@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}, series = {Adv. Appl. Probab.}, volume = {42}, journal = {Adv. Appl. Probab.}, pages = {246 -- 267}, year = {2010}, language = {en} } @inproceedings{BrandtBrandtRugeletal.2005, author = {Brandt, Andreas and Brandt, Manfred and Rugel, S. and Weber, D.}, title = {Admission control for realtime traffic}, series = {Proc. IEEE Wireless Communications \& Networking Conference WCNC 2005}, booktitle = {Proc. IEEE Wireless Communications \& Networking Conference WCNC 2005}, address = {New Orleans, USA}, year = {2005}, 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}, series = {Proc. 15th Int. Teletraffic Cong. (ITC 15)}, booktitle = {Proc. 15th Int. Teletraffic Cong. (ITC 15)}, address = {Washington D.C., USA}, pages = {133 -- 144}, year = {1997}, language = {en} } @misc{Brandt, 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}, 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} } @misc{BrandtBrandt, 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}, 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} } @article{BrandtBrandt, author = {Brandt, Manfred and Brandt, Andreas}, title = {Additive functionals with application to sojourn times in infinite-server and processor sharing systems}, series = {Queueing Systems}, volume = {70}, journal = {Queueing Systems}, number = {4}, doi = {10.1007/s11134-012-9282-1}, pages = {369 -- 409}, 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{BrandtBrandt2010, author = {Brandt, Manfred and Brandt, Andreas}, title = {On sojourn times in \$M/GI\$ systems under state-dependent processor sharing}, series = {Queueing Syst.}, volume = {64}, journal = {Queueing Syst.}, pages = {167 -- 201}, year = {2010}, language = {en} } @misc{ManfredAndreas, 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}, 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} }