Modeling Hybrid Systems in SIMTHESys

In this paper, we present a Fluid Stochastic Petri Net (FSPN) based model which captures the behavior of aging software systems with checkpointing, rejuvenation and self-restoration, three well known techniques of software fault tolerance. The proposed FSPN based modeling framework is novel in many aspects. First, the FSPN formalism itself is extended by adding flush-out arcs. Second, the three techniques are simultaneously captured in a single model for the first time. Third, the formalism enables modeling dependencies of the three techniques on various system features such as failure, load and time in the same framework. Further, our base FSPN model can be viewed as a generalization of most previous models in the literature. We show that these FSPNs can not only mimic previously published models but can also extend them. For one FSPN model, we present numerical results to illustrate their usage in deriving measures of interest.

In this paper we focused our attention on a particular hybrid PN model called FOHPN, that is based on the fluidification of discrete PNs, and whose main feature is that the instantaneous firing speed of continuous transitions keeps constant during each macroperiod. In the first part of the paper we discuss in detail the advantages of fluidification, and provide a brief survey of the most important formalisms within the hybrid PN framework. Finally, we showed how FOHPNs can be efficiently used to model SC, and how interesting optimization problems can be solved via numerical simulation, by simply solving on-line a certain number of LPPs.

Performance regulation of event-driven dynamical systems using infinitesimal perturbation analysis

Speculative execution is one of the key issues to boost the performance of future generation microprocessors. In this paper, we introduce a novel approach to evaluate the effects of branch and value prediction, which allow the processor to execute instructions beyond the limits of control and true data dependences. Until now, almost all the estimations of their performance potential under different scenarios have been obtained using trace-driven or execution-driven simulation. Occasionally, some simple deterministic models have been used. We employ an analytical model based on recently introduced Fluid Stochastic Petri Nets (FSPNs) in order to capture the dynamic behavior of an ILP processor with aggressive use of prediction techniques and speculative execution. Here we define the FSPN model, derive the state equations for the underlying stochastic process and present performance evaluation results to illustrate its usage in deriving measures of interest. Our implementation-independent stochastic modeling framework reveals considerable potential for further research in this area using numerical solution of systems of partial differential equations and/or discrete-event simulation of FSPN models.

Stochastic Petri Net Package (SPNP) is a software package whose goal is to compute performance, availability or performability measures from Stochastic Petri Nets (SPN) and Fluid Stochastic Petri nets (FSPN). This software can use either analytic numeric methods, or simulation methods. Unfortunately, the standard discrete event simulation is inefficient to estimate the probabilities of rare events. For such rare event simulations, importance splitting technique is a good method to speedup the simulation. In the literature, two diffierent importance splitting techniques are known: RESTART and splitting. In this paper, we describe the application of these methods to (both fluid and discrete) Petri nets, their implementation in SPNP and we give some illustrations of the speed-up. The RESTART technique has already been applied in another Petri net package, TimeNet, but here we implement both RESTART and splitting, and we apply them to a more general class of Petri nets including the fluid ones.

This paper presents a new framework for traffic flow control based on an integrated model description by means of a Hybrid Dynamical System (HDS). The geometrical information on the traffic network is characterized by a Hybrid Petri Net (HPN). Then, the algebraic behavior of traffic flow is transformed into a Mixed Logical Dynamical System (MLDS) in order to introduce an optimization technique. These expressions involve both continuous evolution of traffic flow and event driven behavior of traffic light. HPN allows us to formulate the problem easily for complicated and large-scale traffic network due to its graphical understanding. MLDS enables us to optimize the control policy for traffic light by means of its algebraic manipulability and use of model predictive control framework. Since the behavior represented by HPN can be directly transformed into corresponding MLDS, the seamless incorporation of two different modeling schemes provide a systematic design for traffic flow control.

This paper presents a new framework for traffic flow control based on an integrated model description by means of a Hybrid Dynamical System (HDS). The geometrical information on the traffic network is characterized by a Hybrid Petri Net (HPN). Then, the algebraic behavior of traffic flow is transformed into a Mixed Logical Dynamical System (MLDS) form in order to introduce an optimization technique. These expressions involve both continuous evolution of traffic flow and event driven behavior of traffic light. HPN allows us to formulate the problem easily for complicated and large-scale traffic network due to its graphical understanding. MLDS enables us to optimize the control policy for traffic light by means of its algebraic manipulability and use of model predictive control framework. Since the behavior represented by HPN can be directly transformed into corresponding MLDS form, the seamless incorporation of two different modeling schemes provide a systematic design scenario for traffic flow control.

On fluidification of Petri Nets: from discrete to hybrid and continuous models

Hybrid Petri nets (HPN) are frequently used to model and study hybrid systems, i.e. those with both discrete (events, state changes) and continuous (flows, changing physical variables) dynamics. In recent years, their field of application has been extended to energy systems, and in particular to the modeling of electrical microgrids. These microgrids are often powered by renewable energy sources, whose operation is variable and subject to environmental constraints. With this in mind, the present study suggests an HPN-based modeling technique to illustrate the operation of an electrical microgrid that integrates a photovoltaic installation associated with a battery storage system. The idea is to capture the complex interaction between various energy sources, storage units, energy demand and possible complementary sources. The associated mathematical model is based on a linear algebraic representation of type Ax ≤ b, which formalizes the system's constraints, tracks the evolution of its state over time, and anticipates its future behavior. Simultaneously, a state observer is developed with the intention of evaluating the current Petri net marking exclusively from observable results, in order to deduce an accurate estimate of the internal state. By comparing the simulation results with the estimation results, we are able to assess the robustness and accuracy of the suggested model. This research is part of an approach aimed at optimizing and intelligently supervising sustainable energy systems.

We propose a generalized concept of hybrid Petri nets as a modeling and design tool for the distributed dynamical systems involving asynchronous event-based communication. The proposed Petri net machine represents an extension of the standard discrete-event Petri nets, but differs conceptually from the conventional hybrid Petri nets in that dynamical processes are stored in the continuous places, similarly as in hybrid automata. In particular, the firing semantics of the corresponding continuous arcs is defined to capture the information exchange between the network nodes. While being primarily a graphical modeling tool, the proposed formalism of hybrid Petri nets infers a hybrid dynamical setting of event-triggered control problems, an approach that has been missing in the literature. The usability of our approach is illustrated in the context of the distributed event-triggered LQR and consensus problems, in conjunction with a systematic methodology for designing optimal adaptive scheduling schemes, governing the communication resources.

In this paper jump transitions for the fluid model part in second order fluid stochastic Petri nets (FSPNs) are introduced. The extended formalism is defined, and the underlying partial differential equations are derived. With a jump a certain amount of fluid is added to a fluid place or taken out at once. This amount, the jump height, is sampled from a probability distribution. The dynamics of a model are described by second order partial differential equations that include integrals. As examples the virtual waiting time (or unfinished work) in a queueing system and a performability model of a multi-processor system are modelled.

In this chapter, modeling of GRNs using Petri net theory is considered. It aims at providing a conceptual understanding of Petri nets to enable the reader to explore GRNs applying Petri net modeling and analysis techniques. Starting with an overview on modeling biochemical networks using Petri nets, the state-of-the-art with focus on GRNs is described. Other modeling techniques, for example, hybrid Petri nets are discussed. Basic concepts of Petri net theory are introduced involving special analysis techniques for modeling biochemical systems, for example, MCT-sets, T-clusters, and Mauritius maps. To illustrate these Petri net concepts, a more complex case study–the gene regulation in Duchenne Muscular Dystrophy–is explained in detail, considering the biological background and the interpretation of analysis results. Considering both, advantages and disadvantages, the chapter demonstrates the usefulness of Petri net modeling, in particular for GRNs.

Control Design of Elementary Hybrid Petri Nets via Model Predictive Control

The problem of providing a modular representation of urban traffic and roadways by means of Petri nets is addressed. A modular modelling methodology is presented. After defining the urban traffic model and identifying each one of its components, it is possible to define for each element a hybrid Petri net module (HPNM), and provide an internal representation of it at a desired level of detail. A synthesis procedure allows one to join the single modules and provide the representation of the whole urban roadway considered in terms of hybrid Petri nets. An interesting characteristic proper to the proposed methodology is the complete independence of HPNMs from their internal representations. This allows one to change the model without any effort by increasing/decreasing the specified level of detail of some HPNMs making up the HPN-model of the whole system.

Summary Biological systems are often very complex so that an appropriate formalism is needed for modeling their behavior. Hybrid Petri Nets, consisting of time-discrete Petri Net elements as well as continuous ones, have proven to be ideal for this task. Therefore, a new Petri Net library was implemented based on the object-oriented modeling language Modelica which allows the modeling of discrete, stochastic and continuous Petri Net elements by differential, algebraic and discrete equations. An appropriate Modelica-tool performs the hybrid simulation with discrete events and the solution of continuous differential equations. A special sub-library contains so-called wrappers for specific reactions to simplify the modeling process.The Modelica-models can be connected to Simulink-models for parameter optimization, sensitivity analysis and stochastic simulation in Matlab.The present paper illustrates the implementation of the Petri Net component models, their usage within the modeling process and the coupling between the Modelica-tool Dymola and Matlab/Simulink. The application is demonstrated by modeling the metabolism of Chinese Hamster Ovary Cells.