Stochastic processes and their applications semi-markov analysis of some inventory and queuing problems

This thesis analyses certain problems in Inventories and Queues. There are many situations in real-life where we encounter models as described in this thesis. It analyses in depth various models which can be applied to production, storag¢, telephone traffic, road traffic, economics, business administration, serving of customers, operations of particle counters and others. Certain models described here is not a complete representation of the true situation in all its complexity, but a simplified version amenable to analysis. While discussing the models, we show how a dependence structure can be suitably introduced in some problems of Inventories and Queues. Continuous review, single commodity inventory systems with Markov dependence structure introduced in the demand quantities, replenishment quantities and reordering levels are considered separately. Lead time is assumed to be zero in these models. An inventory model involving random lead time is also considered (Chapter-4). Further finite capacity single server queueing systems with single/bulk arrival, single/bulk services are also discussed. In some models the server is assumed to go on vacation (Chapters 7 and 8). In chapters 5 and 6 a sort of dependence is introduced in the service pattern in some queuing models.

Mathematical models of generalized diffusion

We discuss in this paper equations describing processes involving non-linear and higher-order diffusion. We focus on a particular case (u(t) = 2 lambda (2)(uu(x))(x) + lambda (2)u(xxxx)), which is put into analogy with the KdV equation. A balance of nonlinearity and higher-order diffusion enables the existence of self-similar solutions, describing diffusive shocks. These shocks are continuous solutions with a discontinuous higher-order derivative at the shock front. We argue that they play a role analogous to the soliton solutions in the dispersive case. We also discuss several physical instances where such equations are relevant.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Sample size considerations using mathematical models: an example with Chlamydia trachomatis infection and its sequelae pelvic inflammatory disease.

BACKGROUND The success of an intervention to prevent the complications of an infection is influenced by the natural history of the infection. Assumptions about the temporal relationship between infection and the development of sequelae can affect the predicted effect size of an intervention and the sample size calculation. This study investigates how a mathematical model can be used to inform sample size calculations for a randomised controlled trial (RCT) using the example of Chlamydia trachomatis infection and pelvic inflammatory disease (PID). METHODS We used a compartmental model to imitate the structure of a published RCT. We considered three different processes for the timing of PID development, in relation to the initial C. trachomatis infection: immediate, constant throughout, or at the end of the infectious period. For each process we assumed that, of all women infected, the same fraction would develop PID in the absence of an intervention. We examined two sets of assumptions used to calculate the sample size in a published RCT that investigated the effect of chlamydia screening on PID incidence. We also investigated the influence of the natural history parameters of chlamydia on the required sample size. RESULTS The assumed event rates and effect sizes used for the sample size calculation implicitly determined the temporal relationship between chlamydia infection and PID in the model. Even small changes in the assumed PID incidence and relative risk (RR) led to considerable differences in the hypothesised mechanism of PID development. The RR and the sample size needed per group also depend on the natural history parameters of chlamydia. CONCLUSIONS Mathematical modelling helps to understand the temporal relationship between an infection and its sequelae and can show how uncertainties about natural history parameters affect sample size calculations when planning a RCT.

Oscillatory genetic circuits: new designs and mathematical models

Introduction and motivation: A wide variety of organisms have developed in-ternal biomolecular clocks in order to adapt to cyclic changes of the environment. Clock operation involves genetic networks. These genetic networks have to be mod¬eled in order to understand the underlying mechanism of oscillations and to design new synthetic cellular clocks. This doctoral thesis has resulted in two contributions to the fields of genetic clocks and systems and synthetic biology, generally. The first contribution is a new genetic circuit model that exhibits an oscillatory behav¬ior through catalytic RNA molecules. The second and major contribution is a new genetic circuit model demonstrating that a repressor molecule acting on the positive feedback of a self-activating gene produces reliable oscillations. First contribution: A new model of a synthetic genetic oscillator based on a typical two-gene motif with one positive and one negative feedback loop is pre¬sented. The originality is that the repressor is a catalytic RNA molecule rather than a protein or a non-catalytic RNA molecule. This catalytic RNA is a ribozyme that acts post-transcriptionally by binding to and cleaving target mRNA molecules. This genetic clock involves just two genes, a mRNA and an activator protein, apart from the ribozyme. Parameter values that produce a circadian period in both determin¬istic and stochastic simulations have been chosen as an example of clock operation. The effects of the stochastic fluctuations are quantified by a period histogram and autocorrelation function. The conclusion is that catalytic RNA molecules can act as repressor proteins and simplify the design of genetic oscillators. Second and major contribution: It is demonstrated that a self-activating gene in conjunction with a simple negative interaction can easily produce robust matically validated. This model is comprised of two clearly distinct parts. The first is a positive feedback created by a protein that binds to the promoter of its own gene and activates the transcription. The second is a negative interaction in which a repressor molecule prevents this protein from binding to its promoter. A stochastic study shows that the system is robust to noise. A deterministic study identifies that the oscillator dynamics are mainly driven by two types of biomolecules: the protein, and the complex formed by the repressor and this protein. The main conclusion of this study is that a simple and usual negative interaction, such as degradation, se¬questration or inhibition, acting on the positive transcriptional feedback of a single gene is a sufficient condition to produce reliable oscillations. One gene is enough and the positive transcriptional feedback signal does not need to activate a second repressor gene. At the genetic level, this means that an explicit negative feedback loop is not necessary. Unlike many genetic oscillators, this model needs neither cooperative binding reactions nor the formation of protein multimers. Applications and future research directions: Recently, RNA molecules have been found to play many new catalytic roles. The first oscillatory genetic model proposed in this thesis uses ribozymes as repressor molecules. This could provide new synthetic biology design principles and a better understanding of cel¬lular clocks regulated by RNA molecules. The second genetic model proposed here involves only a repression acting on a self-activating gene and produces robust oscil¬lations. Unlike current two-gene oscillators, this model surprisingly does not require a second repressor gene. This result could help to clarify the design principles of cellular clocks and constitute a new efficient tool for engineering synthetic genetic oscillators. Possible follow-on research directions are: validate models in vivo and in vitro, research the potential of second model as a genetic memory, investigate new genetic oscillators regulated by non-coding RNAs and design a biosensor of positive feedbacks in genetic networks based on the operation of the second model Resumen Introduccion y motivacion: Una amplia variedad de organismos han desarro-llado relojes biomoleculares internos con el fin de adaptarse a los cambios ciclicos del entorno. El funcionamiento de estos relojes involucra redes geneticas. El mo delado de estas redes geneticas es esencial tanto para entender los mecanismos que producen las oscilaciones como para diseiiar nuevos circuitos sinteticos en celulas. Esta tesis doctoral ha dado lugar a dos contribuciones dentro de los campos de los circuitos geneticos en particular, y biologia de sistemas y sintetica en general. La primera contribucion es un nuevo modelo de circuito genetico que muestra un comportamiento oscilatorio usando moleculas de ARN cataliticas. La segunda y principal contribucion es un nuevo modelo de circuito genetico que demuestra que una molecula represora actuando sobre el lazo de un gen auto-activado produce oscilaciones robustas. Primera contribucion: Es un nuevo modelo de oscilador genetico sintetico basado en una tipica red genetica compuesta por dos genes con dos lazos de retroa-limentacion, uno positivo y otro negativo. La novedad de este modelo es que el represor es una molecula de ARN catalftica, en lugar de una protefna o una molecula de ARN no-catalitica. Este ARN catalitico es una ribozima que actua despues de la transcription genetica uniendose y cortando moleculas de ARN mensajero (ARNm). Este reloj genetico involucra solo dos genes, un ARNm y una proteina activadora, aparte de la ribozima. Como ejemplo de funcionamiento, se han escogido valores de los parametros que producen oscilaciones con periodo circadiano (24 horas) tanto en simulaciones deterministas como estocasticas. El efecto de las fluctuaciones es-tocasticas ha sido cuantificado mediante un histograma del periodo y la función de auto-correlacion. La conclusion es que las moleculas de ARN con propiedades cataliticas pueden jugar el misnio papel que las protemas represoras, y por lo tanto, simplificar el diseno de los osciladores geneticos. Segunda y principal contribucion: Es un nuevo modelo de oscilador genetico que demuestra que un gen auto-activado junto con una simple interaction negativa puede producir oscilaciones robustas. Este modelo ha sido estudiado y validado matematicamente. El modelo esta compuesto de dos partes bien diferenciadas. La primera parte es un lazo de retroalimentacion positiva creado por una proteina que se une al promotor de su propio gen activando la transcription. La segunda parte es una interaction negativa en la que una molecula represora evita la union de la proteina con el promotor. Un estudio estocastico muestra que el sistema es robusto al ruido. Un estudio determinista muestra que la dinamica del sistema es debida principalmente a dos tipos de biomoleculas: la proteina, y el complejo formado por el represor y esta proteina. La conclusion principal de este estudio es que una simple y usual interaction negativa, tal como una degradation, un secuestro o una inhibition, actuando sobre el lazo de retroalimentacion positiva de un solo gen es una condition suficiente para producir oscilaciones robustas. Un gen es suficiente y el lazo de retroalimentacion positiva no necesita activar a un segundo gen represor, tal y como ocurre en los relojes actuales con dos genes. Esto significa que a nivel genetico un lazo de retroalimentacion negativa no es necesario de forma explicita. Ademas, este modelo no necesita reacciones cooperativas ni la formation de multimeros proteicos, al contrario que en muchos osciladores geneticos. Aplicaciones y futuras lineas de investigacion: En los liltimos anos, se han descubierto muchas moleculas de ARN con capacidad catalitica. El primer modelo de oscilador genetico propuesto en esta tesis usa ribozimas como moleculas repre¬soras. Esto podria proporcionar nuevos principios de diseno en biologia sintetica y una mejor comprension de los relojes celulares regulados por moleculas de ARN. El segundo modelo de oscilador genetico propuesto aqui involucra solo una represion actuando sobre un gen auto-activado y produce oscilaciones robustas. Sorprendente-mente, un segundo gen represor no es necesario al contrario que en los bien conocidos osciladores con dos genes. Este resultado podria ayudar a clarificar los principios de diseno de los relojes celulares naturales y constituir una nueva y eficiente he-rramienta para crear osciladores geneticos sinteticos. Algunas de las futuras lineas de investigation abiertas tras esta tesis son: (1) la validation in vivo e in vitro de ambos modelos, (2) el estudio del potential del segundo modelo como circuito base para la construction de una memoria genetica, (3) el estudio de nuevos osciladores geneticos regulados por ARN no codificante y, por ultimo, (4) el rediseno del se¬gundo modelo de oscilador genetico para su uso como biosensor capaz de detectar genes auto-activados en redes geneticas.

Cooperative Multi-robot Patrolling: A study of distributed approaches based on mathematical models of game theory to protect infrastructures

El principio de Teoría de Juegos permite desarrollar modelos estocásticos de patrullaje multi-robot para proteger infraestructuras criticas. La protección de infraestructuras criticas representa un gran reto para los países al rededor del mundo, principalmente después de los ataques terroristas llevados a cabo la década pasada. En este documento el termino infraestructura hace referencia a aeropuertos, plantas nucleares u otros instalaciones. El problema de patrullaje se define como la actividad de patrullar un entorno determinado para monitorear cualquier actividad o sensar algunas variables ambientales. En esta actividad, un grupo de robots debe visitar un conjunto de puntos de interés definidos en un entorno en intervalos de tiempo irregulares con propósitos de seguridad. Los modelos de partullaje multi-robot son utilizados para resolver este problema. Hasta el momento existen trabajos que resuelven este problema utilizando diversos principios matemáticos. Los modelos de patrullaje multi-robot desarrollados en esos trabajos representan un gran avance en este campo de investigación. Sin embargo, los modelos con los mejores resultados no son viables para aplicaciones de seguridad debido a su naturaleza centralizada y determinista. Esta tesis presenta cinco modelos de patrullaje multi-robot distribuidos e impredecibles basados en modelos matemáticos de aprendizaje de Teoría de Juegos. El objetivo del desarrollo de estos modelos está en resolver los inconvenientes presentes en trabajos preliminares. Con esta finalidad, el problema de patrullaje multi-robot se formuló utilizando conceptos de Teoría de Grafos, en la cual se definieron varios juegos en cada vértice de un grafo. Los modelos de patrullaje multi-robot desarrollados en este trabajo de investigación se han validado y comparado con los mejores modelos disponibles en la literatura. Para llevar a cabo tanto la validación como la comparación se ha utilizado un simulador de patrullaje y un grupo de robots reales. Los resultados experimentales muestran que los modelos de patrullaje desarrollados en este trabajo de investigación trabajan mejor que modelos de trabajos previos en el 80% de 150 casos de estudio. Además de esto, estos modelos cuentan con varias características importantes tales como distribución, robustez, escalabilidad y dinamismo. Los avances logrados con este trabajo de investigación dan evidencia del potencial de Teoría de Juegos para desarrollar modelos de patrullaje útiles para proteger infraestructuras. ABSTRACT Game theory principle allows to developing stochastic multi-robot patrolling models to protect critical infrastructures. Critical infrastructures protection is a great concern for countries around the world, mainly due to terrorist attacks in the last decade. In this document, the term infrastructures includes airports, nuclear power plants, and many other facilities. The patrolling problem is defined as the activity of traversing a given environment to monitoring any activity or sensing some environmental variables If this activity were performed by a fleet of robots, they would have to visit some places of interest of an environment at irregular intervals of time for security purposes. This problem is solved using multi-robot patrolling models. To date, literature works have been solved this problem applying various mathematical principles.The multi-robot patrolling models developed in those works represent great advances in this field. However, the models that obtain the best results are unfeasible for security applications due to their centralized and predictable nature. This thesis presents five distributed and unpredictable multi-robot patrolling models based on mathematical learning models derived from Game Theory. These multi-robot patrolling models aim at overcoming the disadvantages of previous work. To this end, the multi-robot patrolling problem was formulated using concepts of Graph Theory to represent the environment. Several normal-form games were defined at each vertex of a graph in this formulation. The multi-robot patrolling models developed in this research work have been validated and compared with best ranked multi-robot patrolling models in the literature. Both validation and comparison were preformed by using both a patrolling simulator and real robots. Experimental results show that the multirobot patrolling models developed in this research work improve previous ones in as many as 80% of 150 cases of study. Moreover, these multi-robot patrolling models rely on several features to highlight in security applications such as distribution, robustness, scalability, and dynamism. The achievements obtained in this research work validate the potential of Game Theory to develop patrolling models to protect infrastructures.

Stochastic processes strongly influence HIV-1 evolution during suboptimal protease-inhibitor therapy

It has long been assumed that HIV-1 evolution is best described by deterministic evolutionary models because of the large population size. Recently, however, it was suggested that the effective population size (Ne) may be rather small, thereby allowing chance to influence evolution, a situation best described by a stochastic evolutionary model. To gain experimental evidence supporting one of the evolutionary models, we investigated whether the development of resistance to the protease inhibitor ritonavir affected the evolution of the env gene. Sequential serum samples from five patients treated with ritonavir were used for analysis of the protease gene and the V3 domain of the env gene. Multiple reverse transcription–PCR products were cloned, sequenced, and used to construct phylogenetic trees and to calculate the genetic variation and Ne. Genotypic resistance to ritonavir developed in all five patients, but each patient displayed a unique combination of mutations, indicating a stochastic element in the development of ritonavir resistance. Furthermore, development of resistance induced clear bottleneck effects in the env gene. The mean intrasample genetic variation, which ranged from 1.2% to 5.7% before treatment, decreased significantly (P < 0.025) during treatment. In agreement with these findings, Ne was estimated to be very small (500–15,000) compared with the total HIV-1 RNA copy number. This study combines three independent observations, strong population bottlenecking, small Ne, and selection of different combinations of protease-resistance mutations, all of which indicate that HIV-1 evolution is best described by a stochastic evolutionary model.

Branching Stochastic Processes: History, Theory, Applications

Косто В. Митов - Разклоняващите се стохастични процеси са модели на популационната динамика на обекти, които имат случайно време на живот и произвеждат потомци в съответствие с дадени вероятностни закони. Типични примери са ядрените реакции, клетъчната пролиферация, биологичното размножаване, някои химични реакции, икономически и финансови явления. В този обзор сме се опитали да представим съвсем накратко някои от най-важните моменти и факти от историята, теорията и приложенията на разклоняващите се процеси.

A new method for conditioning stochastic groundwater flow models in fractured media

Many geological formations consist of crystalline rocks that have very low matrix permeability but allow flow through an interconnected network of fractures. Understanding the flow of groundwater through such rocks is important in considering disposal of radioactive waste in underground repositories. A specific area of interest is the conditioning of fracture transmissivities on measured values of pressure in these formations. This is the process where the values of fracture transmissivities in a model are adjusted to obtain a good fit of the calculated pressures to measured pressure values. While there are existing methods to condition transmissivity fields on transmissivity, pressure and flow measurements for a continuous porous medium there is little literature on conditioning fracture networks. Conditioning fracture transmissivities on pressure or flow values is a complex problem because the measurements are not linearly related to the fracture transmissivities and they are also dependent on all the fracture transmissivities in the network. We present a new method for conditioning fracture transmissivities on measured pressure values based on the calculation of certain basis vectors; each basis vector represents the change to the log transmissivity of the fractures in the network that results in a unit increase in the pressure at one measurement point whilst keeping the pressure at the remaining measurement points constant. The fracture transmissivities are updated by adding a linear combination of basis vectors and coefficients, where the coefficients are obtained by minimizing an error function. A mathematical summary of the method is given. This algorithm is implemented in the existing finite element code ConnectFlow developed and marketed by Serco Technical Services, which models groundwater flow in a fracture network. Results of the conditioning are shown for a number of simple test problems as well as for a realistic large scale test case.

Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Mathematical models for describing the shape of the in vitro unstretched human crystalline lens

We developed orthogonal least-squares techniques for fitting crystalline lens shapes, and used the bootstrap method to determine uncertainties associated with the estimated vertex radii of curvature and asphericities of five different models. Three existing models were investigated including one that uses two separate conics for the anterior and posterior surfaces, and two whole lens models based on a modulated hyperbolic cosine function and on a generalized conic function. Two new models were proposed including one that uses two interdependent conics and a polynomial based whole lens model. The models were used to describe the in vitro shape for a data set of twenty human lenses with ages 7–82 years. The two-conic-surface model (7 mm zone diameter) and the interdependent surfaces model had significantly lower merit functions than the other three models for the data set, indicating that most likely they can describe human lens shape over a wide age range better than the other models (although with the two-conic-surfaces model being unable to describe the lens equatorial region). Considerable differences were found between some models regarding estimates of radii of curvature and surface asphericities. The hyperbolic cosine model and the new polynomial based whole lens model had the best precision in determining the radii of curvature and surface asphericities across the five considered models. Most models found significant increase in anterior, but not posterior, radius of curvature with age. Most models found a wide scatter of asphericities, but with the asphericities usually being positive and not significantly related to age. As the interdependent surfaces model had lower merit function than three whole lens models, there is further scope to develop an accurate model of the complete shape of human lenses of all ages. The results highlight the continued difficulty in selecting an appropriate model for the crystalline lens shape.