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.

High-performance switching architectures for optical networks

The need for high bandwidth, due to the explosion of new multi\-media-oriented IP-based services, as well as increasing broadband access requirements is leading to the need of flexible and highly reconfigurable optical networks. While transmission bandwidth does not represent a limit due to the huge bandwidth provided by optical fibers and Dense Wavelength Division Multiplexing (DWDM) technology, the electronic switching nodes in the core of the network represent the bottleneck in terms of speed and capacity for the overall network. For this reason DWDM technology must be exploited not only for data transport but also for switching operations. In this Ph.D. thesis solutions for photonic packet switches, a flexible alternative with respect to circuit-switched optical networks are proposed. In particular solutions based on devices and components that are expected to mature in the near future are proposed, with the aim to limit the employment of complex components. The work presented here is the result of part of the research activities performed by the Networks Research Group at the Department of Electronics, Computer Science and Systems (DEIS) of the University of Bologna, Italy. In particular, the work on optical packet switching has been carried on within three relevant research projects: the e-Photon/ONe and e-Photon/ONe+ projects, funded by the European Union in the Sixth Framework Programme, and the national project OSATE funded by the Italian Ministry of Education, University and Scientific Research. The rest of the work is organized as follows. Chapter 1 gives a brief introduction to network context and contention resolution in photonic packet switches. Chapter 2 presents different strategies for contention resolution in wavelength domain. Chapter 3 illustrates a possible implementation of one of the schemes proposed in chapter 2. Then, chapter 4 presents multi-fiber switches, which employ jointly wavelength and space domains to solve contention. Chapter 5 shows buffered switches, to solve contention in time domain besides wavelength domain. Finally chapter 6 presents a cost model to compare different switch architectures in terms of cost.

Internal Model Principle: extension to the switching case and applications

This thesis deals with a novel control approach based on the extension of the well-known Internal Model Principle to the case of periodic switched linear exosystems. This extension, inspired by power electronics applications, aims to provide an effective design method to robustly achieve the asymptotic tracking of periodic references with an infinite number of harmonics. In the first part of the thesis the basic components of the novel control scheme are described and preliminary results on stabilization are provided. In the second part, advanced control methods for two applications coming from the world high energy physics are presented.

Methodologies for Synthesizable Programmable Devices based on Multi-Stage Switching Networks

Nowadays the rise of non-recurring engineering (NRE) costs associated with complexity is becoming a major factor in SoC design, limiting both scaling opportunities and the flexibility advantages offered by the integration of complex computational units. The introduction of embedded programmable elements can represent an appealing solution, able both to guarantee the desired flexibility and upgradabilty and to widen the SoC market. In particular embedded FPGA (eFPGA) cores can provide bit-level optimization for those applications which benefits from synthesis, paying on the other side in terms of performance penalties and area overhead with respect to standard cell ASIC implementations. In this scenario this thesis proposes a design methodology for a synthesizable programmable device designed to be embedded in a SoC. A soft-core embedded FPGA (eFPGA) is hence presented and analyzed in terms of the opportunities given by a fully synthesizable approach, following an implementation flow based on Standard-Cell methodology. A key point of the proposed eFPGA template is that it adopts a Multi-Stage Switching Network (MSSN) as the foundation of the programmable interconnects, since it can be efficiently synthesized and optimized through a standard cell based implementation flow, ensuring at the same time an intrinsic congestion-free network topology. The evaluation of the flexibility potentialities of the eFPGA has been performed using different technology libraries (STMicroelectronics CMOS 65nm and BCD9s 0.11μm) through a design space exploration in terms of area-speed-leakage tradeoffs, enabled by the full synthesizability of the template. Since the most relevant disadvantage of the adopted soft approach, compared to a hardcore, is represented by a performance overhead increase, the eFPGA analysis has been made targeting small area budgets. The generation of the configuration bitstream has been obtained thanks to the implementation of a custom CAD flow environment, and has allowed functional verification and performance evaluation through an application-aware analysis.