Prof. Dr. Andreas Brandt

• Optimal Admission Control

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  The IP based transport of real time services like for example of speech and moving pictures in mobile networks is considered as an utmost promising application of third generation wireless networks (UMTS). An essential condition for offering these services in satisfactory quality is a functioning admission control. Aim of the admission control is to reject service requests if their acceptance could lead to an overload of the transport system. The following admission control strategy is a simple and effective overflow defense mechanism: A new arriving request is accepted by the network node if and only if the actual packet arrival rate - generated by the just active requests - is not greater than some threshold. However, also if this threshold is less than the processor capacity, buffer overflow may occur due to the on-off structure of the requests. Therefore token bucket algorithms are often used for preventing overload for the processors handling the packets. The probability of packet loss and the probability of a non accepted request depending on the threshold value are of interest.

• Processor Scheduling

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  <p>In telecommunication systems different processes consisting of requests have to be served. Various scheduling disciplines are applied for overload control, prioritizing the different processes, efficient use of the caches and for ensuring performance characteristics. The aim of this project is a performance analysis of a multi-processor system under a round robin discipline. In the round robin discipline the requests receive consecutively a fixed quantum of service in a cyclic manner. As limiting case, where the quanta tend to zero, the processor sharing discipline is obtained.</p><p>Firstly, we analyze a multi-queue multi-processor system where the requests at the head of the queues are served under processor sharing (head of the line multi-processor sharing). The stability of the system depends strongly on the input streams, as some processors may idle although work load is present in the system. For a general stationary input, necessary as well as sufficient stability conditions for the different queues and for the whole system are derived, which are also tight within this class.</p><p>Secondly, we analyze systems where requests are served under the state-dependent processor sharing discipline with Poisson arrivals, in particular with multi-processor system under processor sharing. For general service time distributions we found expressions for the Laplace-Stieltjes Transform of the conditional sojourn times (being the basis for numerical algorithms) as well as tight insensitive upper bounds for their moments. For special cases, including exponential as well as deterministic service times, representations for the moments of the sojourn times of the requests are derived. Thus higher moments of the sojourn time in a multi-processor system under processor sharing are computable in a non-phase-type model for the first time. By means of these results, approximations for the moments of the conditional sojourn times in case of generally distributed service times are given. In case of a two server system for particular service time distributions we were able to derive explicite expressions for the Laplace Stieltjes Transform and the variance of the conditional sojourn times.</p>

• Delay of IP Packets

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  <p>For ensuring a given Quality of Service in IP networks for the various packet streams arriving at a link, they are assigned to different queues. By prioritizing the queues or by an implementation of weighted Round Robin or weighted packet-wise Head of the Line Processor Sharing (weighted fair queueing: WFQ) a corresponding partition of the link capacity to the packet streams to be transported can be ensured.</p><p>Firstly, we analyze the delay of the IP packets of a single queue. The arrival process of the packets at the link is modeled by an Interrupted Poisson Process (IPP), where exponentially distributed on and off phases alternate. During the on phases there arrive packets of random length according to a Poisson process, during the off phases there is no packet arrival. The packets are served with constant speed (link rate) according to the FIFO discipline. From a mathematical point of view, the sojourn time in the queueing system IPP/GI/1 is analyzed.</p><p>Secondly, we analyze the delay of the IP packets of a marked queue under priority scheduling. For the marked queue priority scheduling means that its service is interrupted after a random duration U for a random duration D, where the service of a just served packet is finished. As an approximation of this mechanism, phase-type distributed random variables may be chosen for U and D. The arrival process is modeled by an IPP again, and the length of the packets is modeled by a phase-type distribution. From a mathematical point of view, the sojourn time in the queueing system IPP/PH/1 with a random environment is analyzed.</p><p>Thirdly, we deal with the problem of an efficient modelling of the superposition process of different IPP's and the splitting of an IPP in different IPP's, which is crucial in deriving algorithms for computing performance measures in IP networks.</p>

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