The purpose of this project was to develop a delivery system optimized for laser interstitial thermotherapy of small tumors of the breast. The proposed approach is to combine laser interstitial thermotherapy with stereotactic imaging for fiber guidance and treatment monitoring. The goals of the dissertation were to design a fiber insertion system for cylindrical diffusing tip optical fibers and to derive optimal laser parameters for coagulation of 1 cm tumor plus a surrounding 1 cm thick rim of healthy tissue. A fiber insertion system compatible with a high resolution stereotactic digital X-ray biopsy system was designed to guide the fiber into the tumor site in similar fashion to the insertion of the biopsy needle. An optical-thermal model consisting of a radiation model, a thermal model, and a coagulation model was developed and validated using ex-vivo porcine tissue. A single integrating sphere optical property measurement system and an inverse Monte Carlo algorithm were developed to measure the optical properties of ex-vivo porcine tissue at 830, 940, and 980 nm. An experimental method was developed to determine the parameters of the Arrhenius model (frequency factor (A) and activation energy (Ea)). The optical-thermal model was validated by comparing the predicted temperature and coagulation to results of laser irradiation experiments at 830, 940, and 980 nm. Using published values of the optical properties of the breast, the model predicts that a 3 cm coagulation size can be produced without vaporization in 10 min with 10.4 W at 980 and 940 nm and 13.2 W at 830 nm. The same outcome can be achieved in 20 min with 4.5 W at 980 and 940 nm and 6.1 W at 830 nm.

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In this work we introduce a novel queueing model with two classes of users in which, instead of accessing a resource, users wait in the system to match with a candidate from the other class. The users are selective and the matchings occur probabilistically. This new model is useful for analysing the traffic in web portals that match people who provide a service with people who demand the same service, e.g. employment portals, matrimonial and dating sites and rental portals. We first provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We then prove that if no control mechanism is employed these systems are unstable for any set of parameters. We suggest four different classes of control policies to assure stability and conduct analysis on performance measures under the control policies. Contrary to the intuition that the rejection rate should decrease as the users become more likely to be matched, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times. Further, to gain more insight into the behaviour of probabilistic matching systems, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyse the basic properties of these approximations and show that some performance measures are insensitive to the matching probability agreeing with the results found by the exact analysis. Finally we study the optimal control and revenue management for the systems with the objective of profit maximization. We formulate mathematical models for both unobservable and observable systems. For an unobservable system we suggest a deterministic optimal control, while for an observable system we develop an optimal myopic state dependent pricing.

In this thesis, we study the queueing systems with heterogeneous servers and service rate uncertainty under the Halfin-Whitt heavy traffic regime. First, we analyse many server queues with abandonments when service rates are i.i.d. random variables. We derive a diffusion approximation using a novel method. The diffusion has a random drift, and hence depending on the realisations of service rates, the system can be in Quality Driven (QD), Efficiency Driven (ED) or Quality-Efficiency-Driven (QED) regime. When the system is under QD or QED regime, the abandonments are negligible in the fluid limit, but when it is under ED regime, the probability of abandonment will converge to a non-zero value. We then analyse the optimal staffing levels to balance holding costs with staffing costs combining these three regimes. We also analyse how the variance of service rates influence abandonment rate. Next, we focus on the state space collapse (SSC) phenomenon. We prove that under some assumptions, the system process will collapse to a lower dimensional process without losing essential information. We first formulate a general method to prove SSC results inside pools for heavy traffic systems using the hydrodynamic limit idea. Then we work on the SSC in multi-class queueing networks under the Halfin-Whitt heavy traffic when service rates are i.i.d. random variables within pools. For such systems, exact analysis provides limited insight on the general properties. Alternatively, asymptotic analysis by diffusion approximation proves to be effective. Further, limit theorems, which state the diffusively scaled system process weakly converges to a diffusion process, are usually the central part in such asymptotic analysis. The SSC result is key to proving such a limit. We conclude by giving examples on how SSC is applied to the analysis of systems.

In the first half of this thesis, we extend the results of Halfin and Whitt to generally distributed service times. This is accomplished by first writing the system equations for the G/GI/N queue in a manner similar to the system equations for G/GI/Infinity queue. We next identify a key relationship between these two queues. This relationship allows us to leverage several existing results for the G/GI/Infinity queue in order to prove our main result. Our main result in the first part of this thesis is to show that the diffusion scaled queue length process for the G/GI/N queue in the Halfin-Whitt regime converges to a limiting stochastic process which is driven by a Gaussian process and satisfies a stochastic convolution equation. We also show that a similar result holds true for the fluid scaled queue length process under general initial conditions. Customer abandonment is also a common feature of many call centers. Some researchers have even gone so far as to suggest that the level of customer abandonment is the single most important metric with regards to a call center's performance. In the second half of this thesis, we improve upon a result of Ward and Glynn's concerning the GI/GI/1+GI queue in heavy traffic. Whereas Ward and Glynn obtain a diffusion limit result for the GI/GI/1+GI queue in heavy traffic which incorporates only the density the abandonment distribution at the origin, our result incorporate the entire abandonment distribution. This is accomplished by a scaling the hazard rate function of the abandonment distribution as the system moves into heavy traffic. Our main results are to obtain diffusion limits for the properly scaled workload and queue length processes in the GI/GI/1+GI queue. The limiting diffusions we obtain are reflected at the origin with a negative drift which is dependent upon the hazard rate of the abandonment distribution. Because these diffusions have an analytically tractable steady-state distribution, they can be used to provide a closed-form approximation for the steady-state distribution of the queue length and workload processes in a GI/GI/1+GI queue. We demonstrate the accuracy of these approximations through simulation.

Es wurde eine SP3-Transportmethode entwickelt, die neutronenkinetische Rechnungen für die Kerne von Leichtwasserreaktoren mit höherer Genauigkeit als die gegenwärtig in der Kernauslegung angewandten Standardmethoden auf Basis der Zweigruppendiffusionsnäherung er-laubt. Eine Verbesserung der Genauigkeit von Abbrandrechnungen und der Berechnung von Tran-sienten ist für heterogene Kerne notwendig, in denen neben UO2-Brennelementen auch Mischoxyd – Brennelemente eingesetzt werden. In einem ersten Schritt wird die in dem Rechenprogramm DYN3D verwendete Zweigruppendiffusi-onsmethode auf viele Energiegruppen erweitert. Auf der Basis von Untersuchungen zu einer optima-len Gruppenstruktur wird die Verwendung von 8-10 Energiegruppen der Neutronen als optimal erach-tet. Das Verfahren wurde anhand von stationären und transienten Rechnungen für das OECD/NEA und US NRC PWR MOX/UO2 Core Transient Benchmark verifiziert. In den nächsten Schritten erfolgte die Entwicklung und Implementierung einer SP3-Näherung in DYN3D. Dabei besteht die Möglichkeit, ein feineres Gitter im BE zu benutzen. Das Verfahren wurde zunächst durch pinweise Berechnung stationärer Zustände des obigen Benchmarks verifiziert. Untersuchungen für das Benchmarkproblem zeigen, dass das Verhältniss des 2-ten Momentes zum 0-ten Moment des Flusses klein ist. Die beiden SP3-Gleichungen können deshalb separat in iterativer Weise gelöst werden. Dies reduziert den benötigten Speicherplatz und erfordert weniger CPU-Zeit. Dieses vereinfachte Verfahren wurde deshalb ebenfalls in das Programm implementiert. Es wird ge-zeigt, dass mit diesem Verfahren eine vergleichbare Genauigkeit erreicht wird. Stabweise Rechnun-gen mit 4, 8 und 16 Energiegrupppen wurden für einen stationären Zustand des Benchmarks durch-geführt. Eine 3-dimensionale Aufgabe des Benchmarks mit Rückkopplung und Vollleistung wurde mit dem optimierten SP3-Verfahren gerechnet. A SP3 transport approximation was developed for neutron kinetic calculations of cores of light water reactors with a higher accuracy than the present standard methods of core design based on the two group diffusion approximation. An improvement of accuracy for burnup and transient calculations is required for cores loaded with UO2 and MOX fuel assemblies. In the first step, the two group diffusion method applied in the computer code DYN3D was extended to an arbitrary number of groups. Investigations for an optimal group structure have shown that a number of 8 to 10 energy groups of neutrons seems to be reasonable. The multi-group technique was verified for steady states and transients of the OECD/NEA und US NRC PWR MOX/UO2 Core Tran-sient Benchmark. In the next steps, a SP3-approximation was developed and implemented into DYN3D. The possibility of using finer meshes inside the fuel assemblies is involved in this method. The technique was veri-fied by pinwise calculations for steady states of the above mentioned benchmark. The investigations to the benchmark problem have shown that ratio of the 2nd moment of flux to the 0th moment is small. Therefore the two coupled SP3 equations can be solved separately in an iterative way. The required computer memory and the CPU time can be reduced by this technique. This sim-pler method was also implemented in the code. It is shown that the reached accuracy is comparable to accuracy of the original technique. Pinwise calculations with 4, 8 and 16 energy groups were per-formed for a steady state of this benchmark. A three-dimensional problem of the benchmark at full power and with feedback was calculated with the optimized SP3 technique. The optimized method was used for the time integration of the transient SP3 equations. The pinwise calculation of a control rod ejection was tested for a simple system and the results were compared with the diffusion solution.

We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.

The present thesis is located within the context of the diffusion approximation of Wright–Fisher models and the Kolmogorov equations describing their evolution. On the one hand, a full account of recombinational Wright–Fisher model is developed as well as their enhancement by other evolutionary mechanisms, including some information geometrical analysis. On the other hand, the thesis addresses several issues arising in the context of analytical solution schemes for such Kolmogorov equations, namely the inclusion of the entire boundary of the state space. For this, a hierarchical extension scheme is developed, both for the forward and the backward evolution, and the uniqueness of such extensions is proven. First, a systematic approach to the diffusion approximation of recombinational two- or more loci Wright–Fisher models is presented. As a point of departure a specific Kolmogorov backward equation for the diffusion approximation of a recombinational two-loci Wright–Fisher model is chosen, to which – with the help of some information geometrical methods, i. e. by calculating the sectional curvatures of the corresponding statistical manifold (which is the domain equipped with the corresponding Fisher metric) – one succeeds to identify the underlying Wright–Fisher model. Accompanying this, for all methods and tools involved a suitable introduction is presented. Furthermore, the considerations span a separate analysis for the two most common underlying models (RUZ and RUG) as well as a comparison of the two models. Finally, transferring corresponding results for a simpler model described by Antonelli and Strobeck, solutions of the Kolmogorov equations are contrasted with Brownian motion in the same domain. Furthermore, the perspective of the diffusion approximation of recombinational Wright–Fisher models is widened as the model underlying the Ohta–Kimura formula is subsequently extended by an integration of the concepts of natural fitness and mutation. Simultaneously, the corresponding extensions of the Ohta–Kimura formula are stated. Crucial for this is the development of a suitable fitness scheme, which is accomplished by a multiplicative aggregation of fitness values for pairs of gametes/zygotes. Furthermore, the model is generalised to have an arbitrary number of alleles and – in the following step – an arbitrary number of loci respectively. The latter involves an increased number of recombination modes, for which the concept of recombination masks is also implemented into the model. Another generalisation in terms of coarse-graining is performed via an application of schemata; this also affects the previously introduced concepts, specifically mask recombination, which are adapted accordingly. Eventually, a geometric analysis of linkage equilibrium states of the multi-loci Wright–Fisher models is carried out, relating to the concept of hierarchical probability distributions in information geometry, which concludes the considerations of recombinational Wright–Fisher models and their extensions. Subsequently, the discussion of analytical solution schemes for the Kolmogorov equations corresponding to the diffusion approximation of Wright–Fisher models is ushered in, which represents the second part of the thesis. This is started with the simplest setting of a 1-dimensional Wright–Fisher model, for which the solution strategy for the corresponding Kolmogorov forward equation given by M. Kimura is recalled. From this, one may construct a unique extended solution which also accounts for the dynamics of the model on lower-dimensional entities of the state space, i. e. configurations of the model where one of the alleles no longer exists in the population, utilising the concept of (boundary) flux of a solution; a discussion of the moments of the distribution confirms the findings. A similar treatment is then carried out for the corresponding Kolmogorov backward equation, yielding analogous results of existence and uniqueness for an extended solution. For the latter in particular, a corresponding account of the configuration on the boundary turns out to be crucial, which is also reflected in the probabilistic interpretation of the backward solution. Additionally, the long-term behaviour of solutions is analysed, and a comparison between such solutions of the forward and the backward equation is made. Next, it is basically aimed to transfer the results obtained in the previous chapter to the subsequent increasingly complicated setting of a Wright–Fisher model with 1 locus and an arbitrary number of alleles: With solution schemes for the interior of the state space (i. e. not encompassing the boundary) already existing in the literature, an extension scheme for a successive determination of the solution on lower-dimensional entities of the domain is developed. This scheme, again, makes use of the concept of the (boundary) flux of solutions, and one may therefore show that this extended solution fulfils additional properties regarding the completeness of the diffusion approximation with respect to the boundary. These properties may be formulated in terms of the moments of the distribution, and their connection to the underlying Wright–Fisher model is illustrated. Altogether, stipulating such a moments condition, existence and uniqueness of an extended solution on the entire domain are shown. Furthermore, the corresponding Kolmogorov backward equation is examined, for which similarly a (backward) extension scheme is presented, which allows extending a solution in a domain (perceived as a boundary instance of a larger domain) to all adjacent higher-dimensional entities of the larger domain along a certain path. This generalises the integration of boundary data observed in the previous chapter; in total, the existence of a solution of the Kolmogorov backward equation in the entire domain is shown for arbitrary boundary data. Of particular interest to the discussion are stationary solutions of the Kolmogorov backward equation as they describe eventual hit probabilities for a certain target set of the model (in accordance with the probabilistic interpretation of solutions of the backward equation). The presented backward extension scheme allows the construction of solutions for all relevant cases, reconfirming some results by R. A. Littler for the stationary case, but now providing a previously missing systematic derivation. Eventually, the hitherto missing uniqueness assertion for this type of solutions is established by means of a specific iterated transformation which resolves the critical incompatibilities of solutions by a successive blow-up while the domain is converted from a simplex into a cube. Then – under certain additional assumptions on the regularity of the transformed solution – the uniqueness directly follows from general principles. Lastly, several other aspects of the blow-up scheme are discussed; in particular, it is illustrated in what way the required extra regularity relates to reasonable additional properties of the underlying Wright–Fisher model.

Since the first cases of HIV/AIDS disease were recognised in the early 1980s, a large number of mathematical models have been proposed. However, the mobility of people among regions, which has an obvious impact on the spread of the disease, has not been much considered in the modelling studies. One of the main reasons is that the models for the spread of the disease in multiple populations are very complex and, as a consequence, they can easily become intractable. In this thesis we provide various new results pertaining to the spread of the disease in mobile populations, including epidemic intervention in multiple populations. We first develop stochastic models for the spread of the disease in a single heterosexual population, considering both constant and varying population sizes. In particular, we consider a class of continuous-time Markov chains (CTMCs). We establish deterministic and Gaussian diffusion analogues of these stochastic processes by applying the theory of density dependent processes. A range of numerical experiments are provided to show how well the deterministic and Gaussian counterparts approximate the dynamic behaviour of the processes. We derive threshold parameters, known as basic reproduction numbers, for both cases above the threshold which the disease is uniformly persistent and below the threshold which disease-free equilibrium is locally attractive. We find that the threshold conditions for both constant and varying population sizes have the same form. In order to take into account the mobility of people among regions, we extend the stochastic models to multiple populations. Various stochastic models for multiple populations are formulated as CTMCs. The deterministic and Gaussian diffusion counterparts of the corresponding stochastic processes for the multiple populations are also established. Threshold parameters for the persistence of the disease in the multiple population models are derived by applying the concept of next generation matrices. The results of this study can serve as a basic framework how to formulate and analyse a more realistic stochastic model for the spread of HIV in mobile heterogeneous populations—classifying all individuals by age, risk, and level of infectivities, and at the same time considering different modes of the disease transmission. Assuming an accurate mathematical model for the spread of HIV/AIDS disease, another question that we address in this thesis is how to control the spread of the disease in a mobile population. Most previous studies for the spread of the disease focus on identifying the most significant parameters in a model. In contrast, we study these problems as optimal epidemic intervention problems. The study is mostly motivated by the fact that more and more local governments allocate budgets over a certain period of time to combat the disease in their areas. The question is how to allocate this limited budget to minimise the number of new HIV cases, say on a country level, over a finite time horizon as people move among regions. The mathematical models developed in the first part of this thesis are used as dynamic constraints of the optimal control problems. In this thesis, we also introduce a novel approach to solve quite general optimal control problems using the Cross-Entropy (CE) method. The effectiveness of the CE method is demonstrated through several illustrative examples in optimal control. The main application is the optimal epidemic intervention problems discussed above. These are highly non-linear and multidimensional problems. Many existing numerical techniques for solving such optimal control problems suffer from the curse of dimensionality. However, we find that the CE technique is very efficient in solving such problems. The numerical results of the optimal epidemic strategies obtained via the CE method suggest that the structure of the optimal trajectories are highly synchronised among patches but the trajectories do not depend much on the structure of the models. Instead, the parameters of the models (such as the time horizon, the amount of available budget, infection rates) much affect the form of the solution.