TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the Two-Class M/M/1 System under Preemptive Resume and Impatience of the Prioritized Customers N2 - The paper deals with the two-class priority M/M/1 system, where the prioritized class-1 customers are served under FCFS preemptive resume discipline and may become impatient during their waiting for service with generally distributed maximal waiting times but finite expectation. The class-2 customers have no impatience. The required mean service times may depend on the class of the customer. As the dynamics of class-1 customers are related to the well analyzed M/M/1+GI system, our aim is to derive characteristics for class-2 customers and for the whole system. The solution of the balance equations for the partial probability generating functions of the detailed system state process is given in terms of the weak solution of a family of boundary value problems for ordinary differential equations. By means of this solution formulae for the joint occupancy distribution and for the sojourn and waiting times of class-2 customers are derived generalizing results recently obtained by Choi et al. in case of deterministic maximal waiting times. For deterministic maximal waiting times partially new explicit formulae are given. T3 - ZIB-Report - 02-28 KW - two-class priority M/M/1 system KW - preemptive resume KW - impatient customers KW - occupancy distribution KW - waiting time KW - sojourn time Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6953 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 - 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 -