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This thesis is in two parts. Part I concerns simulation of random fields using the circulant embedding method, and Part II studies the numerical solution of stochastic differential equations (SDEs).

Please read the abstract in the front section of this document / Dissertation (MSc (Mathematics))--University of Pretoria, 2005. / Mathematics and Applied Mathematics / unrestricted

Two well-known methods of improving the reliability of a system are (i) provision of redundants units, and (ii) repair maintenance. In a redundant system more units are made available for performing the system function when fewer are required exactly. There are two major types of redundancy- parallel and standby. In this thesis we confine to both these redundant systems. A series system is also studied. Some of the typical assumptions made in the analysis of redundant systems are (i) the repair times are assumed to be exponential (ii) the system measures are modeled but not estimated (iii) the system is available continuously (iv) environmental factors not affecting the system (v) the failures take place only in one stage (vi) the switching device is perfect (vii) system reliability for given chance constraints (viii) the time required to transfer a unit from the standby state to the operating stage is negligible (instantaneous switchover) (ix) the failures and repairs are independent. However, we frequently come across systems where one or more of these assumptions have to be dropped. This is the motivation for the detailed study of the models presented in this thesis. In this thesis we present several models of redundant systems relaxing one or more of these assumptions simultaneously. More specifically it is a study of stochastic models of redundant repairable systems with ‘rest period’ for the operator, non-instantaneous switchover, imperfect switch, intermittent use, and series system optimization. The thesis contains seven chapters. Chapter 1 is introductory in nature and contains a brief description of the mathematical techniques used in the analysis of redundant systems. In chapter 2, a two unit system with Erlangian repair time is studied by relaxing the assumptions (i) and (ii). The difference- differential equations are formulated for the state probabilities, and the system measures like reliability and the availability are obtained over a long run. The asymptotic interval estimation is studied for these system measures. The model has been illustrated numerically. In chapter 3, an n unit system operating intermittently, and in a random environment is studied, by relaxing the assumptions (iii) and ( iv). In an intermittently used system, the mean number of disappointments is one of the important measures, which has been obtained for this system in the steady state. In chapter 4, the assumption (v) and (vi) are relaxed. In most of the models studied earlier in reliability analysis is the study of system measures like reliability and availability. In this chapter, profit analysis of a single unit system with three possible modes of the failure of the unit is studied .This chapter consists of two models: in model 1, the unit goes under repair (if a repairman is available) the moment it fails partially, whereas in model 2 the unit goes under repair at complete failure. The repairman appears in, and disappears from, the system randomly. A comparison between these two models has been studied, after calculating numerically the profit and the MTSF. Contrary to the previous chapters, stochastic optimization is studied using the Branch and Bound technique in chapter 5 (relaxing the assumption (vii)). In this chapter, an n unit system operating in a random environment is considered. The environment determines the number of units required for the satisfactory performance of the system. Assuming that a unit in standby can fail and that the environment is described by a Markov process, we obtained expressions for the distribution and the moments of the time to the first disappointment, and the expected number of disappointments over an arbitrary interval (0, t]. In chapter 6, the assumption (viii) is relaxed. The reliability, availability and the busy period analysis is studied with the assumption of the non-instantaneous switchover (the time taken from standby state to the operating state is non-negligible random variable). It is also assumed that the unit has three possible failure modes (normal, partial and total failure). Numerical example illustrated the results obtained. The assumption (ix) is relaxed in chapter 7, and a two-unit cold standby system with the provision of rest for the operating unit is studied. Also, the failure and repair times of each unit assumed to be correlated by taking their joint density as bivariate exponential. The system is observed at suitable regenerative epochs to obtain various reliability characteristics of interest, such as the distribution of time to system failure and its mean, and the steady-state probabilities of the system being in up or down states or under repair. Earlier results are verified as particular cases. Numerical example illustrated the results obtained. / Thesis (PhD (Systems Engineering))--University of Pretoria, 2007. / Industrial and Systems Engineering / unrestricted

Adaptive modulation has been actively researched as a means to increase spectral efficiency of wireless communications systems. In general, analytic closed form models have been derived for the performance of the communications system as a function of the control parameters. However, in systems where general error correction coding is employed, it may be difficult to derive closed form performance functions of the communications systems. In addition, in closed form optimization, real time adaptation is not possible. Systems designed with deterministic state optimization are developed offline for a certain set of parameters and hardwired into mobile devices. In this thesis we present stochastic learning algorithms for adaptive modulation design. The algorithms presented allow for adaptive modulation system design in-dependent of error correction coding and modulation constellation requirements. In real time, the performance of the system is measured and stochastic approximation techniques are used to learn the optimal transmission parameters of the system. The technique is applied to Software Defined Radio (SDR) platforms, an emerging wireless technology which is currently being researched as a means of designing intelligent communications devices. The fundamental property that sets SDR apart from traditional radios is that the communications parameters are controlled in software, allowing for real-time control of physical layer communications. Our treatment begins by modeling the time evolution of the adaptive modulation process as a general state space Markov chain. We show the existence and uniqueness of the invariant measure and model performance functions as expectations with respect to the invariant measure. We consider constrained and unconstrained throughput optimization. We show that the cost functions considered are convex. Next we present stochastic approximation algorithms that are used to estimate the gradient of the cost function given only noisy estimates. We conclude by presenting simulation results obtained by the presented method. The learning based method is able to achieve the maximum throughput as dictated by exhaustive Monte Carlo simulation of the communications system, which provide an upper bound on performance. In addition, the learning algorithm is able to optimize communications under various error correction schemes. The tracking abilities of the algorithm are also demonstrated. We see that the proposed method is able to track optimal throughput settings as constraints are changed in time. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Transportation systems such as mail, freight, passenger and even telecommunication systems most often employ a hub and spoke network structure since correctly designed they give a strong balance between high service quality and low costs resulting in an economically competitive operation. In addition, consumers are increasingly demanding fast and reliable transportation services, with services such as next day deliveries and fast business and pleasure trips becoming highly sought after. This makes finding an efficient design of a hub and spoke network of the utmost importance for any competing transportation company. However real life situations are complicated, dynamic and often require responses to many different fixed and random events. Therefore modeling the question of what is an optimal hub and spoke network structure and finding an optimal solution is very difficult. Due to this, many researchers and practitioners alike make several assumptions and simplifications on the behavior of such systems to allow mathematical models to be formulated and solved optimally or near optimally within a practical timeframe. Some assumptions and simplifications can however result in practically poor network design solutions being found. This thesis contributes to the research of hub and spoke networks by introducing new stochastic models and fast solution algorithms to help bridge the gap between theoretical solutions and designs that are useful in practice. Three main contributions are made in the thesis. First, in Chapter 2, a new formulation and solution algorithms are proposed to find exact solutions to a stochastic p-hub center problem. The stochastic p-hub center problem is about finding a network structure, where travel times on links are stochastic, which minimizes the longest path in the network to give fast delivery guarantees which will hold for some given probability. Second, in Chapter 3, the stochastic p-hub center problem is looked at using a new methodological approach which gives more realistic solutions to the network structures when applied to real life situations. In addition a new service model is proposed where volume of flow is also accounted for when considering the stochastic nature of travel times on links. Third, in Chapter 4, stochastic volume is considered to account for capacity constraints at hubs and, de facto, reduce the costs embedded in excessive hub volumes. Numerical experiments and results are conducted and reported for all models in all chapters which demonstrate the efficiency of the new proposed approaches.

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Integrated robust planning systems that cover all levels of SC hierarchy have become increasingly important. Strategic, tactical, and operational SC plans should not be generated in isolation to avoid infeasible and conflicting decisions. On the other hand, enterprise planning systems contain over millions of records that are processed in each planning iteration. In such enterprises, the ability to generate robust plans is vital to their success because such plans can save the enterprise resources that may otherwise have to be reserved for likely SC plan changes. A robust SC plan is valid in all circumstances and does not need many corrections in the case of interruption, error, or disturbance. Such a reliable plan is proactive as well as reactive. Proactivity can be achieved by forecasting the future events and taking them into account in planning. Reactivity is a matter of agility, the capability of keeping track of system behaviour and capturing alarming signals from its environment, and the ability to respond quickly to the occurrence of an unforeseen event. Modeling such a system behaviour and providing solutions after such an event is extremely important for a SC. This study focuses on integrated supply chain planning with stochastic dynamic considerations. An integrated tactical-operational model is developed and then segregated into two sub-models which are solved iteratively. A SC is a stochastic dynamic system whose state changes over time often in an unpredictable manner. As a result, the customer demand is treated as an uncertain parameter and is handled by exploiting scenario-based stochastic programming. The increase in the number of scenarios makes it difficult to obtain quick and good solutions. As such, a Branch and Fix algorithm is developed to segregate the stochastic model into isolated islands so as to make the computationally intractable problem solvable. However not all the practitioners, planners, and managers are risk neutral. Some of them may be concerned about the risky extreme scenarios. In view of this, the robust optimization approach is also adopted in this thesis. Both the solution robustness and model robustness are taken into account in the tactical model. Futhermore, the dynamic behaviour of a SC system is handled with the concept of Model Predictive Control (MPC).

In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex-concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large-scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution. / Business, Sauder School of / Graduate