This study explores optimal strategies for maximizing scores and winning in the popular dice game Qwixx, analyzing both single and multiplayer gameplay scenarios. Through extensive simulations, various strategies were tested and compared, including a scorebased approach that uses a formula tuned by MCMC random walks, and race-to-lock approaches which use absorbing Markov chain qualities of individual score sheet rows to find ways to lock rows as quickly as possible. Results indicate that employing a scorebased strategy, considering gap, count, position, skip, and likelihood scores, significantly improves performance in single player games, while move restrictions based on specific dice roll sums in the race-to-lock strategy were found to enhance winning and scoring points in multiplayer games. While the results do not achieve the optimal scores attained by prior informal work, the study provides valuable insights into decision-making processes and gameplay optimization for Qwixx enthusiasts, offering practical guidance for players seeking to enhance their performance and strategic prowess in the game. It also serves as a lesson for how to approach optimization problems in the future.

The objective of the study is to apply Markov chain analysis to analyze student flow through King Abdul Aziz University (KAU) in Saudi Arabia, and to predict important metrics such as graduation and dropout rates. This objective arises from examination of the policies of KAU University. We begin with background information detailing the subject of study, then move into a general outline of stochastic processes. We then use these methods to construct a specific matrix of transition probabilities with data from the university student population. Finally, we discuss the calculation of the possibilities of transition between each level of study and the average time a student takes to complete each stage. The study uses Markov chains with these outcomes to analyze student retention data from the Department of Mathematics at KAU. From this analysis, the study will provide university policy recommendations that can be generalized to examine other universities.

Les véhicules aériens sans pilote (UAVs), utilisés et développés pour la première fois dans le domaine militaire, ont connu de profonds changements ces dernières années et sont de plus en plus utilisés dans le domaine civil. Etant plus connus sous le nom des drones, ils sont le plus souvent utilisés dans les domaines civiles et militaires. Ils sont employés pour : la lutte contre les incendies, le sauvetage ainsi que dans des applications spécifiques comme la surveillance et l’attaque. Le vol en formation est de loin le plus utilisé car il permet une répartition judicieuse des tâches et améliore grandement l’efficacité des drones (principe de l’attaque en meute, des animaux carnassiers). Cela pose alors la problématique de la coordination et de la stratégie, ainsi que du type de fonctionnement (maitre/esclave,…).Le type et la qualité d’informations optimums restent aussi à définir.L'utilisation accrue de ces systèmes coopératifs dans des environnements dangereux rend leur fiabilité essentielle pour prévenir tout événement catastrophique. Une performance globale de la flotte des drones doit être garantie, malgré une possible dégradation des composants ou de toute modification du réseau et de l'environnement. Il est nécessaire de détecter les comportements anormaux pouvant contribuer aux collisions et ainsi affecter la mission. Compte tenu des performances et du coût, les systèmes à tolérance de pannes et à redondance ne représentent pas toujours la solution la plus efficace pour ce type de vol de flotte en formation. Différentes méthodes telles que l'analyse par arbre de défaillance (ADD), l'analyse des modes de défaillance, de leurs effets et de leurs criticités (AMDEC) ont été utilisées dans le monde des hélicoptères.Dans une première partie, une méthode statique basée sur l’ADD est proposée, pour assurer la fiabilité de la communication entre les drones d’un côté et entre les drones et la station de base de l’autre côté en accentuant l’échange de flux d’informations. Nous utilisons des arbres de défaillance pour représenter les différentes conditions d’erreur de ce système complexe.Dans une deuxième partie, nous analysons les différents états de défaillance des communications et leurs probabilités. Ce processus étant stochastique, une approche par chaîne de Markov absorbante est développée. L’approche proposée peut être utilisée pour trouver les scenarios les plus risqués et les éléments à prendre en compte pour améliorer la fiabilité.Enfin, dans une troisième partie, nous étudions le problème de réception des messages d’un drone en proposant un protocole basé sur le nombre de retransmissions. La réception est assurée avec une certaine probabilité de fiabilité, en fonction de plusieurs attributs tels que la modulation, le taux d’erreur des bits (BER) caractérisant les drones. / Unmanned aerial vehicles, used and developed initially in the military field, have experienced profound changes in recent years and are increasingly used in the civilian field. Recognized as drones, they are most often used in the civil and military domains. They are used for firefighting, rescue as well as in specific applications such as surveillance and attack. The formation flight is the most used because it allows a judicious distribution of the tasks and greatly improves the efficiency of the drones (principle of the attack in pack, carnivorous animals). This will raise the issue of coordination and strategy, as well as the type of operation (master /slave, ...). The type and quality of optimal information also remain to be defined.The increased use of these cooperative systems in hazardous environments makes their reliability essential to prevent any catastrophic event. Overall performance of the drone fleet should be ensured, despite possible degradation of components or any changes that occur to the network and the environment. It is necessary to detect the anomalous behaviors that might contribute to collisions and thus affect the mission. Taking into consideration performance and cost, the fault-tolerant system and redundant systems are not always the most efficient solution for the formation fleet flight. Different methods like the fault tree analysis (FTA), Failure Modes and Effects Analysis (FMEA) have been used in the helicopter field.In the first part, we propose a static method based on FTA, to ensure a successful communication between the drones from one side, and between the drones and the ground station from the other side by emphasizing on the exchange of information flows. It uses various fault trees to represent the different error conditions of this complex system.In the second part, we analyze the different fault states and their probabilities. As this process is stochastic, an absorbing Markov chain approach is developed. The proposed approach can be used to find the most risky scenarios and considerations for improving reliability.Finally, in the third part, we put the emphasis on the message receipt problem in a drone’s communication network by proposing a protocol based on number of retransmissions. The reception of a message is provided with a certain probability of reliability depending on several attributes such as modulation and bit error rate (BER) characterizing the UAVs.

This dissertation presents a theoretical study of arbitrary discretizations of general nonequilibrium and non-steady-state systems. It will be shown that, without requiring the partitions of the phase-space to fulfill certain assumptions, such as culminating in Markovian partitions, a Markov chain can be constructed which has the same macro-change of probability of the occupation of the states as the original process. This is true for any classical and semiclassical system under any discrete or continuous, deterministic or stochastic, Markovian or non-Markovian dynamics. Restricted to classical and semi-classical systems, a formalism is developed which treats the projection of arbitrary (multidimensional) complex systems onto a discrete set of states of an abstract state-space using time and ensemble sampled transitions between the states of the trajectories of the original process. This formalism is then used to develop expressions for the mean first passage time and (in the case of projections resulting in pseudo-one-dimensional motion) for the individual residence times of the states using just the time and ensemble sampled transition rates. The theoretical work is illustrated by several numerical examples of non-linear diffusion processes. Those include the escape over a Kramers potential and a rough energy barrier, the escape from an entropic barrier, the folding process of a toy model of a linear polymer chain and the escape over a fluctuating barrier. The latter is an example of a non- Markovian dynamics of the original process. The results for the mean first passage time and the residence times (using both physically meaningful and non-meaningful partitions of the phase-space) confirms the theory. With an accuracy restricted only by the resolution of the measurement and/or the finite sampling size, the values of the mean first passage time of the projected process agree with those of a direct measurement on the original dynamics and with any available semi-analytical solution.