Queue-Based Random-Access Algorithms: Fluid Limits and Stability Issues

We use fluid limits to explore the (in)stability properties of wireless networks with queue-based random-access algorithms. Queue-based random-access schemes are simple and inherently distributed in nature, yet provide the capability to match the optimal throughput performance of centralized scheduling mechanisms in a wide range of scenarios. Unfortunately, the type of activation rules for which throughput optimality has been established, may result in excessive queue lengths and delays. The use of more aggressive/persistent access schemes can improve the delay performance, but does not offer any universal maximum-stability guarantees. In order to gain qualitative insight and investigate the (in)stability properties of more aggressive/persistent activation rules, we examine fluid limits where the dynamics are scaled in space and time. In some situations, the fluid limits have smooth deterministic features and maximum stability is maintained, while in other scenarios they exhibit random oscillatory characteristics, giving rise to major technical challenges. In the latter regime, more aggressive access schemes continue to provide maximum stability in some networks, but may cause instability in others. Simulation experiments are conducted to illustrate and validate the analytical results

Functional limit theorems, branching processes and stochastic networks

This manuscript describes some of the work I have been doing since 2010 and the end of my PhD. As the title suggests, it contains three main parts. 1. Functional limit theorems: Chapter 2 presents two theoretical results on the weak convergence of stochastic processes: one is a sufficient condition for the tightness of a sequence of stochastic processes and the other provides a sufficient condition for the weak convergence of a sequence of regenerative processes; 2. Branching processes: in Chapter 3, scaling limits of three particular types of branching processes are discussed: 1) Galton-Watson processes in varying environments, 2) binary and homogeneous Crump-Mode-Jagers processes and 3) Crump-Mode-Jagers processes with short edges;3. Stochastic networks: Chapter 4 presents three results on stochastic networks: 1) scaling limits of the M/G/1 Processor-Sharing queue length process, 2) study of a model of stochastic network with mobile customers and 3) heavy traffic delay performance of queue-based scheduling algorithms

Stochastic hybrid system : modelling and verification

Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult. One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization. In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis. We propose a powerful unifying paradigm that combines analytical and formal methods. Its applications vary from air traffic control to communication networks and healthcare systems. The stochastic hybrid system paradigm has an explosive development. This is because of its very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems. Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability analysis, model checking). These constitute the major motivations of our research. We investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural way. The concrete contribution of this thesis has four major milestones: 1. A new and a very general model for stochastic hybrid systems; 2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities; 3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated. 4. Considering the communication issue, we extend the modelling paradigm

A Control-Theoretic Methodology for Adaptive Structured Parallel Computations

Adaptivity for distributed parallel applications is an essential feature whose impor- tance has been assessed in many research fields (e.g. scientific computations, large- scale real-time simulation systems and emergency management applications). Especially for high-performance computing, this feature is of special interest in order to properly and promptly respond to time-varying QoS requirements, to react to uncontrollable environ- mental effects influencing the underlying execution platform and to efficiently deal with highly irregular parallel problems. In this scenario the Structured Parallel Programming paradigm is a cornerstone for expressing adaptive parallel programs: the high-degree of composability of parallelization schemes, their QoS predictability formally expressed by performance models, are basic tools in order to introduce dynamic reconfiguration processes of adaptive applications. These reconfigurations are not only limited to imple- mentation aspects (e.g. parallelism degree modifications), but also parallel versions with different structures can be expressed for the same computation, featuring different levels of performance, memory utilization, energy consumption, and exploitation of the memory hierarchies. Over the last decade several programming models and research frameworks have been developed aimed at the definition of tools and strategies for expressing adaptive parallel applications. Notwithstanding this notable research effort, properties like the optimal- ity of the application execution and the stability of control decisions are not sufficiently studied in the existing work. For this reason this thesis exploits a pioneer research in the context of providing formal theoretical tools founded on Control Theory and Game Theory techniques. Based on these approaches, we introduce a formal model for control- ling distributed parallel applications represented by computational graphs of structured parallelism schemes (also called skeleton-based parallelism). Starting out from the performance predictability of structured parallelism schemes, in this thesis we provide a formalization of the concept of adaptive parallel module per- forming structured parallel computations. The module behavior is described in terms of a Hybrid System abstraction and reconfigurations are driven by a Predictive Control ap- proach. Experimental results show the effectiveness of this work, in terms of execution cost reduction as well as the stability degree of a system reconfiguration: i.e. how long a reconfiguration choice is useful for targeting the required QoS levels. This thesis also faces with the issue of controlling large-scale distributed applications composed of several interacting adaptive components. After a panoramic view of the existing control-theoretic approaches (e.g. based on decentralized, distributed or hierar- chical structures of controllers), we introduce a methodology for the distributed predictive control. For controlling computational graphs, the overall control problem consists in a set of coupled control sub-problems for each application module. The decomposition is- sue has a twofold nature: first of all we need to model the coupling relationships between control sub-problems, furthermore we need to introduce proper notions of negotiation and convergence in the control decisions collectively taken by the parallel modules of the application graph. This thesis provides a formalization through basic concepts of Non-cooperative Games and Cooperative Optimization. In the notable context of the dis- tributed control of performance and resource utilization, we exploit a formal description of the control problem providing results for equilibrium point existence and the compari- son of the control optimality with different adaptation strategies and interaction protocols. Discussions and a first validation of the proposed techniques are exploited through exper- iments performed in a simulation environment

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result