Synchronization and Linearity: An Algebra for Discrete Event Systems
01 Jan 1994-Journal of the Operational Research Society (John Wiley & Sons Ltd)-Vol. 45, Iss: 1, pp 118-119
TL;DR: This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints, which is shown to parallel the classical linear system theory in several ways.
Abstract: This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints. The linearity has to be understood with respect to nonstandard algebraic structures, e.g. the 'max-plus algebra'. A calculus is developed based on such structures, which is followed by tools for computing the time behaviour to such systems. This algebraic vision lays the foundation of a bona fide 'discrete event system theory', which is shown to parallel the classical linear system theory in several ways.
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18 Dec 1992TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.
Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.
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06 Jul 2001TL;DR: The application of Network Calculus to the Internet and basic Min-plus and Max-plus Calculus and Optimal Multimedia Smoothing and Adaptive and Packet Scale Rate Guarantees are studied.
Abstract: Network Calculus.- Application of Network Calculus to the Internet.- Basic Min-plus and Max-plus Calculus.- Min-plus and Max-plus System Theory.- Optimal Multimedia Smoothing.- FIFO Systems and Aggregate Scheduling.- Adaptive and Packet Scale Rate Guarantees.- Time Varying Shapers.- Systems with Losses.
1,666 citations
Cites background from "Synchronization and Linearity: An A..."
...Min‐Plus Residuation Theory [Baccelli et al. 1992] is the set of wide‐sense increasing functions and ....
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Book•
01 Jan 1998
TL;DR: Survey of iterated random functions offers a method for studying the steady state distribution of a Markov chain, and presents useful bounds on rates of convergence in a variety of examples.
Abstract: Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average.
693 citations
TL;DR: Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system, and a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is express as a multiple linear mapping.
Abstract: A new matrix product, called semi-tensor product of matrices, is reviewed Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor The corresponding algorithms are developed and used to some examples
589 citations
TL;DR: An overview of the various models and problems formulated in the literature focusing on two particular models, the controlled Petri nets and the labeled nets, and two efficient techniques for the on-line computation of the control law.
Abstract: This paper surveys recent research on the application of Petri net models to the analysis and synthesis of controllers for discrete event systems. Petri nets have been used extensively in applications such as automated manufacturing, and there exists a large body of tools for qualitative and quantitative analysis of Petri nets. The goal of Petri net research in discrete event systems is to exploit the structural properties of Petri net models in computationally efficient algorithms for computing controls. We present an overview of the various models and problems formulated in the literature focusing on two particular models, the controlled Petri nets and the labeled nets. We describe two basic approaches for controller synthesis, based on state feedback and event feedback. We also discuss two efficient techniques for the on-line computation of the control law, namely the linear integer programming approach which takes advantage of the linear structure of the Petri net state transition equation, and path-based algorithms which take advantage of the graphical structure of Petri net models. Extensions to timed models are briefly described. The paper concludes with a discussion of directions for future research.
441 citations
Cites methods from "Synchronization and Linearity: An A..."
...The theory and applications of timed Petri net models for simulation, performance evaluation, and system optimization are outside the scope of this survey (see Baccelli et al. (1992) , Cohen et al. (1989) and references therein for recent research on Petri net models for timed DESs)....
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References
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01 Apr 1989
TL;DR: The author proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis, and one section is devoted to marked graphs, the concurrent system model most amenable to analysis.
Abstract: Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets. >
10,755 citations
"Synchronization and Linearity: An A..." refers background in this paper
...The section on timed Petri nets, however, is rather brief in [96]....
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...In the discussion to come we confine ourselves to event graphs, although the extension to Petri nets is quite straightforward, see [96]....
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...In terms of ‘modeling power’, see [96] and [108] for a definition, this generalization is not more powerful than the rules which will beused here....
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...An excellent overview of the theory of Petri nets is given in [96], where also many other references can be found....
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01 Mar 1973
TL;DR: This paper gives a tutorial exposition of the Viterbi algorithm and of how it is implemented and analyzed, and increasing use of the algorithm in a widening variety of areas is foreseen.
Abstract: The Viterbi algorithm (VA) is a recursive optimal solution to the problem of estimating the state sequence of a discrete-time finite-state Markov process observed in memoryless noise. Many problems in areas such as digital communications can be cast in this form. This paper gives a tutorial exposition of the algorithm and of how it is implemented and analyzed. Applications to date are reviewed. Increasing use of the algorithm in a widening variety of areas is foreseen.
5,995 citations
Additional excerpts
...The Viterbi algorithm was found in [59]....
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4,258 citations
TL;DR: In this paper, the control of a class of discrete event processes, i.e., processes that are discrete, asynchronous and possibly non-deterministic, is studied. And the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language, where the control process is described as the generator of a formal language, while the supervisor is constructed from the grammar of a specified target language that incorporates the desired closed-loop system behavior.
Abstract: This paper studies the control of a class of discrete event processes, i.e. processes that are discrete, asynchronous and possibly nondeter-ministic. The controlled process is described as the generator of a formal language, while the controller, or supervisor, is constructed from the grammar of a specified target language that incorporates the desired closed-loop system behavior. The existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language. Two examples are provided.
3,432 citations
"Synchronization and Linearity: An A..." refers background in this paper
...Among the first class, we would quote theoretical computer science languages like CSPor CCSand recent system-theoretic extensions of automata theory [114]....
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