TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Waiting times for M/M systems under generalized processor sharing N2 - We consider a system where the arrivals form a Poisson process and the required service times of the requests are exponentially distributed. According to the generalized processor sharing discipline, each request in the system receives a fraction of the capacity of one processor which depends on the actual number of requests in the system. We derive systems of ordinary differential equations for the LST and for the moments of the conditional waiting time of a request with given required service time as well as a stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time, which in particular yields the second moment of the unconditional waiting time. Moreover, asymptotically tight upper bounds for the moments of the conditional waiting time are given. The presented numerical results for the first two moments of the sojourn times in the $M/M/m-PS$ system show that the proposed algorithms work well. T3 - ZIB-Report - 06-46 KW - Poisson arrivals KW - exponential service times KW - generalized processor sharing KW - many-server KW - permanent customers KW - M/M/m-PS KW - waiting time KW - sojourn t Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9404 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the stability of the multi-queue multi-server processor sharing with limited service N2 - We consider a multi-queue multi-server system with $n$ servers (processors) and $m$ queues. At the system there arrives a stationary and ergodic stream of $m$ different types of requests with service requirements which are served according to the following $k$-limited head of the line processor sharing discipline: The first $k$ requests at the head of the $m$ queues are served in processor sharing by the $n$ processors, where each request may receive at most the capacity of one processor. By means of sample path analysis and Loynes' monotonicity method, a stationary and ergodic state process is constructed, and a necessary as well as a sufficient condition for the stability of the $m$ separate queues are given, which are tight within the class of all stationary ergodic inputs. These conditions lead to tight necessary and sufficient conditions for the whole system, also in case of permanent customers, generalizing an earlier result by the authors for the case of $n$=$k$=1. T3 - ZIB-Report - 06-25 KW - head of the line processor sharing KW - thread pools KW - many queues KW - many servers KW - general input KW - batch arrivals KW - permanent customers KW - marked point p Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9199 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 -