@misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5801}, number = {00-12}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the Two-Class M/M/1 System under Preemptive Resume and Impatience of the Prioritized Customers}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6953}, number = {02-28}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the Moments of the Overflow and Freed Carried Traffic for the GI/M/C/0 System}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6355}, number = {01-09}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, 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}, 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} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {A sample path relation for the sojourn times in G/G/1-PS systems and its applications}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7401}, number = {03-18}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, 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}, 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{BrandtBrandt, 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}, 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{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {Waiting times for M/M systems under generalized processor sharing}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9404}, number = {06-46}, abstract = {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.}, language = {en} } @misc{BrandtBrandt, author = {Brandt, Andreas and Brandt, Manfred}, title = {On the stability of the multi-queue multi-server processor sharing with limited service}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9199}, number = {06-25}, abstract = {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.}, language = {en} } @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} }