PERFORMANCE EVALUATION OF RESOURCE-AWARE BUSINESS PROCESSES USING STOCHASTIC AUTOMATA NETWORKS

In this work, we study the performance evaluation of resource-aware business process models. We define a new framework that allows the generation of analytical models for performance evaluation from business process models annotated with resource management information. This framework is composed of a new notation that allows the specification of resource management constraints and a method to convert a business process specification and its resource constraints into Stochastic Automata Networks (SANs). We show that the analysis of the generated SAN model provides several performance indices, such as average throughput of the system, average waiting time, average queues size, and utilization rate of resources. Using the BP2SAN tool - our implementation of the proposed framework - and a SAN solver (such as the PEPS tool) we show through a simple use-case how a business specialist with no skills in stochastic modeling can easily obtain performance indices that, in turn, can help to identify bottlenecks on the model, to perform workload characterization, to define the provisioning of resources, and to study other performance related aspects of the business process.

Stochastic quantization of the spherical model and supersymmetry.

In this work, we reported some results about the stochastic quantization of the spherical model. We started by reviewing some basic aspects of this method with emphasis in the connection between the Langevin equation and the supersymmetric quantum mechanics, aiming at the application of the corresponding connection to the spherical model. An intuitive idea is that when applied to the spherical model this gives rise to a supersymmetric version that is identified with one studied in Phys. Rev. E 85, 061109, (2012). Before investigating in detail this aspect, we studied the stochastic quantization of the mean spherical model that is simpler to implement than the one with the strict constraint. We also highlight some points concerning more traditional methods discussed in the literature like canonical and path integral quantization. To produce a supersymmetric version, grounded in the Nicolai map, we investigated the stochastic quantization of the strict spherical model. We showed in fact that the result of this process is an off-shell supersymmetric extension of the quantum spherical model (with the precise supersymmetric constraint structure). That analysis establishes a connection between the classical model and its supersymmetric quantum counterpart. The supersymmetric version in this way constructed is a more natural one and gives further support and motivations to investigate similar connections in other models of the literature.

Multimodality and flexibility of stochastic gene expression

We consider a general class of mathematical models for stochastic gene expression where the transcription rate is allowed to depend on a promoter state variable that can take an arbitrary (finite) number of values. We provide the solution of the master equations in the stationary limit, based on a factorization of the stochastic transition matrix that separates timescales and relative interaction strengths, and we express its entries in terms of parameters that have a natural physical and/or biological interpretation. The solution illustrates the capacity of multiple states promoters to generate multimodal distributions of gene products, without the need for feedback. Furthermore, using the example of a three states promoter operating at low, high, and intermediate expression levels, we show that using multiple states operons will typically lead to a significant reduction of noise in the system. The underlying mechanism is that a three-states promoter can change its level of expression from low to high by passing through an intermediate state with a much smaller increase of fluctuations than by means of a direct transition.

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.

Model reduction of stochastic groundwater flow and transport processes

This work presents a comprehensive methodology for the reduction of analytical or numerical stochastic models characterized by uncertain input parameters or boundary conditions. The technique, based on the Polynomial Chaos Expansion (PCE) theory, represents a versatile solution to solve direct or inverse problems related to propagation of uncertainty. The potentiality of the methodology is assessed investigating different applicative contexts related to groundwater flow and transport scenarios, such as global sensitivity analysis, risk analysis and model calibration. This is achieved by implementing a numerical code, developed in the MATLAB environment, presented here in its main features and tested with literature examples. The procedure has been conceived under flexibility and efficiency criteria in order to ensure its adaptability to different fields of engineering; it has been applied to different case studies related to flow and transport in porous media. Each application is associated with innovative elements such as (i) new analytical formulations describing motion and displacement of non-Newtonian fluids in porous media, (ii) application of global sensitivity analysis to a high-complexity numerical model inspired by a real case of risk of radionuclide migration in the subsurface environment, and (iii) development of a novel sensitivity-based strategy for parameter calibration and experiment design in laboratory scale tracer transport.

Forward implied volatility expansions in LSV models

In this work we address the problem of finding formulas for efficient and reliable analytical approximation for the calculation of forward implied volatility in LSV models, a problem which is reduced to the calculation of option prices as an expansion of the price of the same financial asset in a Black-Scholes dynamic. Our approach involves an expansion of the differential operator, whose solution represents the price in local stochastic volatility dynamics. Further calculations then allow to obtain an expansion of the implied volatility without the aid of any special function or expensive from the computational point of view, in order to obtain explicit formulas fast to calculate but also as accurate as possible.

Estimation of Quasi-Rational DSGE Models

Small-scale dynamic stochastic general equilibrium have been treated as the benchmark of much of the monetary policy literature, given their ability to explain the impact of monetary policy on output, inflation and financial markets. One cause of the empirical failure of New Keynesian models is partially due to the Rational Expectations (RE) paradigm, which entails a tight structure on the dynamics of the system. Under this hypothesis, the agents are assumed to know the data genereting process. In this paper, we propose the econometric analysis of New Keynesian DSGE models under an alternative expectations generating paradigm, which can be regarded as an intermediate position between rational expectations and learning, nameley an adapted version of the "Quasi-Rational" Expectatations (QRE) hypothesis. Given the agents' statistical model, we build a pseudo-structural form from the baseline system of Euler equations, imposing that the length of the reduced form is the same as in the `best' statistical model.

Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input

A field of computational neuroscience develops mathematical models to describe neuronal systems. The aim is to better understand the nervous system. Historically, the integrate-and-fire model, developed by Lapique in 1907, was the first model describing a neuron. In 1952 Hodgkin and Huxley [8] described the so called Hodgkin-Huxley model in the article “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”. The Hodgkin-Huxley model is one of the most successful and widely-used biological neuron models. Based on experimental data from the squid giant axon, Hodgkin and Huxley developed their mathematical model as a four-dimensional system of first-order ordinary differential equations. One of these equations characterizes the membrane potential as a process in time, whereas the other three equations depict the opening and closing state of sodium and potassium ion channels. The membrane potential is proportional to the sum of ionic current flowing across the membrane and an externally applied current. For various types of external input the membrane potential behaves differently. This thesis considers the following three types of input: (i) Rinzel and Miller [15] calculated an interval of amplitudes for a constant applied current, where the membrane potential is repetitively spiking; (ii) Aihara, Matsumoto and Ikegaya [1] said that dependent on the amplitude and the frequency of a periodic applied current the membrane potential responds periodically; (iii) Izhikevich [12] stated that brief pulses of positive and negative current with different amplitudes and frequencies can lead to a periodic response of the membrane potential. In chapter 1 the Hodgkin-Huxley model is introduced according to Izhikevich [12]. Besides the definition of the model, several biological and physiological notes are made, and further concepts are described by examples. Moreover, the numerical methods to solve the equations of the Hodgkin-Huxley model are presented which were used for the computer simulations in chapter 2 and chapter 3. In chapter 2 the statements for the three different inputs (i), (ii) and (iii) will be verified, and periodic behavior for the inputs (ii) and (iii) will be investigated. In chapter 3 the inputs are embedded in an Ornstein-Uhlenbeck process to see the influence of noise on the results of chapter 2.

Statistical analysis of the error associated with the simplification of the stratigraphy in geotechnical models

The uncertainties in the determination of the stratigraphic profile of natural soils is one of the main problems in geotechnics, in particular for landslide characterization and modeling. The study deals with a new approach in geotechnical modeling which relays on a stochastic generation of different soil layers distributions, following a boolean logic – the method has been thus called BoSG (Boolean Stochastic Generation). In this way, it is possible to randomize the presence of a specific material interdigitated in a uniform matrix. In the building of a geotechnical model it is generally common to discard some stratigraphic data in order to simplify the model itself, assuming that the significance of the results of the modeling procedure would not be affected. With the proposed technique it is possible to quantify the error associated with this simplification. Moreover, it could be used to determine the most significant zones where eventual further investigations and surveys would be more effective to build the geotechnical model of the slope. The commercial software FLAC was used for the 2D and 3D geotechnical model. The distribution of the materials was randomized through a specifically coded MatLab program that automatically generates text files, each of them representing a specific soil configuration. Besides, a routine was designed to automate the computation of FLAC with the different data files in order to maximize the sample number. The methodology is applied with reference to a simplified slope in 2D, a simplified slope in 3D and an actual landslide, namely the Mortisa mudslide (Cortina d’Ampezzo, BL, Italy). However, it could be extended to numerous different cases, especially for hydrogeological analysis and landslide stability assessment, in different geological and geomorphological contexts.

Models and Algorihtm for the Optimization of Real-World Routing and Logistics Problems

Logistics involves planning, managing, and organizing the flows of goods from the point of origin to the point of destination in order to meet some requirements. Logistics and transportation aspects are very important and represent a relevant costs for producing and shipping companies, but also for public administration and private citizens. The optimization of resources and the improvement in the organization of operations is crucial for all branches of logistics, from the operation management to the transportation. As we will have the chance to see in this work, optimization techniques, models, and algorithms represent important methods to solve the always new and more complex problems arising in different segments of logistics. Many operation management and transportation problems are related to the optimization class of problems called Vehicle Routing Problems (VRPs). In this work, we consider several real-world deterministic and stochastic problems that are included in the wide class of the VRPs, and we solve them by means of exact and heuristic methods. We treat three classes of real-world routing and logistics problems. We deal with one of the most important tactical problems that arises in the managing of the bike sharing systems, that is the Bike sharing Rebalancing Problem (BRP). We propose models and algorithms for real-world earthwork optimization problems. We describe the 3DP process and we highlight several optimization issues in 3DP. Among those, we define the problem related to the tool path definition in the 3DP process, the 3D Routing Problem (3DRP), which is a generalization of the arc routing problem. We present an ILP model and several heuristic algorithms to solve the 3DRP.

Convergence results for stochastic particle systems with social interaction

We consider stochastic individual-based models for social behaviour of groups of animals. In these models the trajectory of each animal is given by a stochastic differential equation with interaction. The social interaction is contained in the drift term of the SDE. We consider a global aggregation force and a short-range repulsion force. The repulsion range and strength gets rescaled with the number of animals N. We show that for N tending to infinity stochastic fluctuations disappear and a smoothed version of the empirical process converges uniformly towards the solution of a nonlinear, nonlocal partial differential equation of advection-reaction-diffusion type. The rescaling of the repulsion in the individual-based model implies that the corresponding term in the limit equation is local while the aggregation term is non-local. Moreover, we discuss the effect of a predator on the system and derive an analogous convergence result. The predator acts as an repulsive force. Different laws of motion for the predator are considered.

Chargaff symmetric stochastic processes

Scopo della modellizzazione delle stringhe di DNA è la formulazione di modelli matematici che generano sequenze di basi azotate compatibili con il genoma esistente. In questa tesi si prendono in esame quei modelli matematici che conservano un'importante proprietà, scoperta nel 1952 dal biochimico Erwin Chargaff, chiamata oggi "seconda regola di Chargaff". I modelli matematici che tengono conto delle simmetrie di Chargaff si dividono principalmente in due filoni: uno la ritiene un risultato dell'evoluzione sul genoma, mentre l'altro la ipotizza peculiare di un genoma primitivo e non intaccata dalle modifiche apportate dall'evoluzione. Questa tesi si propone di analizzare un modello del secondo tipo. In particolare ci siamo ispirati al modello definito da da Sobottka e Hart. Dopo un'analisi critica e lo studio del lavoro degli autori, abbiamo esteso il modello ad un più ampio insieme di casi. Abbiamo utilizzato processi stocastici come Bernoulli-scheme e catene di Markov per costruire una possibile generalizzazione della struttura proposta nell'articolo, analizzando le condizioni che implicano la validità della regola di Chargaff. I modelli esaminati sono costituiti da semplici processi stazionari o concatenazioni di processi stazionari. Nel primo capitolo vengono introdotte alcune nozioni di biologia. Nel secondo si fa una descrizione critica e prospettica del modello proposto da Sobottka e Hart, introducendo le definizioni formali per il caso generale presentato nel terzo capitolo, dove si sviluppa l'apparato teorico del modello generale.

Fluctuation properties in random walks on networks and simple integrate and fire models

In questa tesi si è studiato l’insorgere di eventi critici in un semplice modello neurale del tipo Integrate and Fire, basato su processi dinamici stocastici markoviani definiti su una rete. Il segnale neurale elettrico è stato modellato da un flusso di particelle. Si è concentrata l’attenzione sulla fase transiente del sistema, cercando di identificare fenomeni simili alla sincronizzazione neurale, la quale può essere considerata un evento critico. Sono state studiate reti particolarmente semplici, trovando che il modello proposto ha la capacità di produrre effetti "a cascata" nell’attività neurale, dovuti a Self Organized Criticality (auto organizzazione del sistema in stati instabili); questi effetti non vengono invece osservati in Random Walks sulle stesse reti. Si è visto che un piccolo stimolo random è capace di generare nell’attività della rete delle fluttuazioni notevoli, in particolar modo se il sistema si trova in una fase al limite dell’equilibrio. I picchi di attività così rilevati sono stati interpretati come valanghe di segnale neurale, fenomeno riconducibile alla sincronizzazione.

Markers Models in Survival Analysis and Applications to Issues Associated with AIDS

Jewell and Kalbfleisch (1992) consider the use of marker processes for applications related to estimation of the survival distribution of time to failure. Marker processes were assumed to be stochastic processes that, at a given point in time, provide information about the current hazard and consequently on the remaining time to failure. Particular attention was paid to calculations based on a simple additive model for the relationship between the hazard function at time t and the history of the marker process up until time t. Specific applications to the analysis of AIDS data included the use of markers as surrogate responses for onset of AIDS with censored data and as predictors of the time elapsed since infection in prevalent individuals. Here we review recent work on the use of marker data to tackle these kinds of problems with AIDS data. The Poisson marker process with an additive model, introduced in Jewell and Kalbfleisch (1992) may be a useful "test" example for comparison of various procedures.