Closed-form waiting time approximations for polling systems

A typical polling system consists of a number of queues, attended by a single server in a fixed order. The present study derives closed-form approximations for the mean waiting times and mean marginal queue lengths of polling systems with renewal arrival processes, which can be computed by simple calculations. The results of the present research may be very suitable for the design and optimisation phase in many application areas, such as telecommunication, maintenance, manufacturing and transportation

Pooling and polling : creation of pooling in inventory and queueing models

The subject of the present monograph is the ‘Creation of Pooling in Inventory and Queueing Models’. This research consists of the study of sharing a scarce resource (such as inventory, server capacity, or production capacity) between multiple customer classes. This is called pooling, where the goal is to achieve cost or waiting time reductions. For the queueing and inventory models studied, both theoretical, scientific insights, are generated, as well as strategies which are applicable in practice. This monograph consists of two parts: pooling and polling. In both research streams, a scarce resource (inventory or server capacity, respectively production capacity) has to be shared between multiple users. In the first part of the thesis, pooling is applied to multi-location inventory models. It is studied how cost reduction can be achieved by the use of stock transfers between local warehouses, so-called lateral transshipments. In this way, stock is pooled between the warehouses. The setting is motivated by a spare parts inventory network, where critical components of technically advanced machines are kept on stock, to reduce down time durations. We create insights into the question when lateral transshipments lead to cost reductions, by studying several models. Firstly, a system with two stock points is studied, for which we completely characterize the structure of the optimal policy, using dynamic programming. For this, we formulate the model as a Markov decision process. We also derived conditions under which simple, easy to implement, policies are always optimal, such as a hold back policy and a complete pooling policy. Furthermore, we identified the parameter settings under which cost savings can be achieved. Secondly, we characterize the optimal policy structure for a multi-location model where only one stock point issues lateral transshipments, a so-called quick response warehouse. Thirdly, we apply the insights generated to the general multi-location model with lateral transshipments. We propose the use of a hold back policy, and construct a new approximation algorithm for deriving the performance characteristics. It is based on the use of interrupted Poisson processes. The algorithm is shown to be very accurate, and can be used for the optimization of the hold back levels, the parameters of this class of policies. Also, we study related inventory models, where a single stock point servers multiple customers classes. Furthermore, the pooling of server capacity is studied. For a two queue model where the head-of-line processor sharing discipline is applied, we derive the optimal control policy for dividing the servers attention, as well as for accepting customers. Also, a server farm with an infinite number of servers is studied, where servers can be turned off after a service completion in order to save costs. We characterize the optimal policy for this model. In the second part of the thesis polling models are studied, which are queueing systems where multiple queues are served by a single server. An application is the production of multiple types of products on a single machine. In this way, the production capacity is pooled between the product types. For the classical polling model, we derive a closedform approximation for the mean waiting time at each of the queues. The approximation is based on the interpolation of light and heavy traffic results. Also, we study a system with so-called smart customers, where the arrival rate at a queue depends on the position of the server. Finally, we invent two new service disciplines (the gated/exhaustive and the ??-gated discipline) for polling models, designed to yield ’fairness and efficiency’ in the mean waiting times. That is, they result in almost equal mean waiting times at each of the queues, without increasing the weighted sum of the mean waiting times too much

Performance analysis of networks on chips

Modules on a chip (such as processors and memories) are traditionally connected through a single link, called a bus. As chips become more complex and the number of modules on a chip increases, this connection method becomes inefficient because the bus can only be used by one module at a time. Networks on chips are an emerging technology for the connection of on-chip modules. In networks on chips, switches are used to transmit data from one module to another, which entails that multiple links can be used simultaneously so that communication is more efficient. Switches consist of a number of input ports to which data arrives and output ports from which data leaves. If data at multiple input ports has to be transmitted to the same output port, only one input port may actually transmit its data, which may lead to congestion. Queueing theory deals with the analysis of congestion phenomena caused by competition for service facilities with scarce resources. Such phenomena occur, for example, in traffic intersections, manufacturing systems, and communication networks like networks on chips. These congestion phenomena are typically analysed using stochastic models, which capture the uncertain and unpredictable nature of processes leading to congestion (such as irregular car arrivals to a traffic intersection). Stochastic models are useful tools for the analysis of networks on chips as well, due to the complexity of data traffic on these networks. In this thesis, we therefore study queueing models aimed at networks on chips. The thesis is centred around two key models: A model of a switch in isolation, the so-called single-switch model, and a model of a network of switches where all traffic has the same destination, the so-called network of polling stations. For both models we are interested in the throughput (the amount of data transmitted per time unit) and the mean delay (the time it takes data to travel across the network). Single-switch models are often studied under the assumption that the number of ports tends to infinity and that traffic is uniform (i.e., on average equally many packets arrive to all buffers, and all possible destinations are equally likely). In networks on chips, however, the number of buffers is typically small. We introduce a new approximation specifically aimed at small switches with (memoryless) Bernoulli arrivals. We show that, for such switches, this approximation is more accurate than currently known approximations. As traffic in networks on chips is usually non-uniform, we also extend our approximation to non-uniform switches. The key difference between uniform and nonuniform switches is that in non-uniform switches, all queues have a different maximum throughput. We obtain a very accurate approximation of this throughput, which allows us to extend the mean delay approximation. The extended approximation is derived for Bernoulli arrivals and correlated arrival processes. Its accuracy is verified through a comparison with simulation results. The second key model is that of concentrating tree networks of polling stations (polling stations are essentially switches where all traffic has the same output port as destination). Single polling stations have been studied extensively in literature, but only few attempts have been made to analyse networks of polling stations. We establish a reduction theorem that states that networks of polling stations can be reduced to single polling stations while preserving some information on mean waiting times. This reduction theorem holds under the assumption that the last node of the network uses a so-called HoL-based service discipline, which means that the choice to transmit data from a certain buffer may only depend on which buffers are empty, but not on the amount of data in the buffers. The reduction theorem is a key tool for the analysis of networks of polling stations. In addition to this, mean waiting times in single polling stations have to be calculated, either exactly or approximately. To this end, known results can be used, but we also devise a new single-station approximation that can be used for a large subclass of HoL-based service disciplines. Finally, networks on chips typically implement flow control, which is a mechanism that limits the amount of data in the network from one source. We analyse the division of throughput over several sources in a network of polling stations with flow control. Our results indicate that the throughput in such a network is determined by an interaction between buffer sizes, flow control limits, and service disciplines. This interaction is studied in more detail by means of a numerical analysis

Contributions to the development of an integrated toolbox of solvers in Derivative-Free Optimization

This dissertation is framed on the ongoing research project BoostDFO - Improving the performance and moving to newer dimensions in Derivative-Free Optimization. The final goal of this project is to develop efficient and robust algorithms for Global and/or Multiobjective Derivative-free Optimization. This type of optimization is typically required in complex scientific/industrial applications, where the function evaluation is time-consuming and derivatives are not available for use, neither can be numerically approximated. Often problems present several conflicting objectives or users aspire to obtain global solutions. Inspired by successful approaches used in single objective local Derivative-free Optimization, we intend to address the inherent problem of the huge execution times by resorting to parallel/cloud computing and carrying a detailed performance analysis. As result, an integrated toolbox for solving single/multi objective, local/global Derivativefree Optimization problems is made available, with recommendations for taking advantage of parallelization and cloud computing, providing easy access to several efficient and robust algorithms and allowing to tackle harder Derivative-free Optimization problems.Esta dissertação insere-se no projecto científico BoostDFO - Improving the performance and moving to newer dimensions in Derivative-Free Optimization. O objectivo final desta investigação é desenvolver algoritmos robustos e eficientes para problemas de Optimização Sem Derivadas Globais e/ou Multiobjectivo. Este tipo de optimização é tipicamente requerido em aplicações científicas/industriais complexas, onde a avaliação da função é bastante demorada e as derivadas não se encontram disponíveis, nem podem ser aproximadas numericamente. Os problemas apresentam frequentemente vários objectivos divergentes ou os utilizadores procuram obter soluções globais. Tendo por base abordagens prévias bem-sucedidas utilizadas em Optimização Sem Derivadas local e uniobjectivo, pretende-se abordar o problema inerente aos grandes tempos de execução, recorrendo ao paralelismo/computação em cloud e efectuando uma detalhada análise de desempenho. Como resultado, é disponibilizada uma ferramenta integrada destinada a problemas de Optimização Sem Derivadas uni/multiobjectivo, com optimização local/global, incluindo recomendações que permitam tirar partido do paralelismo e computação em cloud, facilitando o acesso a vários algoritmos robustos e eficientes e permitindo abordar problemas mais difíceis nesta classe