An efficient algorithm for the parallel solution of high-dimensional differential equations

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16

Approximation methods for stochastic petri nets

Stochastic Marked Graphs are a concurrent decision free formalism provided with a powerful synchronization mechanism generalizing conventional Fork Join Queueing Networks. In some particular cases the analysis of the throughput can be done analytically. Otherwise the analysis suffers from the classical state explosion problem. Embedded in the divide and conquer paradigm, approximation techniques are introduced for the analysis of stochastic marked graphs and Macroplace/Macrotransition-nets (MPMT-nets), a new subclass introduced herein. MPMT-nets are a subclass of Petri nets that allow limited choice, concurrency and sharing of resources. The modeling power of MPMT is much larger than that of marked graphs, e.g., MPMT-nets can model manufacturing flow lines with unreliable machines and dataflow graphs where choice and synchronization occur. The basic idea leads to the notion of a cut to split the original net system into two subnets. The cuts lead to two aggregated net systems where one of the subnets is reduced to a single transition. A further reduction leads to a basic skeleton. The generalization of the idea leads to multiple cuts, where single cuts can be applied recursively leading to a hierarchical decomposition. Based on the decomposition, a response time approximation technique for the performance analysis is introduced. Also, delay equivalence, which has previously been introduced in the context of marked graphs by Woodside et al., Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay equivalence often fails to converge, while flow equivalent aggregation can lead to potentially bad results if a strong dependence of the mean completion time on the interarrival process exists

Modelling safety critical systems with ageing components, with application to underground railway risk and hazards

In this thesis methodologies for modelling risk on ageing systems are developed. In the first stages of the thesis, two systems on an underground railway are used to demonstrate the modelling approach. In the latter stages of this thesis the modelling approach is expanded further, presenting a method for optimisation of a phased maintenance strategy, an inclusion of uncertainty in model outputs and an approach to model size reduction. Initially, a Petri net modelling approach is proposed to predict the derailment caused by component failures on a Switch and Crossing (S&C). A holistic methodology is adopted such that components of the system are divided into subsets of interconnected modules at a system level. Degradation within each module is idealized through a sequence of discrete states of wear until final failure occurs. Monte Carlo analysis is used to numerically evaluate the resulting Petri net. Through this methodology, different maintenance strategies, such as partial replacement, complete replacement, and opportunistic maintenance, are tested, to evaluate their influence on the final risk of derailment and predicted system state over time. This work includes a more in-depth modelling approach for S&C than that available in literature. This improves on the state of the art by removing assumptions of perfect maintenance and inspection. In addition, the approach includes modelling of dependencies between components, that are introduced through shared maintenance actions. Secondly, a Petri net modelling approach is applied to an automatic fire protection system to assess the probability of system failure, throughout the system life. Components are modelled with individual Petri nets, which are connected by a phased asset management strategy. The model is solved numerically via Monte Carlo simulation and component failure probabilities are combined using logic developed through Fault Tree analysis. For each time period, this application gives the probability of detection, deluge and alarm system failure, along with the number of maintenance actions, system tests and false system activations. The key contributions from this work include a detailed model for the interlocking fire protection systems and the application of a phased asset management strategy. This phased strategy allows the modelling of different maintenance approaches that are applied at different times depending on the system age. This approach demonstrates an increased functionality in comparison to modelling approaches currently available for fire protection systems, In addition, the modelling approach is extended further towards an optimal risk-based asset management decision making tool. The model for the fire protection systems is used as an application and is extended to give a measure of risk and whole-life cost. This extended model forms the basis of a two-stage optimisation approach within the framework of a phased asset management strategy. A Simulated Annealing algorithm is combined with a Genetic Algorithm to reduce system level risk and whole-life cost. A method for the incorporation of uncertainty in predicted model outputs is also presented. Novel aspects within this work include: the development of the optimisation approach for a phased asset management strategy and the developed algorithm for quantifying model output uncertainty given uncertain input parameters. The optimization of a phased system shows improvements on current model optimisation examples as it allows different strategies to be applied at different phases of the system lifecycle. It allows these phases to be determined in an automatic manner. The inclusion of uncertainty estimates on model outputs improves current Petri net modelling approaches, where uncertainty in input parameters is not included, as it allows decisions based on modelling outcomes to be more fully informed. Finally, a method is presented that can be applied to large system level Petri net models to produce equivalent model at a reduced computational cost. The method consists of generating a reduced Petri net which approximates the behaviour of its larger counterpart with a shorter simulation time. Parameters in this reduced structure are updated following a combined Approximate Bayesian Computation and Subset Simulation framework. Novel contributions from this work include: the proposed reduction approach, a method for using this reduction approach to improve model optimisation efficiency and the exploration of the reduction approach to justify model structure selection. These improve on approaches for model reduction available in literature, which are commonly rule based and so less flexible. In addition, model choice is typically user defined without quantifiable evidence for the suitability of the selected model structure

Modelling safety critical systems with ageing components, with application to underground railway risk and hazards

In this thesis methodologies for modelling risk on ageing systems are developed. In the first stages of the thesis, two systems on an underground railway are used to demonstrate the modelling approach. In the latter stages of this thesis the modelling approach is expanded further, presenting a method for optimisation of a phased maintenance strategy, an inclusion of uncertainty in model outputs and an approach to model size reduction. Initially, a Petri net modelling approach is proposed to predict the derailment caused by component failures on a Switch and Crossing (S&C). A holistic methodology is adopted such that components of the system are divided into subsets of interconnected modules at a system level. Degradation within each module is idealized through a sequence of discrete states of wear until final failure occurs. Monte Carlo analysis is used to numerically evaluate the resulting Petri net. Through this methodology, different maintenance strategies, such as partial replacement, complete replacement, and opportunistic maintenance, are tested, to evaluate their influence on the final risk of derailment and predicted system state over time. This work includes a more in-depth modelling approach for S&C than that available in literature. This improves on the state of the art by removing assumptions of perfect maintenance and inspection. In addition, the approach includes modelling of dependencies between components, that are introduced through shared maintenance actions. Secondly, a Petri net modelling approach is applied to an automatic fire protection system to assess the probability of system failure, throughout the system life. Components are modelled with individual Petri nets, which are connected by a phased asset management strategy. The model is solved numerically via Monte Carlo simulation and component failure probabilities are combined using logic developed through Fault Tree analysis. For each time period, this application gives the probability of detection, deluge and alarm system failure, along with the number of maintenance actions, system tests and false system activations. The key contributions from this work include a detailed model for the interlocking fire protection systems and the application of a phased asset management strategy. This phased strategy allows the modelling of different maintenance approaches that are applied at different times depending on the system age. This approach demonstrates an increased functionality in comparison to modelling approaches currently available for fire protection systems, In addition, the modelling approach is extended further towards an optimal risk-based asset management decision making tool. The model for the fire protection systems is used as an application and is extended to give a measure of risk and whole-life cost. This extended model forms the basis of a two-stage optimisation approach within the framework of a phased asset management strategy. A Simulated Annealing algorithm is combined with a Genetic Algorithm to reduce system level risk and whole-life cost. A method for the incorporation of uncertainty in predicted model outputs is also presented. Novel aspects within this work include: the development of the optimisation approach for a phased asset management strategy and the developed algorithm for quantifying model output uncertainty given uncertain input parameters. The optimization of a phased system shows improvements on current model optimisation examples as it allows different strategies to be applied at different phases of the system lifecycle. It allows these phases to be determined in an automatic manner. The inclusion of uncertainty estimates on model outputs improves current Petri net modelling approaches, where uncertainty in input parameters is not included, as it allows decisions based on modelling outcomes to be more fully informed. Finally, a method is presented that can be applied to large system level Petri net models to produce equivalent model at a reduced computational cost. The method consists of generating a reduced Petri net which approximates the behaviour of its larger counterpart with a shorter simulation time. Parameters in this reduced structure are updated following a combined Approximate Bayesian Computation and Subset Simulation framework. Novel contributions from this work include: the proposed reduction approach, a method for using this reduction approach to improve model optimisation efficiency and the exploration of the reduction approach to justify model structure selection. These improve on approaches for model reduction available in literature, which are commonly rule based and so less flexible. In addition, model choice is typically user defined without quantifiable evidence for the suitability of the selected model structure