@misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the M(n)/M(m)/s Queue with impatient Calls}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2885}, number = {SC-97-19}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the Sojourn Times for Many-Queue Head-of-the-Line Processor-Sharing Systems with Permanent Customers}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2005}, number = {SC-95-34}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On a Two-Queue Priority System with Impatience and its Application to a Call Center}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3641}, number = {SC-98-21}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, 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}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2659}, number = {SC-96-55}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, 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}, 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{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} } @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{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {Workload and busy period for M/GI/1 with a general impatience mechanism}, series = {Queueing Systems}, volume = {75}, journal = {Queueing Systems}, number = {2-4}, doi = {10.1007/s11134-013-9373-7}, pages = {189 -- 209}, 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{BrandtBrandt, 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 = {Advances in Applied Probability}, volume = {42}, journal = {Advances in Applied Probability}, number = {1}, doi = {10.1239/aap/1269611152}, pages = {246 -- 267}, 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{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}, series = {Queueing Syst.}, volume = {32}, journal = {Queueing Syst.}, pages = {363 -- 381}, year = {1999}, language = {en} }