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 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 - 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 - 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 - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the stability of the multi-queue multi-server processor sharing with limited service N2 - We consider a multi-queue multi-server system with $n$ servers (processors) and $m$ queues. At the system there arrives a stationary and ergodic stream of $m$ different types of requests with service requirements which are served according to the following $k$-limited head of the line processor sharing discipline: The first $k$ requests at the head of the $m$ queues are served in processor sharing by the $n$ processors, where each request may receive at most the capacity of one processor. By means of sample path analysis and Loynes' monotonicity method, a stationary and ergodic state process is constructed, and a necessary as well as a sufficient condition for the stability of the $m$ separate queues are given, which are tight within the class of all stationary ergodic inputs. These conditions lead to tight necessary and sufficient conditions for the whole system, also in case of permanent customers, generalizing an earlier result by the authors for the case of $n$=$k$=1. T3 - ZIB-Report - 06-25 KW - head of the line processor sharing KW - thread pools KW - many queues KW - many servers KW - general input KW - batch arrivals KW - permanent customers KW - marked point p Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9199 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the M(n)/M(m)/s Queue with impatient Calls N2 - The paper is concerned with the analysis of an $s$ server queueing system in which the calls become impatient and leave the system if their waiting time exceeds their own patience. The individual patience times are assumed to be i.i.d.\ and arbitrary distributed. The arrival and service rate may depend on the number of calls in the system and in service, respectively. For this system, denoted by $M(n)/M(m)/s+GI$, where $m=\min(n,s)$ is the number of busy servers in the system, we derive a system of integral equations for the vector of the residual patience times of the waiting calls and their original maximal patience times. By solving these equations explicitly we get the stability condition and, for the steady state of the system, the occupancy distribution and various waiting time distributions. As an application of the \mbox{$M(n)/M(m)/s+GI$} system we give a performance analysis of an Automatic Call Distributor system (ACD system) of finite capacity with outbound calls and impatient inbound calls, especially in case of patience times being the minimum of constant and exponentially distributed times. T3 - ZIB-Report - SC-97-19 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2885 ER - TY - GEN A1 - Brandt, Andreas A1 - Brandt, Manfred T1 - On the Sojourn Times for Many-Queue Head-of-the-Line Processor-Sharing Systems with Permanent Customers N2 - We consider a single server system consisting of $n$ queues with different types of customers (Poisson streams) 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). The stability condition and a pseudo work conservation law will be given for arbitrary service time distributions; for exponential service times a pseudo conservation law for the mean sojourn times can be derived. In case of two queues and exponential service times, the generating function of the stationary distribution satisfies a functional equation being a Riemann-Hilbert problem which can be reduced to a Dirichlet problem for a circle. The solution yields the mean sojourn times as an elliptic integral, which can be computed numerically very efficiently. In case $n\ge 2$ a numerical algorithm for computing the performance measures is presented, which is efficient for $n=2,3$. Since for $n\ge 4$ an exact analytical or/and numerical treatment is too complex a heuristic approximation for the mean sojourn times of the different types of customers is given, which in case of a (complete) symmetric system is exact. The numerical and simulation results show that, over a wide range of parameters, the approximation works well. T3 - ZIB-Report - SC-95-34 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2005 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 -