The aim of this thesis is to first give a brief review of waiting line problems which often is a subject related to queueing theory. Simple counting processes such as the Poisson process and the duration of service time of each customer being exponentially distributed are often taught in a undergraduate or graduate stochastic process course. In this thesis, we will continue discussing such waiting line problems with priority assignment on each customer. This type of queueing processes are called priority queueing models. Patients requiring ER service are triaged and the order of providing service to patients more than often reflects early symptoms and complaints than final diagnoses. Triage systems used in hospitals vary from country to country and region to region. However, the goal of using a triage system is to ensure that the sickest patients are seen first. Such wait line system is much comparable to a priority queueing system in our study. The finite Markov chain imbedding technique is very effective in obtaining the waiting time distribution of runs and patterns. Applying this technique, we are able to obtain the probability distribution of customer wait time of priority queues. The results of this research can be applied directly when studying patient wait time of emergency medical service. Lengthy ER wait time issue often is studied from the view of limited spacing and complications in hospital administration and allocation of resources. In this thesis, we would like to study priority queueing systems by mathematical and probabilistic modeling.

An optimal stochastic multi-crop irrigation scheduling algorithm was developed which was able to incorporate complex farm system models, and constraints on daily and seasonal water use, with the objective of maximising farm profit. This scheduling method included a complex farm simulation model in the objective function, used decision variables to describe general management decisions, and used a custom heuristic method for optimisation. Existing optimal schedulers generally use stochastic dynamic programming which relies on time independence of all parameters except state variables, thereby requiring over-simplistic crop models. An alternative scheduling method was therefore proposed which allows for the inclusion of complex farm system models. Climate stochastic properties are modelled within the objective function through the simulation of several years of historical data. The decoupling of the optimiser from the objective function allows easy interchanging of farm model components. The custom heuristic method, definition of decision variables, and use of the Markov chain equation (relating an irrigation management strategy to mean water use) considerably increases optimisation efficiency. The custom heuristic method used simulated annealing with continuous variables. Two extensions to this method were the efficient incorporation of equality constraints and utilisation of population information. A case study comparison between the simulated annealing scheduler and scheduling using stochastic dynamic programming, using a simplistic crop model, showed that the two methods resulted in similar performance. This demonstrates the ability of the simulated annealing scheduler to produce close to optimal schedules. A second case study demonstrates the ability of the simulated annealing scheduler to incorporate complex farm system models by including the FarmWi$e model by CSIRO in the objective function. This case study indicates that under conditions of limited seasonal water, the simulated annealing scheduler increases pasture yield returns by an average of 10%, compared with scheduling irrigation using best management practice. Alternatively expressed, this corresponds to a 20-25% reduction in seasonal water use (given no change in yield return).

The aim of this thesis is to first give a brief review of waiting line problems which often is a subject related to queueing theory. Simple counting processes such as the Poisson process and the duration of service time of each customer being exponentially distributed are often taught in a undergraduate or graduate stochastic process course. In this thesis, we will continue discussing such waiting line problems with priority assignment on each customer. This type of queueing processes are called priority queueing models. Patients requiring ER service are triaged and the order of providing service to patients more than often reflects early symptoms and complaints than final diagnoses. Triage systems used in hospitals vary from country to country and region to region. However, the goal of using a triage system is to ensure that the sickest patients are seen first. Such wait line system is much comparable to a priority queueing system in our study. The finite Markov chain imbedding technique is very effective in obtaining the waiting time distribution of runs and patterns. Applying this technique, we are able to obtain the probability distribution of customer wait time of priority queues. The results of this research can be applied directly when studying patient wait time of emergency medical service. Lengthy ER wait time issue often is studied from the view of limited spacing and complications in hospital administration and allocation of resources. In this thesis, we would like to study priority queueing systems by mathematical and probabilistic modeling.

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This study tests the size effect in the London Stock Exchange, using data for all nonfinancial listed firms from January 1985 to December 1995. The initial tests indicate that average stock returns are negatively related to firm size and that small firm portfolios earn returns in excess of the market risk. Further, the study tests whether the size effect is a proxy for variables such as the Book-to- Market Value and the Borrowing Ratio, as well as the impact of the dividend and the Bid- Ask spread on the return of the extreme size portfolios. The originality of this study is in the application of the Markov Chain Model to testing the Random Walk and Bubbles hypotheses, and the Vector Autoregression (VAR) framework for testing the relationship of macroeconomic variables with size portfolio returns.

Master of Science / Department of Agricultural Economics / Terry Griffin / Farmers and agricultural lenders often seek the ability to identify positive or negative characteristics to improve farm operations. Determining these characteristics has been the goal of many research studies. More often than not, a unique set of uncontrollable events was credited for contributing the majority of one farm’s success relative to their peers. The goal of this study was to evaluate the assumption that farmers can control their financial persistence defined as remaining in their current financial category, based upon a farm’s debt to asset ratio (D/A), and net farm income per acre (NFI acre⁻¹). Financial categories give agricultural producers a concrete answer to the question of one farm’s ability to maintain their financial persistence during market downturns and poor growing conditions and include Favorable, Marginal Income, Marginal Solvency, and Vulnerable. Farmers across the United States are subject to many uncontrollable variables (temperature, precipitation, market volatility, land value fluctuations, interest rates) leaving them vulnerable to agricultural market downturns, such as the one that began in 2014. Seasonal cash inflows and outflows of farms and their profitability create a difficult situation for farmers and agricultural lenders alike to predict the future. Identifying and estimating the likelihood of financial persistence has become an area of interest for farmers, their advisors, and their financial lenders. Currently, agricultural lenders rely on loan assessment techniques, such as net present values and loss-based methods. These techniques fail to account for the unique and often long-term investment nature of farming. If an additional method for identifying at-risk farms or at least understanding the likelihood of persistence in farms could be found, it would provide an insight into the riskiness of lending to a farm and provide agricultural lenders with an additional analysis tool. The dynamic nature of farm financials and the ever-changing variables of farming limit traditional statistical methods. Considering the difficulty associated with predicting farm default rates due to the complexity of the question, a secondary approach is possible. This study utilized an approach in determining farm financial persistence by estimating the Markov Chain probabilities of four financial categories ranging from Favorable, solvent with positive income to Vulnerable, an insolvent and negative income financial position. Kansas Farm Management Association (KFMA) data from 1993 to 2014 were used to estimate the probability of transitioning between financial categories. This thesis combines transition probabilities of Kanas farms and a multinomial logit model (MNL) to identify farm characteristics of significance. The matrix of probabilities generated, when interpreted, provide information about Kansas farms and their probability of financial persistence, and the MNL model allows for insights into favorable or un-favorable farm characteristics. Farms were found to transition easily between financial categories that had the same debt to asset ratio (D/A), but different net farm income per acre (NFI acre⁻¹, positive or negative) indicating that farm income is more easily changed than farm D/A ratios. Farms in the Favorable category (D/A < 0.4, + NFI acre⁻¹) had the largest probability of financial persistence at 0.83, whereas Vulnerable farms (D/A > 0.4, - NFI acre⁻¹) were most likely to transition to the Marginal Solvency category (D/A > 0.4, + NFI acre⁻¹) with a probability of transitioning of 0.55 versus the probability of remaining in the Vulnerable category of 0.33. It was also found that crop mixture and age were not statistically significant in the MNL model, but gross profit margin and a farm’s percentage of owned land out of total crop acres were statistically significant in explaining why farms were in each category.

Many modern-day Bioinformatics algorithms rely heavily on statistical models to analyze their biological data. Some of these statistical models lend themselves nicely to standard high performance computing optimizations such as parallelism, while others do not. One such algorithm is Markov Chain Monte Carlo (MCMC). In this thesis, we present a heterogeneous compute solution for optimizing GenSel, a genetic selection analysis tool. GenSel utilizes a MCMC algorithm to perform Bayesian inference using Gibbs sampling. Optimizing an MCMC algorithm is a difficult problem because it is inherently sequential, containing a loop carried dependence between each Markov Chain iteration. The optimization presented in this thesis utilizes GPU computing to exploit the data-level parallelism within each of these iterations. In addition, it allows for the efficient management of memory, the pipelining of CUDA kernels, and the use of multiple GPUs. The optimizations presented show performance improvements of up to 1.84 times that of the original algorithm.

Markov Chain Monte Carlo algorithms are often used to sample combinatorial structures such as matchings and independent sets in graphs. A Markov chain is defined whose state space includes the desired sample space, and which has an appropriate stationary distribution. By simulating the chain for a sufficiently large number of steps, we can sample from a distribution arbitrarily close to the stationary distribution. The number of steps required to do this is known as the mixing time of the Markov chain. In this thesis, we consider a number of Markov chains for sampling matchings, both in general and more restricted classes of graphs, and also for sampling independent sets in claw-free graphs. We apply techniques for showing rapid mixing based on two main approaches: coupling and conductance. We consider chains using single-site moves, and also chains using large block moves. Perfect matchings of bipartite graphs are of particular interest in our community. We investigate the mixing time of a Markov chain for sampling perfect matchings in a restricted class of bipartite graphs, and show that its mixing time is exponential in some instances. For a further restricted class of graphs, however, we can show subexponential mixing time. One of the techniques for showing rapid mixing is coupling. The bound on the mixing time depends on a contraction ratio b. Ideally, b < 1, but in the case b = 1 it is still possible to obtain a bound on the mixing time, provided there is a sufficiently large probability of contraction for all pairs of states. We develop a lemma which obtains better bounds on the mixing time in this case than existing theorems, in the case where b = 1 and the probability of a change in distance is proportional to the distance between the two states. We apply this lemma to the Dyer-Greenhill chain for sampling independent sets, and to a Markov chain for sampling 2D-colourings.

For the purpose of modelling and prediction of volatility, the family of Stochastic Volatility (SV) models is an alternative to the extensively used ARCH type models. SV models differ in their assumption that volatility itself follows a latent stochastic process. This reformulation of the volatility process makes however model estimation distinctly more complicated for the SV type models, which in this paper is conducted through Markov Chain Monte Carlo methods. The aim of this paper is to assess the standard SV model and the SV model assuming t-distributed errors and compare the results with their corresponding GARCH(1,1) counterpart. The data examined cover daily closing prices of the Swedish stock index OMXS30 for the period 2010-01-05 to 2016- 03-02. The evaluation show that both SV models outperform the two GARCH(1,1) models, where the SV model with assumed t-distributed error distribution give the smallest forecast errors.

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