The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research.

In simulation, time averages are important for estimating equilibrium parameters. In particular, we would like to have the variance, bias and mean-square error for time averages. First, we will discuss various factors and their effect on the bias, the variance and the mean-square error. We will use the Markovian Event System to model various systems, including M/M/1 queues, M/E_k/1 queues, M/M/c queues, sequential queues, inventory systems and queueing networks. We use a numerical method for the computation of the variance, the bias and the mean-square error of the time average. The effectiveness of the method is tested by experimenting with models of various stochastic systems. The contribution of this thesis is to use numerical and graphical interpretations to study the general characteristics of the measures. The important characteristics included in our study are decomposability and periodicity.

The electron transport through a molecular wire under the influence of an external laser field is studied using a reduced density matrix formalism. The full system is partitioned into the relevant part, i.e. the wire, electron reservoirs and a phonon bath. An earlier second-order perturbation theory approach of Meier and Tannor for bosonic environments which employs a numerical decomposition of the spectral density is used to describe the coupling to the phonon bath and is extended to deal with the electron transfer between the reservoirs and the molecular wire. Furthermore, from the resulting time-nonlocal (TNL) scheme a time-local (TL) approach can be determined. Both are employed to propagate the reduced density operator in time for an arbitrary time-dependent system Hamiltonian which incorporates the laser field non-perturbatively. Within the TL formulation, one can extract a current operator for the open quantum system. This enables a more general formulation of the problem which is necessary to employ an optimal control algorithm for open quantum systems in order to compute optimal control fields for time-distributed target states, e.g. current patterns. Thus, we take a fundamental step towards optimal control in molecular electronics. Numerical examples of the population dynamics, laser controlled current, TNL vs. TL and optimal control fields are presented to demonstrate the diverse applicability of the derived formalism.

Understanding energy transfer at the molecular scale is both essential for the design of novel molecular level devices and vital for uncovering the fundamental properties of non-equilibrium open quantum systems. In this thesis, we first establish the connection between molecular scale devices -- molecular electronics and phononics -- and open quantum system models. We then develop theoretical tools to study various properties of these models. We extend the standard master equation method to calculate the steady state thermal current and conductance coefficients. We then study the scaling laws of the thermal current with molecular chain size and energy, and apply this tool to investigate the onset of nonlinear thermal current - temperature characteristics, thermal rectification and negative differential conductance. Our master equation technique is valid in the ``on-resonance" regime, referring to the situation in which bath modes in resonance with the subsystem modes are thermally populated. In the opposite ``off-resonance" limit, we develop the Energy Transfer Born-Oppenheimer method to obtain the thermal current scaling without the need to solve for the subsystem dynamics. Finally, we develop a mapping scheme that allows the dynamics of a class of open quantum systems containing coupled subsystems to be treated by considering the separate dynamics in different subsections of the Hilbert space. We combine this mapping scheme with path integral numerical simulations to explore the rich phenomenon of entanglement dynamics within a dissipative two-qubit model. The formalisms developed in this thesis could be applied for the study of energy transfer in different realizations, including molecular electronic junctions, donor-acceptor molecules, artificial solid state qubits and cold-atom lattices.

The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research.

Understanding energy transfer at the molecular scale is both essential for the design of novel molecular level devices and vital for uncovering the fundamental properties of non-equilibrium open quantum systems. In this thesis, we first establish the connection between molecular scale devices -- molecular electronics and phononics -- and open quantum system models. We then develop theoretical tools to study various properties of these models. We extend the standard master equation method to calculate the steady state thermal current and conductance coefficients. We then study the scaling laws of the thermal current with molecular chain size and energy, and apply this tool to investigate the onset of nonlinear thermal current - temperature characteristics, thermal rectification and negative differential conductance. Our master equation technique is valid in the ``on-resonance" regime, referring to the situation in which bath modes in resonance with the subsystem modes are thermally populated. In the opposite ``off-resonance" limit, we develop the Energy Transfer Born-Oppenheimer method to obtain the thermal current scaling without the need to solve for the subsystem dynamics. Finally, we develop a mapping scheme that allows the dynamics of a class of open quantum systems containing coupled subsystems to be treated by considering the separate dynamics in different subsections of the Hilbert space. We combine this mapping scheme with path integral numerical simulations to explore the rich phenomenon of entanglement dynamics within a dissipative two-qubit model. The formalisms developed in this thesis could be applied for the study of energy transfer in different realizations, including molecular electronic junctions, donor-acceptor molecules, artificial solid state qubits and cold-atom lattices.

Most of the conventional communication systems use channel interleaving as well as hard decision decoding in their designs, which lead to discarding channel memory and soft-decision information. This simplification is usually done since the complexity of handling the memory or soft-decision information is rather high. In this work, we design two lossy joint source-channel coding (JSCC) schemes that do not use explicit algebraic channel coding for a recently introduced channel model, in order to take advantage of both channel memory and soft-decision information. The channel model, called the non-binary noise discrete channel with queue based noise (NBNDC-QB), obtains closed form expressions for the channel transition distribution, correlation coefficient, and many other channel properties. The channel has binary input and $2^q$-ary output and the noise is a $2^q$-ary Markovian stationary ergodic process, based on a finite queue, where $q$ is the output's soft-decision resolution. We also numerically show that the NBNDC-QB model can effectively approximate correlated Rayleigh fading channels without losing its analytical tractability. The first JSCC scheme is the so called channel optimized vector quantizer (COVQ) and the second scheme consists of a scalar quantizer, a proper index assignment, and a sequence maximum a posteriori (MAP) decoder, designed to harness the redundancy left in the quantizer's indices, the channel's soft-decision output, and noise time correlation. We also find necessary and sufficient condition when the sequence MAP decoder is reduced to an instantaneous symbol-by-symbol decoder, i.e., a simple instantaneous mapping. / Thesis (Master, Electrical & Computer Engineering) -- Queen's University, 2011-09-25 19:43:28.785

The purpose of this research is to reduce maintenance resources and improve Escravos–Gas–to–Liquids plant availability (EGTL) in Escravos, Nigeria using wireless condition monitoring. Secondary to the above is to justify the use of this technology over other conventional condition monitoring methods in petrochemical plants with specific reference to cost, reliability and security of the system. Wireless and continuous condition monitoring provides the means to evaluate current conditions of equipment and detect abnormalities. It allows for corrective measures to be taken to prevent upcoming failures. Continuous monitoring and event recording provides information on the energized equipment's response to normal and emergency conditions. Wireless/remote monitoring helps to coordinate equipment specifications and ratings, determine the real limits of the monitored equipment and optimize facility operations. Bentley N, (2005). Using wireless techniques eliminate any need for special cables and wires with lower installation costs if compared to other types of condition monitoring systems. In addition to this, wireless condition monitoring works well under difficult conditions in strategically important locations. The Escravos gas–to–liquid plant in Nigeria, located in a remote and offshore area where accommodation and space for offices is a factor for monitoring plant conditions in every office, is a typical example. Wireless technology for condition monitoring of energized equipment is applicable to both standalone and remote systems. In the research work of Meyer and Brambley (2002), they characterized the current problem with regards to cost effectiveness and availability of wireless condition monitoring. Maintenance of rotating equipment provides probability estimates of the total impact of the problem, cost implication of plant equipment maintenance and describes a generic system in which these developing technologies are used to provide real–time wireless/remote condition monitoring for rotating main air compressor (MAC) units and their components as a case study. Costs with today’s technology are provided and future costs are estimated, showing that benefits will greatly exceed costs in many cases, particularly if low–cost wireless monitoring is used. With management trends such as “re–engineering” and “downsizing” of the available workforce, wireless condition–monitoring of critical machines has been given more importance as a way to ensure quality production with fewer personnel. Wireless condition–monitoring using inexpensive wireless communication technology frees up existing plant maintenance personnel work on machines that are signaling problems and focusing the maintenance efforts away from attempting to work on a large population of machines to only those machines requiring immediate attention. Lloyd and Buddy (200) suggested that Point–to–point wireless data transmission systems, an excellent example of recent technological advances in communication systems, are now practical and cost–effective for industrial use. While both complex infrastructures and complex protocols are required for cellular communications, non– cellular communication systems, such as the point–to–point wireless data transmission system example, require no elaborate infrastructure. Limited research was done on the immediate benefits of implementing wireless condition monitoring systems in plants. All papers on the subject have been drawn up by manufacturers of such equipment. This research will thus also deliver a "third–party" perspective on the effectiveness of such devices, justifying their impact on data gathering security, cost and reliability. / Thesis (M.Ing. (Development and Management Engineering))--North-West University, Potchefstroom Campus, 2012.

The purpose of this research is to reduce maintenance resources and improve Escravos–Gas–to–Liquids plant availability (EGTL) in Escravos, Nigeria using wireless condition monitoring. Secondary to the above is to justify the use of this technology over other conventional condition monitoring methods in petrochemical plants with specific reference to cost, reliability and security of the system. Wireless and continuous condition monitoring provides the means to evaluate current conditions of equipment and detect abnormalities. It allows for corrective measures to be taken to prevent upcoming failures. Continuous monitoring and event recording provides information on the energized equipment's response to normal and emergency conditions. Wireless/remote monitoring helps to coordinate equipment specifications and ratings, determine the real limits of the monitored equipment and optimize facility operations. Bentley N, (2005). Using wireless techniques eliminate any need for special cables and wires with lower installation costs if compared to other types of condition monitoring systems. In addition to this, wireless condition monitoring works well under difficult conditions in strategically important locations. The Escravos gas–to–liquid plant in Nigeria, located in a remote and offshore area where accommodation and space for offices is a factor for monitoring plant conditions in every office, is a typical example. Wireless technology for condition monitoring of energized equipment is applicable to both standalone and remote systems. In the research work of Meyer and Brambley (2002), they characterized the current problem with regards to cost effectiveness and availability of wireless condition monitoring. Maintenance of rotating equipment provides probability estimates of the total impact of the problem, cost implication of plant equipment maintenance and describes a generic system in which these developing technologies are used to provide real–time wireless/remote condition monitoring for rotating main air compressor (MAC) units and their components as a case study. Costs with today’s technology are provided and future costs are estimated, showing that benefits will greatly exceed costs in many cases, particularly if low–cost wireless monitoring is used. With management trends such as “re–engineering” and “downsizing” of the available workforce, wireless condition–monitoring of critical machines has been given more importance as a way to ensure quality production with fewer personnel. Wireless condition–monitoring using inexpensive wireless communication technology frees up existing plant maintenance personnel work on machines that are signaling problems and focusing the maintenance efforts away from attempting to work on a large population of machines to only those machines requiring immediate attention. Lloyd and Buddy (200) suggested that Point–to–point wireless data transmission systems, an excellent example of recent technological advances in communication systems, are now practical and cost–effective for industrial use. While both complex infrastructures and complex protocols are required for cellular communications, non– cellular communication systems, such as the point–to–point wireless data transmission system example, require no elaborate infrastructure. Limited research was done on the immediate benefits of implementing wireless condition monitoring systems in plants. All papers on the subject have been drawn up by manufacturers of such equipment. This research will thus also deliver a "third–party" perspective on the effectiveness of such devices, justifying their impact on data gathering security, cost and reliability. / Thesis (M.Ing. (Development and Management Engineering))--North-West University, Potchefstroom Campus, 2012.

In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics