This paper describes the analysis tool DSPNexpress which has been developed at the Technische Universität Berlin since 1991. The development of DSPNexpress has been motivated by the lack of a powerful software package for the numerical solution of deterministic and stochastic Petri nets (DSPNs) and the complexity requirements imposed by evaluating memory consistency models for multicomputer systems. The development of DSPNexpress has gained by the author's experience with the version 1.4 of the software package GreatSPN. However, opposed to GreatSPN, the software architecture of DSPNexpress is particularly tailored to the numerical evaluation of DSPNs. Furthermore, DSPNexpress contains a graphical interface running under the X11 window system. To the best of the author's knowledge, DSPNexpress is the first software package which contains an efficient numerical algorithm for computing steady-state solutions of DSPNs.

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the gt different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t ×n submatrices, we consider only submatrices of dimension t ×n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.

Vector born disease account for about one third of all cases of emerging diseases. Culex sp., particularly, is one of the most important mosquito vectors transmitting important diseases such as the West Nile virus, filariasis and related encephalitis. Because there are no vaccines available the most effectual means to prevent infections from the above diseases, is to target mosquitos to prevent bites and disease transmission. However, to be effective such a strategy, it is important to predict the temporal change in mosquito abundance as well as to study how it is affected by weather conditions. This dissertation is devoted on the development of new methods to predict arthropod vector dynamics and with emphasis on the development of stochastic models and computational methods for predicting Culex sp. abundance in Northern Greece. The current dissertation is divided in three parts. The first part explores the non-trivial associations between Culex sp. mosquito abundance and weather variables using traditional and straightforward novel techniques. The information from the first part was a prerequisite for developing a series of stochastic prediction models based on the most detrimental factors affecting mosquito abundance. In the second part, a series of conventional and conditional stochastic Markov chain models are applied for the first time to predict the non-linear dynamics of Culex sp. adult abundance. In the third part of the dissertation a soft computing approach is introduced to model the population dynamics of Culex sp. and a series of autoregressive artificial neural networks are implied. Finally, the information of the models is extrapolated and a machine learning algorithm is proposed to be used for predicting arthropod vector dynamics having practical implications for public health decision making. Based on the current results there was a high and positive correlation between temperature and mosquito abundance during both observation years (r = 0.6). However, a very poor correlation was observed between rain and weekly mosquito abundances (r = 0.29), as well as between wind speed (r = 0.29), respectively. Additionally, according to the multiple linear regression model the effect of temperature, was significant. The continuous power spectrum of the mosquito abundance counts and mean temperatures depict in most cases similar power for periods which are close to 1 week, indicating the point of the lowest variance of the time series, although appearing on slightly different moments of time. The cross wavelet coherent analysis showed that inter weekly cycles with a period between 2 and 3 weeks between mosquito abundance and temperature were coherent mostly during the first and the last weeks of the season. Hence, the wavelet analysis shows a progressive oscillation in mosquito occurrences with time, which is higher at the start and the end of the season. Moreover, in contrast with standard methods of analysis, wavelets can provide useful insights into the time-resolved oscillation structure of mosquito data and accompanying revealing a non-stationary association with temperature. According to the correlation results a climate-conditioned Markov Chain (CMC) model was developed and applied for the first time to predict the dynamics of vectors of important medical diseases. Temporal changes in mosquito population profiles were generated to simulate the probabilities of a high population impact. The probabilities achieved from the trained model are very near to the observed data and the CMC model satisfactorily describes the temporal evolution of the mosquito population process. In general, our numerical results indicate that it is more likely for the population system to move into a state of high population level, when the former is a state of a low population level than the opponent. Field data on frequencies of successive mosquito population levels, which were not used for the data inferred MC modeling, were assembled to obtain an empirical intensity transition matrix and the observed frequencies. The findings match to a certain degree the empirical results in which the probabilities follow analogous patterns while no significant differences were observed between the transition matrices of the CMC model and the validation data (ChiSq=14.58013, df=24, p=0.9324451). Furter, a soft system computing modeling approach was followed to simulate and predict Culex sp. abundances. Three dynamic artificial neural network (ANNs) models were developed and applied to describe and predict the non-linear incidence and time evolution of a medical important mosquito species Culex sp. in Northern Greece. The first is a simple nonlinear autoregressive ANN model that used lagged population values as inputs, the second is an exogenous non-linear autoregressive recurrent neural network (NARX), which is designed to take as inputs the temperature as exogenous variable and mosquito abundance as endogenous. Finally, the third model is a focused time-delay neural network (FTD), which takes in to account only the temperature variable as input to provide forecasts of the mosquito abundance as target variable. All three models behaved well considering the non-linear nature of the adult mosquito abundance data. However, the NARX model, which takes in to account temperature, showed the best overall modelling performances. Nevertheless, although, the NARX model predicted slight better (R=0.623) compared to the FTD model (R=0.534), the advantage of the FTD over the NARX neural network model is that it can be applied in the case where past values of the population system, here mosquito abundance, are not available for their forecasting. This is very important considering that arthropod vector data are not always available as climatic data. Concluding, the proposed methods for simulating and predicting mosquito dynamics are recommended as viable for modeling vector disease population dynamics in order to make real-time recommendations utile for dynamic health policies decision making. The proposed stochastic models, as well as the current computational and machine learning techniques, of this work provide an accurate abstraction of the arthropod vector population progress observed within the dataset used for their generation. Nevertheless, the current study may consider also as a new entry point into the extensive literature of ecological modelling, medical entomology, as well as in simulating arthropod vector diseases epidemics. From a public health standpoint, the current models have the potential to be integrated into a decision support system allowing health policy makers in their planning to initiate specific management actions against the period of high activity of mosquito adults.

Cette thèse porte sur l'étude mathématique de modèles stochastiques de dynamique de populations structurées.Dans le premier chapitre, nous introduisons un modèle stochastique à temps discret prenant en compte les diverses interactions possibles entre les individus, que ce soit de la compétition, de la migration, des mutations, ou bien de la prédation. Nous montrons d'abord un résultat de type ``loi des grands nombres'', où on montre que si la population initiale tend vers l'infini, alors sur un intervalle de temps fini, le processus stochastique converge en probabilité vers un processus déterministe sous-jacent. Nous quantifions aussi les écarts entre ces deux processus par un résultat de type ``théorème central limite''. Enfin, nous donnons un critère de persistance/extinction afin de déterminer le comportement en temps long de notre processus stochastique. Ce critère met en exergue un cas critique qui sera étudié plus en détail dans les chapitres suivants.Dans le deuxième chapitre, nous donnons un critère de croissance illimitée pour des processus vérifiant le cas critique évoqué plus haut. Nous illustrons en particulier ce critère avec l'exemple d'une métapopulation constituée de parcelles de type puits (c'est à dire dont la population s'éteint sans tenir compte de la migration), où l'on montre que la survie de la population est possible.Dans le troisième chapitre, nous nous intéressons au comportement du processus critique lorsqu'il croît vers l'infini. Nous montrons en particulier une convergence en loi vers une loi gamma de notre processus renormalisé et dans un cadre plus général, en renormalisant aussi en temps, nous obtenons une convergence en loi d'une fonction de notre processus vers la solution d'une équation différentielle stochastique appelée un processus de Bessel carré.Dans le quatrième et dernier chapitre, nous nous plac{c}ons dans le cas où le processus critique ne tend pas vers l'infini et étudions le temps d'atteinte de certains ensembles compacts. Nous donnons un encadrement asymptotique de la queue de ce temps d'atteinte. Lorsque le processus s'éteint, ces résultats nous permettent en particulier d'encadrer la queue du temps d'extinction. Dans le cas où notre processus est une chaîne de Markov, nous en déduisons un critère de récurrence nulle ou récurrence positive et dans ce cas, nous obtenons un taux de convergence sous-géométrique du noyau de transition de notre chaîne vers sa mesure de probabilité invariante. / This thesis is devoted to the mathematical study of stochastic modelds of structured populations dynamics.In the first chapter, we introduce a discrete time stochastic process taking into account various ecological interactions between individuals, such as competition, migration, mutation, or predation. We first prove a ``law of large numbers'': where we show that if the initial population tends to infinity, then, on any finite interval of time, the stochastic process converges in probability to an underlying deterministic process. We also quantify the discrepancy between these two processes by a kind of ``central limit theorem''. Finally, we give a criterion of persistence/extinction in order to determine the long time behavior of the process. This criterion highlights a critical case which will be studied in more detail in the following chapters.In the second chapter, we give a criterion for the possible unlimited growth in the critical case mentioned above. We apply this criterion to the example of a source-sink metapopulation with two patches of type source, textit{i.e.} the population of each patch goes to extinction if we do not take into account the migration. We prove that there is a possible survival of the metapopulation.In the third chapter, we focus on the behavior of our critical process when it tends to infinity. We prove a convergence in distribution of the scaled process to a gamma distribution, and in a more general framework, by also rescaling time, we obtain a distribution limit of a function of our process to the solution of a stochastic differential equation called a squared Bessel process.In the fourth and last chapter, we study hitting times of some compact sets when our process does not tend to infinity. We give nearly optimal bounds for the tail of these hitting times. If the process goes to extinction almost surely, we deduce from these bounds precise estimates of the tail of the extinction time. Moreover, if the process is a Markov chain, we give a criterion of null recurrence or positive recurrence and in the latter case, we obtain a subgeometric convergence of its transition kernel to its invariant probability measure.

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Markov chains are a fundamental subject of study in mathematical probability and have found wide application in nearly every branch of science. Of particular interest are finite-state Markov chains; the representation of finite-state Markov chains by a transition matrix facilitates detailed analysis by linear algebraic methods. Previous methods of analyzing finite-state Markov chains have emphasized state events. In this thesis we develop the concept of a transition event and define two types of transition events: cumulative events and time-average events. Transition events generalize state events and provide a more flexible framework for analysis. We derive computable, closed-form expressions for the expectation of these two events, characterize the conditioning of transition events, provide an algorithm for computing the expectation of these events, and analyze the complexity and stability of the algorithm. As an application, we derive a construction of composite Markov chains, which we use to study competitive dynamics.

Generalized hill climbing (GHC) algorithms provide a framework for using local search algorithms to address intractable discrete optimization problems. Many well-known local search algorithms can be formulated as GHC algorithms, including simulated annealing, threshold accepting, Monte Carlo search, and pure local search (among others). This dissertation develops a mathematical framework for simultaneously addressing a set of related discrete optimization problems using GHC algorithms. The resulting algorithms, termed simultaneous generalized hill climbing (SGHC) algorithms, can be applied to a wide variety of sets of related discrete optimization problems. The SGHC algorithm probabilistically moves between these discrete optimization problems according to a problem generation probability function. This dissertation establishes that the problem generation probability function is a stochastic process that satisfies the Markov property. Therefore, given a SGHC algorithm, movement between these discrete optimization problems can be modeled as a Markov chain. Sufficient conditions that guarantee that this Markov chain has a uniform stationary probability distribution are presented. Moreover, sufficient conditions are obtained that guarantee that a SGHC algorithm will visit the globally optimal solution over all the problems in a set of related discrete optimization problems. Computational results are presented with SGHC algorithms for a set of traveling salesman problems. For comparison purposes, GHC algorithms are also applied individually to each traveling salesman problem. These computational results suggest that optimal/near optimal solutions can often be reached more quickly using a SGHC algorithm. / Ph. D.

This thesis deals with the application of Markov chains for the production of concrete products. The theoretical part is focused on clarifying the concepts of risk management and describes the procedures for dealing with classical Markov chains. There are presented basics of fuzzy logic and finally there is explained the procedure using fuzzy logic in calculating of classical Markov chains in the subsection entitled Fuzzy Markov chains. The practical part describes production process, namely concrete pavements. On this production process is applied knowledge from the theoretical part and there is a comparison and evaluation of two methods of Marcov chains calculation (classic and fuzzy approach).

Introduit dans les années 1960, le modèle de la marche aléatoire en milieu aléatoire i.i.d. sur les entiers relatifs (ou MAMA) a récemment été l'objet d'un regain d'intérêt dans la communauté statistique.Divers travaux se sont en particulier intéressés à la question de l'estimation de la loi du milieu à partir de l'observation d'une unique trajectoire de la MAMA.Cette thèse s'inscrit dans cette dynamique.Dans un premier temps, nous considérons le problème d'estimation d'un point de vue fréquentiste. Lorsque la MAMA est transiente à droite ou récurrente, nous construisons le premier estimateur non paramétrique de la densité de la loi du milieu et obtenons une majoration du risque associé mesuré en norme infinie.Dans un deuxième temps, nous envisageons le problème d'estimation sous un angle Bayésien. Lorsque la MAMA est transiente à droite, nous démontrons la consistance à posteriori de l'estimateur Bayésien de la loi du milieu.La principale difficulté mathématique de la thèse a été l'élaboration des outils nécessaires à la preuve du résultat de consistance bayésienne.Nous démontrons pour cela une version quantitative de l'inégalité de concentration de type Mac Diarmid pour chaînes de Markov.Nous étudions également le temps de retour en 0 d'un processus de branchement en milieu aléatoire avec immigration. Nous montrons l'existence d'un moment exponentiel fini uniformément valable sur une classe de processus de branchement en milieu aléatoire. Le processus de branchement en milieu aléatoire constituant une chaîne de Markov, ce résultat permet alors d'expliciter la dépendance des constantes de l'inégalité de concentration en fonction des caractéristiques de ce processus. / Introduced in the 1960s, the model of random walk in i.i.d. environment on integers (or RWRE) raised only recently interest in the statistical community. Various works have in particular focused on the estimation of the environment distribution from a single trajectory of the RWRE.This thesis extends the advances made in those works and offers new approaches to the problem.First, we consider the estimation problem from a frequentist point of view. When the RWRE is transient to the right or recurrent, we build the first non-parametric estimator of the density of the environment distribution and obtain an upper-bound of the associated risk in infinite norm.Then, we consider the estimation problem from a Bayesian perspective. When the RWRE is transient to the right, we prove the posterior consistency of the Bayesian estimator of the environment distribution.The main difficulty of the thesis was to develop the tools necessary to the proof of Bayesian consistency.For this purpose, we demonstrate a quantitative version of a Mac Diarmid's type concentration inequality for Markov chains.We also study the return time to 0 of a branching process with immigration in random environment (or BPIRE). We show the existence of a finite exponential moment uniformly valid on a class of BPIRE. The BPIRE being a Markov chain, this result enables then to make explicit the dependence of the constants of the concentration inequality with respect to the characteristics of the BPIRE.

In this thesis we present a framework for automatic formal analysis of competitive stochastic systems, such as sensor networks, decentralised resource management schemes or distributed user-centric environments. We model such systems as stochastic multi-player games, which are turn-based models where an action in each state is chosen by one of the players or according to a probability distribution. The specifications, such as “sensors 1 and 2 can collaborate to detect the target with probability 1, no matter what other sensors in the network do” or “the controller can ensure that the energy used is less than 75 mJ, and the algorithm terminates with probability at least 0.5'', are provided as temporal logic formulae. We introduce a branching-time temporal logic rPATL and its multi-objective extension to specify such probabilistic and reward-based properties of stochastic multi-player games. We also provide algorithms for these logics that can either verify such properties against the model, providing a yes/no answer, or perform strategy synthesis by constructing the strategy for the players that satisfies the specification. We conduct a detailed complexity analysis of the model checking problem for rPATL and its multi-objective extension and provide efficient algorithms for verification and strategy synthesis. We also implement the proposed techniques in the PRISM-games tool and apply them to the analysis of several case studies of competitive stochastic systems.