Numerical methods for queues with shared service

A queueing system is a mathematical abstraction of a situation where elements, called customers, arrive in a system and wait until they receive some kind of service. Queueing systems are omnipresent in real life. Prime examples include people waiting at a counter to be served, airplanes waiting to take off, traffic jams during rush hour etc. Queueing theory is the mathematical study of queueing phenomena. As often neither the arrival instants of the customers nor their service times are known in advance, queueing theory most often assumes that these processes are random variables. The queueing process itself is then a stochastic process and most often also a Markov process, provided a proper description of the state of the queueing process is introduced. This dissertation investigates numerical methods for a particular type of Markovian queueing systems, namely queueing systems with shared service. These queueing systems differ from traditional queueing systems in that there is simultaneous service of the head-of-line customers of all queues and in that there is no service if there are no customers in one of the queues. The absence of service whenever one of the queues is empty yields particular dynamics which are not found in traditional queueing systems. These queueing systems with shared service are not only beautiful mathematical objects in their own right, but are also motivated by an extensive range of applications. The original motivation for studying queueing systems with shared service came from a particular process in inventory management called kitting. A kitting process collects the necessary parts for an end product in a box prior to sending it to the assembly area. The parts and their inventories being the customers and queues, we get ``shared service'' as kitting cannot proceed if some parts are absent. Still in the area of inventory management, the decoupling inventory of a hybrid make-to-stock/make-to-order system exhibits shared service. The production process prior to the decoupling inventory is make-to-stock and driven by demand forecasts. In contrast, the production process after the decoupling inventory is make-to-order and driven by actual demand as items from the decoupling inventory are customised according to customer specifications. At the decoupling point, the decoupling inventory is complemented with a queue of outstanding orders. As customisation only starts when the decoupling inventory is nonempty and there is at least one order, there is again shared service. Moving to applications in telecommunications, shared service applies to energy harvesting sensor nodes. Such a sensor node scavenges energy from its environment to meet its energy expenditure or to prolong its lifetime. A rechargeable battery operates very much like a queue, customers being discretised as chunks of energy. As a sensor node requires both sensed data and energy for transmission, shared service can again be identified. In the Markovian framework, "solving" a queueing system corresponds to finding the steady-state solution of the Markov process that describes the queueing system at hand. Indeed, most performance measures of interest of the queueing system can be expressed in terms of the steady-state solution of the underlying Markov process. For a finite ergodic Markov process, the steady-state solution is the unique solution of $N-1$ balance equations complemented with the normalisation condition, $N$ being the size of the state space. For the queueing systems with shared service, the size of the state space of the Markov processes grows exponentially with the number of queues involved. Hence, even if only a moderate number of queues are considered, the size of the state space is huge. This is the state-space explosion problem. As direct solution methods for such Markov processes are computationally infeasible, this dissertation aims at exploiting structural properties of the Markov processes, as to speed up computation of the steady-state solution. The first property that can be exploited is sparsity of the generator matrix of the Markov process. Indeed, the number of events that can occur in any state --- or equivalently, the number of transitions to other states --- is far smaller than the size of the state space. This means that the generator matrix of the Markov process is mainly filled with zeroes. Iterative methods for sparse linear systems --- in particular the Krylov subspace solver GMRES --- were found to be computationally efficient for studying kitting processes only if the number of queues is limited. For more queues (or a larger state space), the methods cannot calculate the steady-state performance measures sufficiently fast. The applications related to the decoupling inventory and the energy harvesting sensor node involve only two queues. In this case, the generator matrix exhibits a homogene block-tridiagonal structure. Such Markov processes can be solved efficiently by means of matrix-geometric methods, both in the case that the process has finite size and --- even more efficiently --- in the case that it has an infinite size and a finite block size. Neither of the former exact solution methods allows for investigating systems with many queues. Therefore we developed an approximate numerical solution method, based on Maclaurin series expansions. Rather than focussing on structural properties of the Markov process for any parameter setting, the series expansion technique exploits structural properties of the Markov process when some parameter is sent to zero. For the queues with shared exponential service and the service rate sent to zero, the resulting process has a single absorbing state and the states can be ordered such that the generator matrix is upper-diagonal. In this case, the solution at zero is trivial and the calculation of the higher order terms in the series expansion around zero has a computational complexity proportional to the size of the state space. This is a case of regular perturbation of the parameter and contrasts to singular perturbation which is applied when the service times of the kitting process are phase-type distributed. For singular perturbation, the Markov process has no unique steady-state solution when the parameter is sent to zero. However, similar techniques still apply, albeit at a higher computational cost. Finally we note that the numerical series expansion technique is not limited to evaluating queues with shared service. Resembling shared queueing systems in that a Markov process with multidimensional state space is considered, it is shown that the regular series expansion technique can be applied on an epidemic model for opinion propagation in a social network. Interestingly, we find that the series expansion technique complements the usual fluid approach of the epidemic literature

Modelling and performability evaluation of Wireless Sensor Networks

This thesis presents generic analytical models of homogeneous clustered Wireless Sensor Networks (WSNs) with a centrally located Cluster Head (CH) coordinating cluster communication with the sink directly or through other intermediate nodes. The focus is to integrate performance and availability studies of WSNs in the presence of sensor nodes and channel failures and repair/replacement. The main purpose is to enhance improvement of WSN Quality of Service (QoS). Other research works also considered in this thesis include modelling of packet arrival distribution at the CH and intermediate nodes, and modelling of energy consumption at the sensor nodes. An investigation and critical analysis of wireless sensor network architectures, energy conservation techniques and QoS requirements are performed in order to improve performance and availability of the network. Existing techniques used for performance evaluation of single and multi-server systems with several operative states are investigated and analysed in details. To begin with, existing approaches for independent (pure) performance modelling are critically analysed with highlights on merits and drawbacks. Similarly, pure availability modelling approaches are also analysed. Considering that pure performance models tend to be too optimistic and pure availability models are too conservative, performability, which is the integration of performance and availability studies is used for the evaluation of the WSN models developed in this study. Two-dimensional Markov state space representations of the systems are used for performability modelling. Following critical analysis of the existing solution techniques, spectral expansion method and system of simultaneous linear equations are developed and used to solving the proposed models. To validate the results obtained with the two techniques, a discrete event simulation tool is explored. In this research, open queuing networks are used to model the behaviour of the CH when subjected to streams of traffic from cluster nodes in addition to dynamics of operating in the various states. The research begins with a model of a CH with an infinite queue capacity subject to failures and repair/replacement. The model is developed progressively to consider bounded queue capacity systems, channel failures and sleep scheduling mechanisms for performability evaluation of WSNs. Using the developed models, various performance measures of the considered system including mean queue length, throughput, response time and blocking probability are evaluated. Finally, energy models considering mean power consumption in each of the possible operative states is developed. The resulting models are in turn employed for the evaluation of energy saving for the proposed case study model. Numerical solutions and discussions are presented for all the queuing models developed. Simulation is also performed in order to validate the accuracy of the results obtained. In order to address issues of performance and availability of WSNs, current research present independent performance and availability studies. The concerns resulting from such studies have therefore remained unresolved over the years hence persistence poor system performance. The novelty of this research is a proposed integrated performance and availability modelling approach for WSNs meant to address challenges of independent studies. In addition, a novel methodology for modelling and evaluation of power consumption is also offered. Proposed model results provide remarkable improvement on system performance and availability in addition to providing tools for further optimisation studies. A significant power saving is also observed from the proposed model results. In order to improve QoS for WSN, it is possible to improve the proposed models by incorporating priority queuing in a mixed traffic environment. A model of multi-server system is also appropriate for addressing traffic routing. It is also possible to extend the proposed energy model to consider other sleep scheduling mechanisms other than On-demand proposed herein. Analysis and classification of possible arrival distribution of WSN packets for various application environments would be a great idea for enabling robust scientific research

Performability modelling of homogenous and heterogeneous multiserver systems with breakdowns and repairs

This thesis presents analytical modelling of homogeneous multi-server systems with reconfiguration and rebooting delays, heterogeneous multi-server systems with one main and several identical servers, and farm paradigm multi-server systems. This thesis also includes a number of other research works such as, fast performability evaluation models of open networks of nodes with repairs and finite queuing capacities, multi-server systems with deferred repairs, and two stage tandem networks with failures, repairs and multiple servers at the second stage. Applications of these for the popular Beowulf cluster systems and memory servers are also accomplished. Existing techniques used in performance evaluation of multi-server systems are investigated and analysed in detail. Pure performance modelling techniques, pure availability models, and performability models are also considered. First, the existing approaches for pure performance modelling are critically analysed with the discussions on merits and demerits. Then relevant terminology is defined and explained. Since the pure performance models tend to be too optimistic and pure availability models are too conservative, performability models are used for the evaluation of multi-server systems. Fault-tolerant multi-server systems can continue service in case of certain failures. If failure does not occur at a critical point (such as breakdown of the head processor of a farm paradigm system) the system continues serving in a degraded mode of operation. In such systems, reconfiguration and/or rebooting delays are expected while a processor is being mapped out from the system. These delay stages are also taken into account in addition to failures and repairs, in the exact performability models that are developed. Two dimensional Markov state space representations of the systems are used for performability modelling. Following the critical analysis of the existing solution techniques, the Spectral Expansion method is chosen for the solution of the models developed. In this work, open queuing networks are also considered. To evaluate their performability, existing modelling approaches are expanded and validated by simulations, for performability analysis of multistage open networks with finite queuing capacities. The performances of two extended modelling approaches are compared in terms of accuracy for open networks with various queuing capacities. Deferred repair strategies are becoming popular because of the cost reductions they can provide. Effects of using deferred repairs are analysed and performability models are provided for homogeneous multi-server systems and highly available farm paradigm multi-server systems. Since one of the random variables is used to represent the number of jobs in one of the queues, analytical models for performance evaluation of two stage tandem networks suffer because of numerical cumbersomeness. Existing approaches for modelling these systems are actually pure performance models since breakdowns and repairs cannot be considered. One way of modelling these systems can be to divide one of the random variables to present both the operative and non-operative states of the server in one dimension. However, this will give rise to state explosion problem severely limiting the maximum queue capacity that can be handled. In order to overcome this problem a new approach is presented for modelling two stage tandem networks in three dimensions. An approximate solution is presented to solve such a system. This approach manifests itself as a novel contribution for alleviating the state space explosion problem for large and/or complex systems. When two state tandem networks with feedback are modelled using this approach, the operative states can be handled independently and this makes it possible to consider multiple operative states at the second stage. The analytical models presented can be used with various parameters and they are extendible to consider systems with similar architectures. The developed three dimensional approach is capable to handle two stage tandem networks with various characteristics for performability measures. All the approaches presented give accurate results. Numerical solutions are presented for all models developed. In case the solution presented is not exact, simulations are performed to validate the accuracy of the results obtained

Establishment of a research focus on resilient sustainable climate neutral agricultural production: resilient farming initiative

Efficient and sustainable agricultural production is a pivotal factor in meeting the nutritional needs of an expanding global population. However, it is imperative to optimize national, regional and local effectiveness to support government initiatives towards climate neutrality and resilience, while simultaneously ensuring economic viability for farmers. A significant reduction in the cost of goods must be accompanied by a decrease in their global warming potential contribution to maintain competitiveness in the world market. As such, it is necessary to adopt practices that enhance productivity while minimizing environmental impacts. This paper discusses potential solutions for the sustainable enlargement of botanical product portfolios towards essential oil products and natural extracts for value-added products, such as natural pharmaceuticals, cosmetics, agrochemicals and materials by direct waste valorization. Contributions from the fields of automation and digitalization provide the basic technology for the realization of the approaches presented. Agricultural photovoltaics can contribute to the goal of the reduction of the cost of goods and global warming potential, such as the already established utilization of biogas. The potential of the research initiative described is demonstrated by basic data on key characteristic numbers and costs from the literature. The economic potential for climate neutrality and the reduction of global warming potential contribution is seen in magnitudes of factors 5–10. A research initiative is recommended and exemplified for the industrialization of such integrated processing

Modelling and performance evaluation of wireless and mobile communication systems in heterogeneous environments

It is widely expected that next generation wireless communication systems will be heterogeneous, integrating a wide variety of wireless access networks. Of particular interest recently is the integration of cellular networks (GSM, GPRS, UMTS, EDGE and LTE) and wireless local area networks (WLANs) to provide complementary features in terms of coverage, capacity and mobility support. These different networks will work together using vertical handover techniques and hence understanding how well these mechanisms perform is a significant issue. In this thesis, these networks are modelled to yield performance results such as mean queue lengths and blocking probabilities over a range of different conditions. The results are then analysed using network constraints to yield operational graphs based on handover probabilities to different networks. Firstly, individual networks with horizontal handover are analysed using performability techniques. The thesis moves on to look at vertical handovers between cellular networks using pure performance models. Then the integration of cellular networks and WLAN is considered. While analysing these results it became clear that the common models that were being used were subjected to handover hysteresis resulting from feedback loops in the model. A new analytical model was developed which addressed this issue but was shown to be problematic in developing state probabilities for more complicated scenarios. Guard channels analysis, which is normally used to give priority to handover traffic in mobile networks, was employed as a practical solution to the observed handover hysteresis. Overall, using different analytical techniques as well as simulation, the results of this work form an important part in the design and development of future mobile systems

Iterative learning control of crystallisation systems

Under the increasing pressure of issues like reducing the time to market, managing lower production costs, and improving the flexibility of operation, batch process industries thrive towards the production of high value added commodity, i.e. specialty chemicals, pharmaceuticals, agricultural, and biotechnology enabled products. For better design, consistent operation and improved control of batch chemical processes one cannot ignore the sensing and computational blessings provided by modern sensors, computers, algorithms, and software. In addition, there is a growing demand for modelling and control tools based on process operating data. This study is focused on developing process operation data-based iterative learning control (ILC) strategies for batch processes, more specifically for batch crystallisation systems. In order to proceed, the research took a step backward to explore the existing control strategies, fundamentals, mechanisms, and various process analytical technology (PAT) tools used in batch crystallisation control. From the basics of the background study, an operating data-driven ILC approach was developed to improve the product quality from batch-to-batch. The concept of ILC is to exploit the repetitive nature of batch processes to automate recipe updating using process knowledge obtained from previous runs. The methodology stated here was based on the linear time varying (LTV) perturbation model in an ILC framework to provide a convergent batch-to-batch improvement of the process performance indicator. In an attempt to create uniqueness in the research, a novel hierarchical ILC (HILC) scheme was proposed for the systematic design of the supersaturation control (SSC) of a seeded batch cooling crystalliser. This model free control approach is implemented in a hierarchical structure by assigning data-driven supersaturation controller on the upper level and a simple temperature controller in the lower level. In order to familiarise with other data based control of crystallisation processes, the study rehearsed the existing direct nucleation control (DNC) approach. However, this part was more committed to perform a detailed strategic investigation of different possible structures of DNC and to compare the results with that of a first principle model based optimisation for the very first time. The DNC results in fact outperformed the model based optimisation approach and established an ultimate guideline to select the preferable DNC structure. Batch chemical processes are distributed as well as nonlinear in nature which need to be operated over a wide range of operating conditions and often near the boundary of the admissible region. As the linear lumped model predictive controllers (MPCs) often subject to severe performance limitations, there is a growing demand of simple data driven nonlinear control strategy to control batch crystallisers that will consider the spatio-temporal aspects. In this study, an operating data-driven polynomial chaos expansion (PCE) based nonlinear surrogate modelling and optimisation strategy was presented for batch crystallisation processes. Model validation and optimisation results confirmed this approach as a promise to nonlinear control. The evaluations of the proposed data based methodologies were carried out by simulation case studies, laboratory experiments and industrial pilot plant experiments. For all the simulation case studies a detailed mathematical models covering reaction kinetics and heat mass balances were developed for a batch cooling crystallisation system of Paracetamol in water. Based on these models, rigorous simulation programs were developed in MATLAB®, which was then treated as the real batch cooling crystallisation system. The laboratory experimental works were carried out using a lab scale system of Paracetamol and iso-Propyl alcohol (IPA). All the experimental works including the qualitative and quantitative monitoring of the crystallisation experiments and products demonstrated an inclusive application of various in situ process analytical technology (PAT) tools, such as focused beam reflectance measurement (FBRM), UV/Vis spectroscopy and particle vision measurement (PVM) as well. The industrial pilot scale study was carried out in GlaxoSmithKline Bangladesh Limited, Bangladesh, and the system of experiments was Paracetamol and other powdered excipients used to make paracetamol tablets. The methodologies presented in this thesis provide a comprehensive framework for data-based dynamic optimisation and control of crystallisation processes. All the simulation and experimental evaluations of the proposed approaches emphasised the potential of the data-driven techniques to provide considerable advances in the current state-of-the-art in crystallisation control