TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - A Note on the Stability of the Many-Queue Head-of-the Line Processor-Sharing System with Permanent Customers N2 - We consider a single server system consisting of $n$ queues with different types of customers and $k$ permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). By means of Loynes' monotonicity method a stationary work load process is constructed and using sample path analysis general stability conditions are derived. They allow to decide which queues are stable and moreover to compute the fraction of processor capacity devoted to the permanent customers. In case of a stable system the constructed stationary state process is the only one and for any initial state the system converges pathwise to the steady state. T3 - ZIB-Report - SC-96-55 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2659 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - A note on the stability of the many-queue head-of-the-line processor-sharing system with permanent customers JF - Queueing Syst. Y1 - 1999 VL - 32 SP - 363 EP - 381 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 - 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 - 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 - CHAP A1 - Brandt, Andreas A1 - Brandt, Manfred A1 - Rugel, S. A1 - Weber, D. T1 - Admission control for realtime traffic T2 - Proc. IEEE Wireless Communications & Networking Conference WCNC 2005 Y1 - 2005 CY - New Orleans, USA ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Approximation for overflow moments of a multiservice link with trunk reservation JF - Perform. Eval. Y1 - 2001 VL - 43 SP - 259 EP - 268 ER - TY - GEN A1 - Brandt, Manfred T1 - Approximations for the Distribution Function of the Sum of iid Random Variables with Compact Support in R+ N2 - 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. T3 - ZIB-Report - SC-95-33 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1990 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Approximations for the second moments of sojourn times in M/GI systems under state-dependent processor sharing N2 - 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. T3 - ZIB-Report - 09-25 KW - state-dependent processor sharing KW - M/GI/m-PS KW - sojourn time KW - moments KW - approximations Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11414 SN - 1438-0064 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 - 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 - 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 - Individual overflow and freed carried traffics for a link with trunk reservation JF - Telecommunication Syst. Y1 - 2005 VL - 29 SP - 283 EP - 308 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 - Adv. Appl. Probab. Y1 - 2010 VL - 42 SP - 246 EP - 267 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 - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - Insensitivity bounds for the moments of the sojourn times in M/GI systems under state-dependent processor sharing 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. T3 - ZIB-Report - 09-02 KW - state-dependent processor sharing KW - sojourn time KW - moments KW - bounds KW - M/GI/m-PS Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11092 SN - 1438-0064 ER - TY - CHAP A1 - Brandt, Andreas A1 - Brandt, Manfred A1 - Spahl, G. A1 - Weber, D. T1 - Modelling and optimization of call distribution systems T2 - Proc. 15th Int. Teletraffic Cong. (ITC 15) Y1 - 1997 SP - 133 EP - 144 CY - Washington D.C., USA ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On a Two-Queue Priority System with Impatience and its Application to a Call Center N2 - We consider a $s$-server system with two FCFS queues, where the arrival rates at the queues and the service rate may depend on the number $n$ of customers being in service or in the first queue, but the service rate is assumed to be constant for $n>s$. The customers in the first queue are impatient. If the offered waiting time exceeds a random maximal waiting time $I$, then the customer leaves the first queue after time $I$. If $I$ is less than a given deterministic time then he leaves the system else he transits to the end of the second queue. The customers in the first queue have priority. The service of a customer from the second queue will be started if the first queue is empty and more than a given number of servers become idle. For the model being a generalization of the $M(n)/M(n)/s\!+\!GI$ system balance conditions for the density of the stationary state process are derived yielding the stability conditions and the probabilities that precisely $n$ customers are in service or in the first queue. For obtaining performance measures for the second queue a system approximation basing on fitting impatience intensities is constructed. The results are applied to the performance analysis of a call center with an integrated voice-mail-server. For an important special case a stochastic decomposition is derived illuminating the connection to the dynamics of the $M(n)/M(n)/s\!+\!GI$ system. T3 - ZIB-Report - SC-98-21 KW - two queues KW - many server KW - server reservation KW - impatience KW - occupancy distribution KW - waiting time distribution KW - approximate system KW - $M(n)/M(n)/s+GI Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3641 ER - TY - JOUR A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On a two-queue priority system with impatience and its application to a call center JF - Methodol. Comput. Appl. Probab. Y1 - 1999 VL - 1 SP - 191 EP - 210 ER - TY - GEN A1 - Manfred, Brandt A1 - Andreas, Brandt T1 - On sojourn times for an infinite-server system in random environment and its application to processor sharing 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 $M/GI(n)/\infty$. T3 - ZIB-Report - 11-28 KW - infinite-server KW - random environment KW - time transformation KW - sojourn times KW - state-dependent processor sharing Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13190 ER -