TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System N2 - In this paper for the $M(n)/M(n)/s+GI$ system, i.e.\ for a $s$-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general $M(n)/M/s+GI$ system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct. T3 - ZIB-Report - 00-12 KW - M(n)/M(n)/s+GI system KW - limited waiting times KW - finite buffer KW - virtual waiting times KW - blocking- KW - abandon probability KW - impatience rates KW - asymptotic Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5801 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the Moments of the Overflow and Freed Carried Traffic for the GI/M/C/0 System N2 - In circuit switching networks call streams are characterized by their mean and peakedness (two-moment method). The $GI/M/C/0$ system is used to model a single link, where the $GI$-stream is determined by fitting moments appropriately. For the moments of the overflow traffic of a $GI/M/C/0$ system there are efficient numerical algorithms available. However, for the moments of the freed carried traffic, defined as the moments of a virtual link of infinite capacity to which the process of calls accepted by the link (carried arrival process) is virtually directed and where the virtual calls get fresh exponential i.i.d.\ holding times, only complex numerical algorithms are available. This is the reason why the concept of the freed carried traffic is not used rigorously. The main result of this paper is an efficient algorithm for computing the moments of the freed carried traffic, in particular an explicit formula for its peakedness. This result offers a unified handling of both overflow and carried traffics in networks. Furthermore, some refined characteristics for the overflow and freed carried streams are derived. T3 - ZIB-Report - 01-09 KW - GI/M/C/0 system KW - overflow traffic KW - freed carried traffic KW - carried arrival process KW - factorial moments KW - peakedness KW - two-moment method KW - difference Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6355 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Individual Overflow and Freed Carried Traffics for a Link with Trunk Reservation N2 - Two traffic streams $\Phi_1$, $\Phi_2$ are offered a link. The calls of $\Phi_i$ require exponential holding times with parameter $\mu$ and are accepted if less than $C_i$ trunks are occupied. Approximating the $\Phi_i$ by appropriate renewal processes meeting their first two moments, defined as the moments of the numbers of calls in virtual links of infinite capacity to which the traffic streams as freed traffics are virtually directed and where the calls get fresh exponential i.i.d.\ holding times with parameter $\mu$, stable recursive algorithms of complexity $O(\max(C_1,C_2))$ are derived for the first two defined as above moments of the individual overflow and freed carried traffics. The results offer a unified handling of both overflow and carried traffics in circuit switching networks with trunk reservation, providing a basis for new two-moment network dimensioning algorithms. T3 - ZIB-Report - 01-35 KW - trunk reservation KW - overflow traffic KW - freed carried traffic KW - factorial moments KW - peakedness KW - two-moment method KW - circuit switching network Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6615 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Insensitive bounds for the moments of the sojourn times in M/GI systems under state-dependent processor sharing JF - Advances in Applied Probability 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. Y1 - 2010 U6 - https://doi.org/10.1239/aap/1269611152 VL - 42 IS - 1 SP - 246 EP - 267 ER - TY - JOUR A1 - Brandt, Manfred A1 - Brandt, Andreas T1 - Additive functionals with application to sojourn times in infinite-server and processor sharing systems JF - Queueing Systems N2 - We deal with an infinite-server system where the service speed is governed by a stationary and ergodic process with countably many states. Applying a random time transformation such that the service speed becomes one, the sojourn time of a class of virtual requests with given required service time is equal in distribution to an additive functional defined via a stationary version of the time-changed process. Thus bounds for the expectation of functions of additive functionals yield bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the $GI(n)/GI(n)/\infty$ system or equivalently the $GI(n)/GI$ system under state-dependent processor sharing as an infinite-server system with random states given by the number $n$ of requests in the system provides results for sojourn times of virtual requests. In case of $M(n)/GI(n)/\infty$, the sojourn times of arriving and added requests are equal in distribution to sojourn times of virtual requests in modified systems, which yields many results for the sojourn times of arriving and added requests. In case of integer moments, the bounds generalize earlier results for $M/GI(n)/\infty$. In particular, the mean sojourn times of arriving and added requests in $M(n)/GI(n)/\infty$ are proportional to the required service time, generalizing Cohen's famous result for $M/GI(n)/\infty$. Y1 - 2012 U6 - https://doi.org/10.1007/s11134-012-9282-1 VL - 70 IS - 4 SP - 369 EP - 409 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Workload and busy period for M/GI/1 with a general impatience mechanism JF - Queueing Systems N2 - The paper deals with the workload and busy period for the M/GI/1 system under FCFS discipline, where the customers may become impatient during their waiting for service with generally distributed maximal waiting times and also during their service with generally distributed maximal service times depending on the time waited for service. This general impatience mechanism, originally introduced by Kovalenko (1961) and considered by Daley (1965), too, covers the special cases of impatience on waiting times as well as impatience on sojourn times, for which Boxma et al. (2010), (2011) gave new results and outlined special cases recently. Our unified approach bases on the vector process of workload and busy time. Explicit representations for the LSTs of workload and busy period are given in case of phase-type distributed impatience. Y1 - 2013 U6 - https://doi.org/10.1007/s11134-013-9373-7 VL - 75 IS - 2-4 SP - 189 EP - 209 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 - On the two-class $M/M/1$ system under preemptive resume and impatience of the prioritized customers JF - Queueing Syst. Y1 - 2004 VL - 47 SP - 147 EP - 168 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Asymptotic results and a Markovian approximation for the $M(n)/M(n)/s+GI$ system JF - Queueing Syst. Y1 - 2002 VL - 41 SP - 73 EP - 94 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 - On the moments of overflow and freed carried traffic for the $GI/M/C/0$ system JF - Methodol. Comput. Appl. Probab. Y1 - 2002 VL - 4 SP - 69 EP - 82 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, Manfred A1 - Brandt, Andreas T1 - On sojourn times in $M/GI$ systems under state-dependent processor sharing JF - Queueing Syst. Y1 - 2010 VL - 64 SP - 167 EP - 201 ER -