@misc{BrandtBrandt2001, author = {Brandt, Andreas and Brandt, Manfred}, title = {Individual Overflow and Freed Carried Traffics for a Link with Trunk Reservation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6615}, number = {01-35}, year = {2001}, abstract = {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.}, language = {en} } @article{BrandtBrandt2010, author = {Brandt, Andreas and Brandt, Manfred}, title = {Insensitive bounds for the moments of the sojourn times in M/GI systems under state-dependent processor sharing}, volume = {42}, journal = {Advances in Applied Probability}, number = {1}, doi = {10.1239/aap/1269611152}, pages = {246 -- 267}, year = {2010}, abstract = {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.}, language = {en} } @article{BrandtBrandt2012, author = {Brandt, Manfred and Brandt, Andreas}, title = {Additive functionals with application to sojourn times in infinite-server and processor sharing systems}, volume = {70}, journal = {Queueing Systems}, number = {4}, doi = {10.1007/s11134-012-9282-1}, pages = {369 -- 409}, year = {2012}, abstract = {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\$.}, language = {en} } @article{BrandtBrandt2013, author = {Brandt, Andreas and Brandt, Manfred}, title = {Workload and busy period for M/GI/1 with a general impatience mechanism}, volume = {75}, journal = {Queueing Systems}, number = {2-4}, doi = {10.1007/s11134-013-9373-7}, pages = {189 -- 209}, year = {2013}, abstract = {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.}, language = {en} } @article{BrandtBrandt2007, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the stability of the multi-queue multi-server processor sharing with limited service}, volume = {56}, journal = {Queueing Syst.}, pages = {1 -- 8}, year = {2007}, language = {en} } @article{BrandtBrandt1998, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the sojourn times for many-queue head-of-the-line processor-sharing systems with permanent customers}, volume = {47}, journal = {Math. Methods Oper. Res.}, pages = {181 -- 220}, year = {1998}, language = {en} } @article{BrandtBrandt2004, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the two-class \$M/M/1\$ system under preemptive resume and impatience of the prioritized customers}, volume = {47}, journal = {Queueing Syst.}, pages = {147 -- 168}, year = {2004}, language = {en} } @article{BrandtBrandt2002, author = {Brandt, Andreas and Brandt, Manfred}, title = {Asymptotic results and a Markovian approximation for the \$M(n)/M(n)/s+GI\$ system}, volume = {41}, journal = {Queueing Syst.}, pages = {73 -- 94}, year = {2002}, language = {en} } @article{BrandtBrandt2008, author = {Brandt, Andreas and Brandt, Manfred}, title = {Waiting times for \$M/M\$ systems under state-dependent processor sharing}, volume = {59}, journal = {Queueing Syst.}, pages = {297 -- 319}, year = {2008}, language = {en} } @article{BrandtBrandt2002, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the moments of overflow and freed carried traffic for the \$GI/M/C/0\$ system}, volume = {4}, journal = {Methodol. Comput. Appl. Probab.}, pages = {69 -- 82}, year = {2002}, language = {en} } @article{BrandtBrandt2006, author = {Brandt, Andreas and Brandt, Manfred}, title = {A sample path relation for the sojourn times in \$G/G/1-PS\$ systems and its applications}, volume = {52}, journal = {Queueing Syst.}, pages = {281 -- 286}, year = {2006}, language = {en} } @article{BrandtBrandt2010, author = {Brandt, Manfred and Brandt, Andreas}, title = {On sojourn times in \$M/GI\$ systems under state-dependent processor sharing}, volume = {64}, journal = {Queueing Syst.}, pages = {167 -- 201}, year = {2010}, language = {en} } @article{BrandtBrandt2010, author = {Brandt, Andreas and Brandt, Manfred}, title = {Insensitive bounds for the moments of the sojourn times in \$M/GI\$ systems under state-dependent processor sharing}, volume = {42}, journal = {Adv. Appl. Probab.}, pages = {246 -- 267}, year = {2010}, language = {en} } @article{BrandtBrandt2005, author = {Brandt, Andreas and Brandt, Manfred}, title = {Individual overflow and freed carried traffics for a link with trunk reservation}, volume = {29}, journal = {Telecommunication Syst.}, pages = {283 -- 308}, year = {2005}, language = {en} } @inproceedings{BrandtBrandt1999, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the computation of the probability of a system failure}, booktitle = {Proc. ESREL'99 10th European Conf. on Safety and Reliability, G.I. Schu{\"e}ller, P. Kafka (eds.), Balkema Rotterdam}, address = {M{\"u}nchen-Garching}, pages = {457 -- 461}, year = {1999}, language = {en} } @inproceedings{BrandtBrandtSpahletal.1997, author = {Brandt, Andreas and Brandt, Manfred and Spahl, G. and Weber, D.}, title = {Modelling and optimization of call distribution systems}, booktitle = {Proc. 15th Int. Teletraffic Cong. (ITC 15)}, address = {Washington D.C., USA}, pages = {133 -- 144}, year = {1997}, language = {en} } @article{BrandtBrandt2001, author = {Brandt, Andreas and Brandt, Manfred}, title = {Approximation for overflow moments of a multiservice link with trunk reservation}, volume = {43}, journal = {Perform. Eval.}, pages = {259 -- 268}, year = {2001}, language = {en} } @article{BrandtBrandt1999, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the \$M(n)/M(n)/s\$ queue with impatient calls}, volume = {35}, journal = {Perform. Eval.}, pages = {1 -- 18}, year = {1999}, language = {en} } @misc{BrandtBrandt2011, author = {Brandt, Andreas and Brandt, Manfred}, title = {Workload and busy period for M/GI/1 with a general impatience mechanism}, issn = {1438-0064}, doi = {10.1007/s11134-013-9373-7}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14304}, number = {11-43}, year = {2011}, abstract = {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.}, language = {en} } @misc{ManfredAndreas2011, author = {Manfred, Brandt and Andreas, Brandt}, title = {On sojourn times for an infinite-server system in random environment and its application to processor sharing systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13190}, number = {11-28}, year = {2011}, abstract = {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\$.}, language = {en} } @inproceedings{BrandtBrandtRugeletal.2005, author = {Brandt, Andreas and Brandt, Manfred and Rugel, S. and Weber, D.}, title = {Admission control for realtime traffic}, booktitle = {Proc. IEEE Wireless Communications \& Networking Conference WCNC 2005}, address = {New Orleans, USA}, year = {2005}, language = {en} } @article{BrandtBrandt1999, author = {Brandt, Andreas and Brandt, Manfred}, title = {A note on the stability of the many-queue head-of-the-line processor-sharing system with permanent customers}, volume = {32}, journal = {Queueing Syst.}, pages = {363 -- 381}, year = {1999}, language = {en} } @article{BrandtBrandt1999, author = {Brandt, Andreas and Brandt, Manfred}, title = {On a two-queue priority system with impatience and its application to a call center}, volume = {1}, journal = {Methodol. Comput. Appl. Probab.}, pages = {191 -- 210}, year = {1999}, language = {en} } @article{BrandtBrandt1994, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the distribution of the number of packets in the fluid flow approximation of packet arrival streams}, volume = {17}, journal = {Queueing Syst.}, pages = {275 -- 315}, year = {1994}, language = {en} } @misc{BrandtBrandt2009, author = {Brandt, Manfred and Brandt, Andreas}, title = {On sojourn times in M/GI systems under state-dependent processor sharing}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11366}, number = {09-22}, year = {2009}, abstract = {We consider a system with Poisson arrivals and i.i.d. service times. The requests are served according to the state-dependent processor sharing discipline, where each request receives a service capacity which depends on the actual number of requests in the system. The linear systems of PDEs describing the residual and attained sojourn times coincide for this system, which provides time reversibility including sojourn times for this system, and their minimal non negative solution gives the LST of the sojourn time \$V(\tau)\$ of a request with required service time \$\tau\$. For the case that the service time distribution is exponential in a neighborhood of zero, we derive a linear system of ODEs, whose minimal non negative solution gives the LST of \$V(\tau)\$, and which yields linear systems of ODEs for the moments of \$V(\tau)\$ in the considered neighborhood of zero. Numerical results are presented for the variance of \$V(\tau)\$. In case of an M/GI/2-PS system, the LST of \$V(\tau)\$ is given in terms of the solution of a convolution equation in the considered neighborhood of zero. For bounded from below service times, surprisingly simple expressions for the LST and variance of \$V(\tau)\$ in this neighborhood of zero are derived, which yield in particular the LST and variance of \$V(\tau)\$ in M/D/2-PS.}, language = {en} } @misc{BrandtBrandt2009, author = {Brandt, Andreas and Brandt, Manfred}, title = {Approximations for the second moments of sojourn times in M/GI systems under state-dependent processor sharing}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11414}, number = {09-25}, year = {2009}, abstract = {We consider a system with Poisson arrivals and general service times, where the requests are served according to the State-Dependent Processor Sharing (SDPS) discipline (Cohen's generalized processor sharing discipline), where each request receives a service capacity which depends on the actual number of requests in the system. For this system, denoted by \$M/GI/SDPS\$, we derive approximations for the squared coefficients of variation of the conditional sojourn time of a request given its service time and of the unconditional sojourn time by means of two-moment fittings of the service times. The approximations are given in terms of the squared coefficients of variation of the conditional and unconditional sojourn time in related \$M/D/SDPS\$ and \$M/M/SDPS\$ systems, respectively. The numerical results presented for \$M/GI/m-PS\$ systems illustrate that the proposed approximations work well.}, language = {en} } @misc{BrandtBrandt2009, author = {Brandt, Andreas and Brandt, Manfred}, title = {Insensitivity bounds for the moments of the sojourn times in M/GI systems under state-dependent processor sharing}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11092}, number = {09-02}, year = {2009}, abstract = {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.}, language = {en} } @misc{Brandt1995, author = {Brandt, Manfred}, title = {Approximations for the Distribution Function of the Sum of iid Random Variables with Compact Support in R+}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1990}, number = {SC-95-33}, year = {1995}, abstract = {In this paper a unified approach to central and decentral approximations of the distribution function \$F(x,n)\$ of the sum of \$n\$ iid random variables with compact support in \$I\!\!R_+\$ is given. This approach yields direct Edgeworth expansion (especially the Central limit theorem) and indirect Edgeworth expansion (Theorem of Bahadur-Rao, large deviation results) within a unified framework. An approximative inversion of the LST of \$F(x,n)\$ (approximation of the complex inversion integral over a line by an integral over a proper bounded arc with a proper integrand) allows to get these approximations and moreover explicit error bounds.}, language = {en} }