In this thesis, we consider a risk model which incorporates multiple threshold levels characterizing an insurer's minimal capital requirement, dividend paying situations, and external financial activities. Our model is based on discrete monetary and time units, and the main quantities of interest are the finite-time ruin probabilities and the expected total discounted dividends paid prior to ruin. We mainly focus on the development of computational methods to attain these quantities of interest. One of the popular methods in the current literature used for studying such problems involves a recursive approach which incorporates appropriate conditioning arguments on the claim times and sizes, and we implement this procedure as well. Furthermore, ruin can occur due to both a claim as well as interest expense accumulation as our model allows the insurer to borrow money from an external fund. In this thesis, we consider only non-stochastic interest rates for both lending and borrowing activities. After constructing appropriate recursive formulae for the finite-time ruin probabilities and the expected total discounted dividends paid prior to ruin, we investigate various numerical examples and make some observations concerning the impact our threshold levels have on finite-time ruin probabilities and expected total discounted dividends paid prior to ruin.

In this thesis, we consider a risk model which incorporates multiple threshold levels characterizing an insurer's minimal capital requirement, dividend paying situations, and external financial activities. Our model is based on discrete monetary and time units, and the main quantities of interest are the finite-time ruin probabilities and the expected total discounted dividends paid prior to ruin. We mainly focus on the development of computational methods to attain these quantities of interest. One of the popular methods in the current literature used for studying such problems involves a recursive approach which incorporates appropriate conditioning arguments on the claim times and sizes, and we implement this procedure as well. Furthermore, ruin can occur due to both a claim as well as interest expense accumulation as our model allows the insurer to borrow money from an external fund. In this thesis, we consider only non-stochastic interest rates for both lending and borrowing activities. After constructing appropriate recursive formulae for the finite-time ruin probabilities and the expected total discounted dividends paid prior to ruin, we investigate various numerical examples and make some observations concerning the impact our threshold levels have on finite-time ruin probabilities and expected total discounted dividends paid prior to ruin.

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We consider the optimal pricing problem in a service facility in order to maximize its long-run average profit per unit time. We model the facility as a queueing process that may have finite or infinite capacity. Customers are admitted into the system if it is not full and if they are willing to pay the price posted by the service provider. Moreover, the congestion level in the facility incurs penalties that greatly influence profit. We model congestion penalties in three different manners: holding costs, balking customers and impatient customers. First, we assume that congestion-dependent holding costs are incurred per unit of time. Second, we consider that each customer might be deterred by the system congestion level and might balk upon arrival. Third, customers are impatient and can leave the system with a full refund before being serviced. We are interested in both static and dynamic pricing for all three types of congestion penalties. In the static case, we demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. In the dynamic case, we show the existence of an optimal stationary policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We provide explicit expressions for this policy under certain conditions. We also analyze the structure of this policy and investigate its relationship with our optimal static price.

Near the beginning of the last century, R. A. Fisher and Sewall Wright devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for reasons of mathematical tractability - makes assumptions that are violated in many real-world populations. In particular, the classical models assume fixed population sizes, held constant by (unspecified) sampling mechanisms. Here, we consider an alternative framework that merges Moran’s continuous time Markov chain model of allele frequencies in haploid populations of fixed size with the density dependent models of ecological competition of Lotka, Volterra, Gause, and Kolmogorov. This allows for haploid populations of stochastically varying – but bounded – size. Populations are kept finite by resource limitation. We show the existence of limits that naturally generalize the weak and strong selection regimes of classical population genetics, which allow the calculation of fixation times and probabilities, as well as the long-term stationary allele frequency distribution.

Near the beginning of the last century, R. A. Fisher and Sewall Wright devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for reasons of mathematical tractability - makes assumptions that are violated in many real-world populations. In particular, the classical models assume fixed population sizes, held constant by (unspecified) sampling mechanisms. Here, we consider an alternative framework that merges Moran’s continuous time Markov chain model of allele frequencies in haploid populations of fixed size with the density dependent models of ecological competition of Lotka, Volterra, Gause, and Kolmogorov. This allows for haploid populations of stochastically varying – but bounded – size. Populations are kept finite by resource limitation. We show the existence of limits that naturally generalize the weak and strong selection regimes of classical population genetics, which allow the calculation of fixation times and probabilities, as well as the long-term stationary allele frequency distribution.

The topic of Markov Random Fields (MRFs) has been well studied in the past, and has found practical use in various image processing, and machine learning applications. Where coding is concerned, MRF specific schemes have been largely unexplored. In this thesis, an overview is given of recent developments and challenges in the lossless coding of MRFs. Specifically, we concentrate on difficulties caused by computational intractability due to the partition function of the MRF. One proposed solution to this problem is to segment the MRF with a cutset, and encode the components separately. Using this method, arithmetic coding is possible via the Belief Propagation (BP) algorithm. We consider two cases of the BP algorithm: MRFs with only simple cliques, and MRFs with complex cliques. In the latter case, we study a minimum radius condition requirement for ensuring that all cliques are accounted for during coding. This condition also simplifies the process of conditioning on observed sites. Finally, using these results, we develop a systematic procedure of clustering and choosing cutsets. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2013-08-12 14:50:00.596

This study gives detailed proofs of some limit theorems in probability which are important in theoretical and applied probability, The general introduction contains definitions and theorems that are basic tools of the later development. Included in this first chapter is material concerning normal distributions and characteristic functions, The second chapter introduces lower and upper bounds of the ratio of the binomial distribution to the normal distribution., Then these bound are used to prove the local Deioivre-Laplace limit theorem. The third chapter includes proofs of the central limit theorems for identically distributed and non-identically distributed random variables,

Analyses are addressed for a number of problems in queueing systems and stochastic modeling that arose due to an investigation into techniques that could be used to approximate general closed networks. In Chapter II, a method is presented to calculate the system size distribution at an arbitrary point in time and at departures for a (n)/G/1/N queue. The analysis is carried out using an embedded Markov chain approach. An algorithm is also developed that combines our analysis with the recursive method of Gupta and Rao. This algorithm compares favorably with that of Gupta and Rao and will solve some situations when Gupta and Rao's method fails or becomes intractable. In Chapter III, an approach is developed for generating exact solutions of the time-dependent conditional joint probability distributions for a phase-type renewal process. Closed-form expressions are derived when a class of Coxian distributions are used for the inter-renewal distribution. The class of Coxian distributions was chosen so that solutions could be obtained for any mean and variance desired in the inter-renewal times. In Chapter IV, an algorithm is developed to generate numerical solutions for the steady-state system size probabilities and waiting time distribution functions of the SM/PH/1/N queue by using the matrix-analytic method. Closed form results are also obtained for particular situations of the preceding queue. In addition, it is demonstrated that the SM/PH/1/N model can be implemented to the analysis of a sequential two-queue system. This is an extension to the work by Neuts and Chakravarthy. In Chapter V, principal results developed in the preceding chapters are employed for approximate analysis of the closed network of queues with arbitrary service times. Specifically, the (n)/G/1/N queue is applied to closed networks of a general topology, and a sequential two-queue model consisting of the (n)/G/1/N and SM/PH/1/N queues is proposed for tandem queueing networks.