Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate $1$. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.Comment: 67 pages, 15 figures (some small corrections and improvements, one additional figure); version to be submitte

Requirements modelling of real-time systems

Real-time systems are characterised by the critical nature of their missions, and the demanding environment with which they interact. Real-time systems are used for dedicated applications. Every application is the subject of special requirements enforced by the customer. Considering the vital role that these systems play, it is imperative that a systematic approach be adopted in modelling their unique requirements. In this thesis I propose such a treatment. Real-time systems are time critical. Temporal requirements are the timing restrictions imposed by the application environment. Previous studies in requirements modelling of real-time systems have focused on adding the notion of time to modelling techniques of traditional systems without regard to the realities of requirements modelling. The information should be presented in the way the user handles it, and not the way which is convenient to the software engineer. I attempt to understand the needs of the users better by modelling the real world as close to the user's perspective as possible, and propose the Real World Model (RWM). RWM is assumed to be developed by users, and requirements engineers. An engineering approach to building the model is provided. A real-time system has a well defined use to its community. A requirements model must rely on the user level activities, and aid the human understanding and communication. In the RWM, a real-time system is viewed as a set of concurrently acting automata, each representing a system entity. This model supports temporal reasoning in easily described ways, for all classes of timing properties. A generalised classification of timing constraints is provided. A requirements modelling language facilitates the description of requirements, and serves as a medium of communication among developers and stakeholders. Jarke et al [Jarke 94] observe that there is a need for a requirements language that manages the relationship between the meta-level domain scheme, and the scenarios that actually instantiate the scheme under development. Here I propose Timed Requirements Language (TRL) to bridge this gulf between the world of stakeholders, and the world of specifiers. TRL has natural looking expressions for formulating the needs. TRL has a number of novel features including the treatment of causality, and the description of static, and dynamic constraints all integrated into one uniform framework. TRL has been used with a number of systems. The generality of the language is validated through its application to specific systems

A software testing estimation and process control model

The control of the testing process and estimation of the resource required to perform testing is key to delivering a software product of target quality on budget. This thesis explores the use of testing to remove errors, the part that metrics and models play in this process, and considers an original method for improving the quality of a software product. The thesis investigates the possibility of using software metrics to estimate the testing resource required to deliver a product of target quality into deployment and also determine during the testing phases the correct point in time to proceed to the next testing phase in the life-cycle. Along with the metrics Clear ratio. Chum, Error rate halving. Severity shift, and faults per week, a new metric 'Earliest Visibility' is defined and used to control the testing process. EV is constructed upon the link between the point at which an error is made within development and subsequently found during testing. To increase the effectiveness of testing and reduce costs, whilst maintaining quality the model operates by each test phase being targeted at the errors linked to that test phase and the ability for each test phase to build upon the previous phase. EV also provides a measure of testing effectiveness and fault introduction rate by development phase. The resource estimation model is based on a gradual refinement of an estimate, which is updated following each development phase as more reliable data is available. Used in conjunction with the process control model, which will ensure the correct testing phase is in operation, the estimation model will have accurate data for each testing phase as input. The proposed model and metrics have been developed and tested on a large-scale (4 million LOC) industrial telecommunications product written in C and C++ running within a Unix environment. It should be possible to extend this work to suit other environments and other development life-cycles

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).Comment: minor revision, correcting mistakes in formulation of Lemma 2.3 and A.5 (which are also present in the Journal paper): center of neighborhood must be in $C^1$, which is the case for the main theore