G/GI/1 KW - G/D/1 KW - G/M/1 KW - processor sharing KW - sojourn time KW - waiting time KW - busy period KW - sample path KW - first come first served KW - two-moment ap Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7401 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the stability of the multi-queue multi-server processor sharing with limited service JF - Queueing Syst. Y1 - 2007 VL - 56 SP - 1 EP - 8 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the sojourn times for many-queue head-of-the-line processor-sharing systems with permanent customers JF - Math. Methods Oper. Res. Y1 - 1998 VL - 47 SP - 181 EP - 220 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Waiting times for $M/M$ systems under state-dependent processor sharing JF - Queueing Syst. Y1 - 2008 VL - 59 SP - 297 EP - 319 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - A sample path relation for the sojourn times in $G/G/1-PS$ systems and its applications JF - Queueing Syst. Y1 - 2006 VL - 52 SP - 281 EP - 286 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - A note on the stability of the many-queue head-of-the-line processor-sharing system with permanent customers JF - Queueing Syst. Y1 - 1999 VL - 32 SP - 363 EP - 381 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Insensitivity bounds for the moments of the sojourn times in M/GI systems under state-dependent processor sharing N2 - We consider a system with Poisson arrivals and i.i.d. service times and where the requests are served according to the state-dependent (Cohen's generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive asymptotically tight upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize corresponding results, recently given for the single-server processor sharing system by Cheung et al. and for the state-dependent processor sharing system with exponential service times by the authors. Analogous results hold for the waiting times. T3 - ZIB-Report - 09-02 KW - state-dependent processor sharing KW - sojourn time KW - moments KW - bounds KW - M/GI/m-PS Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11092 SN - 1438-0064 ER -