Smooth flexible models of nonhomogeneous Poisson processes fit to one or more process realizations

Simulation is a technique of creating representations or models of real world systems or processes and conducting experiments to predict behavior of actual systems. Input modeling is a critical aspect of simulation modeling. Stochastic input models are used to model various aspects of the system under uncertainty including process times and interarrival times. This research focuses on input models for nonstationary arrival processes that can be represented as nonhomogeneous Poisson processes (NHPPs). In particular, a smooth flexible model for the mean-value function (or integrated rate function) of a general NHPP is estimated. To represent the mean-value function, the method utilizes a specially formulated polynomial that is constrained in least-squares estimation to be nondecreasing so the corresponding rate function is nonnegative and continuously differentiable. The degree of the polynomial is determined by applying a modified likelihood ratio test to a set of transformed arrival times resulting from a variance stabilizing transformation of the observed data. Given the degree of polynomial, final estimates of the polynomial coefficients are obtained from original arrival times using least-squares estimation. The method is extended to fit an NHPP model to multiple observed realizations of a process. In addition, the method is adapted to a multiresolution procedure that effectively models NHPPs with long term trend and cyclic behavior given multiple process realizations. An experimental performance evaluation is conducted to determine the capabilities and limitations of the NHPP fitting procedure for single and multiple realizations of test processes. The method is implemented in a Java-based programming environment along with a web interface that allows user to upload observed data, fit an NHPP, and generate realizations of the fitted NHPP for use in simulation experiments

Simulations on Lévy subordinators and Lévy driven contagion models

Lévy subordinators have become a fundamental component to be used to construct many useful stochastic processes, which have numerous applications in finance, insurance and many other fields. However, as many applications of Lévy based stochastic models use fairly complicated analytical and probabilistic tools, it has been challenging to implement in practice. Hence, simulation-based study becomes more desirable. In this thesis, we deal with exact simulation on Lévy subordinators and Lévy driven stochastic models. In the first part, we focus on developing more efficient exact simulation schemes for Lévy subordinators with existing simulation algorithms in the literature. Besides, we also introduce a new type of Lévy subordinators, i.e. truncated Lévy subordinators. We study the path properties, develop exact simulation algorithms based on marked renewal representations, and provide relevant applications in finance and insurance. The associated results in this part are later used in the sequel. The second part of this thesis proposes a new type of point processes by generalising the classical self-exciting Hawkes processes and doubly stochastic Poisson processes with Lévy driven Ornstein-Uhlenbeck type intensities. These resulting models are analytically tractable, and intrinsically inherit the great flexibility as well as desirable features from the two original processes, including skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the contagion or feedback effects. These newly constructed processes would then substantially enrich continuous-time models tailored for quantifying the contagion of event arrivals in finance, economics, insurance, queueing and many other fields. In turn, we characterise the distributional properties of this new class of point processes and design an exact simulation algorithm to generate sample paths. This is done by applying the exact distributional decomposition technique. We carry out extensive numerical implementations and tests to demonstrate the accuracy and effectiveness of our scheme and give examples of some financial applications to credit portfolio risk to show the applicability and flexibility of our new model

Efficient Simulation and Performance Stabilization for Time-Varying Single-Server Queues

This thesis develops techniques to evaluate and to improve the performance of single-server service systems with time-varying arrivals. The performance measures considered are the time-varying expected length of the queue and the expected customer waiting time. Time varying arrival rates are considered because they often occur in service systems. For example, arrival rates often vary significantly over the hours of each day and over the days of each week. Stochastic textbook methods do not apply to models with time-varying arrival rates. Hence new techniques are needed to provide high quality of service when stationary steady-state analysis is not appropriate. In contrast to the extensive recent literature on many-server queues with time-varying arrival rates, we focus on single-server queues with time-varying arrival rates. Single-server queues arise in real applications where there is no flexibility in the number of service facilities (servers). Different analysis techniques are required for single-server queues, because the two kinds of models exhibit very different performance. Many-server models are more tractable because methods for highly tractable infinite-server models can be applied. In contrast, single-server models are more complicated because it takes a long time to respond to a build up of workload when there is only one server. The thesis is divided into two parts: simulation algorithms for performance evaluation and service-rate controls for performance stabilization. The first part of the thesis develops algorithms to efficiently simulate the single-server time-varying queue. For the generality considered, no explicit mathematical formulas are available for calculating performance measures, so simulation experiments are needed to calculate and evaluate system performance. Efficient algorithms for both standard simulation and rare-event simulation are developed. The second part of the thesis develops service-rate controls to stabilize performance in the time-varying single-server queue. The performance stabilization problem aims to minimize fluctuations in mean waiting times for customers coming at different times even though the arrival rate is time-varying. A new service rate control is developed, where the service rate at each time is a function of the arrival rate function. We show that a specific service rate control can be found to stabilize performance. In turn, that service rate control can be used to provide guidance for real applications on optimal changes in staffing, processing speed or machine power status over time. Both the simulation experiments to evaluate performance of alternative service-rate controls and the simulation search algorithm to find the best parameters for a damped time-lag service-rate control are based on efficient performance evaluation algorithms in the first part of the thesis. In Chapter Two, we present an efficient algorithm to simulate a general non-Poisson non-stationary point process. The general point process can be represented as a time transformation of a rate-one base process and by exploiting a table of the inverse cumulative arrival rate function outside of simulation, we can efficiently convert the simulated rate-one process into the simulated general point process. The simulation experiments can be conducted in linear time subject to small error bounds. Then we can apply this efficient algorithm to generate the arrival process, the service process and thus to calculate performance measures for the G_t/G_t/1 queues, which are single-server queues with time-varying arrival rates and service rates. Service models are constructed for this purpose where time-varying service rates are specified separately from the rate-one service requirement process, and service times are determined by equating service requirements with integrals of service rates over a time period equal to the service time. In Chapter Three, we develop rare-event simulation algorithms in periodic GI_t/GI/1 queues and further in GI_t/GI_t/1 queues to estimate probabilities of rare but important events as a sanity check of the system, for example, estimating the probability that the waiting time is very long. Importance sampling, specifically exponential tilting, is required to estimate rare-event probabilities because in standard simulation, the number of experiments may blow up to achieve a targeted relative error and for each experiment, it may take a very long time to determine that the rare event does not happen. To extend the rare-event simulation algorithm to periodic queues, we derive a convenient expression for the periodic steady-state virtual waiting time. We apply this expression to establish bounds between the periodic workload and the steady-state workload in stationary queues, so that we can prove that the exponential tilting algorithm with the same parameter efficient in stationary queues is efficient in the periodic setting as well, which has a bounded relative error. We apply this algorithm to compute the periodic steady-state distribution of reflected periodic Brownian motion with support of a heavy-traffic limit theorem and to calculate the periodic steady-state distribution and moments of the virtual waiting time. This algorithm's advantage in calculating these distributions and moments is that it can directly estimate them at a specific position of the cycle without simulating the whole queueing process until steady state is reached for the whole cycle. In Chapter Four, we conduct simulation experiments to validate performance of four service-rate controls: the rate-matching control, which is directly proportional to the arrival rate, two square-root controls related to the square root staffing formula and the square-root control based on the mean stationary waiting time. Simulations show that the rate-matching control stabilizes the queue length distribution but not the virtual waiting time. This is consistent with established theoretical results, which follow from the observation that with rate-matching control, the queueing process becomes a time transformation of the stationary queueing process with constant arrival rates and service rates. Simulation results also show that the two square-root controls analogous to the server staffing formula are not effective in stabilizing performance. On the other hand, the alternative square-root service rate control based on the mean stationary waiting time approximately stabilizes the virtual waiting time when the cycle is long so that the arrival rate changes slowly enough. In Chapter Five, since we are mostly interested in stabilizing waiting times in more common scenarios when the traffic intensity is not close to one or when the arrival rate does not change slowly, we develop a damped time-lag service-rate control that performs fairly well for this purpose. This control is a modification of the rate-matching control involving a time lag and a damping factor. To find the best parameters for this control, we search over reasonable intervals for the most time-stable performance measures, which are computed by the extended rare-event simulation algorithm in GI_t/GI_t/1 queue. We conduct simulation experiments to validate that this control is effective for stabilizing the expected steady-state virtual waiting time (and its distribution to a large extent). We also establish a heavy-traffic limit with periodicity in the fluid scale to provide theoretical support for this control. We also show that there is a time-varying Little's law in heavy-traffic, which implies that this control cannot stabilize the queue length and the waiting time at the same time

Non-stationary processes and their application to financial high-frequency data

The thesis is devoted to non-stationary point process models as generalizations of the standard homogeneous Poisson process. The work can be divided in two parts. In the first part, we introduce a fractional non-homogeneous Poisson process (FNPP) by applying a random time change to the standard Poisson process. We characterize the FNPP by deriving its non-local governing equation. We further compute moments and covariance of the process and discuss the distribution of the arrival times. Moreover, we give both finite-dimensional and functional limit theorems for the FNPP and the corresponding fractional non-homogeneous compound Poisson process. The limit theorems are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified via a Monte-Carlo simulation. In the second part, we analyze statistical point process models for durations between trades recorded in financial high-frequency trading data. We consider parameter settings for models which are non-stationary or very close to non-stationarity which is quite typical for estimated parameter sets of models fitted to financial data. Simulation, parameter estimation and in particular model selection are discussed for the following three models: a non-homogeneous normal compound Poisson process, the exponential autoregressive conditional duration model (ACD) and a Hawkes process model. In a Monte-Carlo simulation, we test the performance of the following information criteria for model selection: Akaike's information criterion, the Bayesian information criterion and the Hannan-Quinn information criterion. We are particularly interested in the relation between the rate of correct model selection and the underlying sample size. Our numerical results show that the model selection for the compound Poisson type model works best for small parameter numbers. Moreover, the results for Hawkes processes confirm the theoretical asymptotic distributions of model selection whereas for the ACD model the model selection exhibits adverse behavior in certain cases

Photodetectors

In this book some recent advances in development of photodetectors and photodetection systems for specific applications are included. In the first section of the book nine different types of photodetectors and their characteristics are presented. Next, some theoretical aspects and simulations are discussed. The last eight chapters are devoted to the development of photodetection systems for imaging, particle size analysis, transfers of time, measurement of vibrations, magnetic field, polarization of light, and particle energy. The book is addressed to students, engineers, and researchers working in the field of photonics and advanced technologies

Handbook of Mathematical Geosciences

This Open Access handbook published at the IAMG's 50th anniversary, presents a compilation of invited path-breaking research contributions by award-winning geoscientists who have been instrumental in shaping the IAMG. It contains 45 chapters that are categorized broadly into five parts (i) theory, (ii) general applications, (iii) exploration and resource estimation, (iv) reviews, and (v) reminiscences covering related topics like mathematical geosciences, mathematical morphology, geostatistics, fractals and multifractals, spatial statistics, multipoint geostatistics, compositional data analysis, informatics, geocomputation, numerical methods, and chaos theory in the geosciences

Flowing matter

This open access book, published in the Soft and Biological Matter series, presents an introduction to selected research topics in the broad field of flowing matter, including the dynamics of fluids with a complex internal structure -from nematic fluids to soft glasses- as well as active matter and turbulent phenomena.Flowing matter is a subject at the crossroads between physics, mathematics, chemistry, engineering, biology and earth sciences, and relies on a multidisciplinary approach to describe the emergence of the macroscopic behaviours in a system from the coordinated dynamics of its microscopic constituents.Depending on the microscopic interactions, an assembly of molecules or of mesoscopic particles can flow like a simple Newtonian fluid, deform elastically like a solid or behave in a complex manner. When the internal constituents are active, as for biological entities, one generally observes complex large-scale collective motions. Phenomenology is further complicated by the invariable tendency of fluids to display chaos at the large scales or when stirred strongly enough. This volume presents several research topics that address these phenomena encompassing the traditional micro-, meso-, and macro-scales descriptions, and contributes to our understanding of the fundamentals of flowing matter.This book is the legacy of the COST Action MP1305 “Flowing Matter”