The home marking problem and some related concepts

In this paper we study the home marking problem for Petri nets, and some related concepts to it like confluence, noetherianity, and state space inclusion. We show that the home marking problem for inhibitor Petri nets is undecidable. We relate then the existence of home markings to confluence and noetherianity and prove that confluent and noetherian Petri nets have an unique home marking. Finally, we define some versions of the state space inclusion problem related to the home marking and sub-marking problems, and discuss their decidability status

Traps characterize home states in free choice systems

AbstractFree choice nets are a subclass of Petri nets allowing to model concurrency and nondeterministic choice, but with the restriction that choices cannot be influenced externally. Home states are ground markings which can be reached from any other reachable marking of a system. A trap is a structurally defined part of a net with the property that once it is marked (that is, carries at least one token), it will remain remarked in any successor marking.The main result of this paper characterizes the home states of a live and bounded free choice system by the property that all traps are marked. This characterization leads to a polynomial-time algorithm for deciding the home state property. Other consequences include the proof that executing all parts of a net at least once necessarily leads to a home state; this has been a long standing conjecture

Performance Bounds for Synchronized Queueing Networks

Las redes de Petri estocásticas constituyen un modelo unificado de las diferentes extensiones de redes de colas con sincronizaciones existentes en la literatura, válido para el diseño y análisis de prestaciones de sistemas informáticos distribuidos. En este trabajo se proponen técnicas de cálculo de cotas superiores e inferiores de las prestaciones de redes de Petri estocásticas en estado estacionario. Las cotas obtenidas son calculables en tiempo polinómico en el tamaño del modelo, por medio de la resolución de ciertos problemas de programación lineal definidos a partir de la matriz de incidencia de la red (en este sentido, las técnicas desarrolladas pueden considerarse estructurales). Las cotas calculadas dependen sólamente de los valores medios de las variables aleatorias que describen la temporización del sistema, y son independientes de los momentos de mayor orden. Esta independencia de la forma de las distribuciones de probabilidad asociadas puede considerarse como una útil generalización de otros resultados existentes para distribuciones particulares, puesto que los momentos de orden superior son, habitualmente, desconocidos en la realidad y difíciles de estimar. Finalmente, las técnicas desarrolladas se aplican al análisis de diferentes ejemplos tomados de la literatura sobre sistemas informáticos distribuidos y sistemas de fabricación. ******* Product form queueing networks have long been used for the performance evaluation of computer systems. Their success has been due to their capability of naturally expressing sharing of resources and queueing, that are typical situations of traditional computer systems, as well as to their efficient solution algorithms, of polynomial complexity on the size of the model. Unfortunately, the introduction of synchronization constraints usually destroys the product form solution, so that general concurrent and distributed systems are not easily studied with this class of models. Petri nets have been proved specially adequate to model parallel and distributed systems. Moreover, they have a well-founded theory of analysis that allows to investigate a great number of qualitative properties of the system. In the original definition, Petri nets did not include the notion of time, and tried to model only the logical behaviour of systems by describing the causal relations existing among events. This approach showed its power in the specification and analysis of concurrent systems in a way independent of the concept of time. Nevertheless the introduction of a timing specification is essential if we want to use this class of models for the performance evaluation of distributed systems. One of the main problems in the actual use of timed and stochastic Petri net models for the quantitative evaluation of large systems is the explosion of the computational complexity of the analysis algorithms. In general, exact performance results are obtained from the numerical solution of a continuous time Markov chain, whose dimension is given by the size of the state space of the model. Structural computation of exact performance measures has been possible for some subclasses of nets such as those with state machine topology. These nets, under certain assumptions on the stochastic interpretation are isomorphic to Gordon and Newell's networks, in queueing theory terminology. In the general case, efficient methods for the derivation of performance measures are still needed. Two complementary approaches to the derivation of exact measures for the analysis of distributed systems are the utilization of approximation techniques and the computation of bounds. Approximate values for the performance parameters are in general more efficiently derived than the exact ones. On the other hand, "exactness" only exists in theory! In other words, numerical algorithms must be applied in practice for the computation of exact values, therefore making errors is inevitable. Performance bounds are useful in the preliminary phases of the design of a system, in which many parameters are not known accurately. Several alternatives for those parameters should be quickly evaluated, and rejected those that are clearly bad. Exact (and even approximate) solutions would be computationally very expensive. Bounds become useful in these instances since they usually require much less computation effort. The computation of upper and lower bounds for the steady-state performance of timed and stochastic Petri nets is considered in this work. In particular, we study the throughput of transitions, defined as the average number of firings per time unit. For this measure we try to compute upper and lower bounds in polynomial time on the size of the net model, by means of proper linear programming problems defined from the incidence matrix of the net (in this sense, we develop structural techniques). These bounds depend only on the mean values and not on the higher moments of the probability distribution functions of the random variables that describe the timing of the system. The independence of the probability distributions can be viewed as a useful generalization of the performance results, since higher moments of the delays are usually unknown for real cases, and difficult to estimate and assess. From a different perspective, the obtained results can be applied to the analysis of queueing networks extended with some synchronization schemes. Monoclass queueing networks can be mapped on stochastic Petri nets. On the other hand, stochastic Petri nets can be interpreted as monoclass queueing networks augmented with synchronization primitives. Concerning the presentation of this manuscript, it should be mentioned that chapter 1 has been written with the object of giving the reader an outline of the stochastic Petri net model: its definition, terminology, basic properties, and related concepts, together with its deep relation with other classic stochastic network models. Chapter 2 is devoted to the presentation of the net subclasses considered in the rest of the work. The classification presented here is quite different from the one which is usual in the framework of Petri nets. The reason lies on the fact that our classification criterion, the computability of visit ratios for transitions, is introduced for the first time in the field of stochastic Petri nets in this work. The significance of that criterion is based on the important role that the visit ratios play in the computation of upper and lower bounds for the performance of the models. Nevertheless, classical important net subclasses are identified here in terms of the computability of their visit ratios from different parameters of the model. Chapter 3 is concerned with the computation of reachable upper and lower bounds for the most restrictive subclass of those presented in chapter 2: marked graphs. The explanation of this fact is easy to understand. The more simple is the model the more accessible will be the techniques an ideas for the development of good results. Chapter 4 provides a generalization for live and bounded free choice nets of the results presented in the previous chapter. Quality of obtained bounds is similar to that for strongly connected marked graphs: throughput lower bounds are reachable for bounded nets while upper bounds are reachable for 1-bounded nets. Chapter 5 considers the extension to other net subclasses, like mono-T-semiflow nets, FRT-nets, totally open deterministic systems of sequential processes, and persistent nets. The results are of diverse colours. For mono-T-semiflow nets and, therefore, for general FRT-nets, it is not possible (so far) to obtain reachable throughput bounds. On the other hand, for bounded ordinary persistent nets, tight throughput upper bounds are derived. Moreover, in the case of totally open deterministic systems of sequential processes the exact steady-state performance measures can be computed in polynomial time on the net size. In chapter 6 bounds for other interesting performance measures are derived from throughput bounds and from classical queueing theory laws. After that, we explore the introduction of more information from the probability distribution functions of service times in order to improve the bounds. In particular, for Coxian service delay of transitions it is possible to improve the throughput upper bounds of previous chapters which held for more general forms of distribution functions. This improvement shows to be specially fruitful for live and bounded free choice nets. Chapter 7 is devoted to case studies. Several examples taken from literature in the fields of distributed computing systems and manufacturing systems are modelled by means of stochastic Petri nets and evaluated using the techniques developed in previous chapters. Finally, some concluding remarks and considerations on possible extensions of the work are presented

Contributions to the deadlock problem in multithreaded software applications observed as Resource Allocation Systems

Desde el punto de vista de la competencia por recursos compartidos sucesivamente reutilizables, se dice que un sistema concurrente compuesto por procesos secuenciales está en situación de bloqueo si existe en él un conjunto de procesos que están indefinidamente esperando la liberación de ciertos recursos retenidos por miembros del mismo conjunto de procesos. En sistemas razonablemente complejos o distribuidos, establecer una política de asignación de recursos que sea libre de bloqueos puede ser un problema muy difícil de resolver de forma eficiente. En este sentido, los modelos formales, y particularmente las redes de Petri, se han ido afianzando como herramientas fructíferas que permiten abstraer el problema de asignación de recursos en este tipo de sistemas, con el fin de abordarlo analíticamente y proveer métodos eficientes para la correcta construcción o corrección de estos sistemas. En particular, la teoría estructural de redes de Petri se postula como un potente aliado para lidiar con el problema de la explosión de estados inherente a aquéllos. En este fértil contexto han florecido una serie de trabajos que defienden una propuesta metodológica de diseño orientada al estudio estructural y la correspondiente corrección física del problema de asignación de recursos en familias de sistemas muy significativas en determinados contextos de aplicación, como el de los Sistemas de Fabricación Flexible. Las clases de modelos de redes de Petri resultantes asumen ciertas restricciones, con significado físico en el contexto de aplicación para el que están destinadas, que alivian en buena medida la complejidad del problema. En la presente tesis, se intenta acercar ese tipo de aproximación metodológica al diseño de aplicaciones software multihilo libres de bloqueos. A tal efecto, se pone de manifiesto cómo aquellas restricciones procedentes del mundo de los Sistemas de Fabricación Flexible se muestran demasiado severas para aprehender la versatilidad inherente a los sistemas software en lo que respecta a la interacción de los procesos con los recursos compartidos. En particular, se han de resaltar dos necesidades de modelado fundamentales que obstaculizan la mera adopción de antiguas aproximaciones surgidas bajo el prisma de otros dominios: (1) la necesidad de soportar el anidamiento de bucles no desplegables en el interior de los procesos, y (2) la posible compartición de recursos no disponibles en el arranque del sistema pero que son creados o declarados por un proceso en ejecución. A resultas, se identifica una serie de requerimientos básicos para la definición de un tipo de modelos orientado al estudio de sistemas software multihilo y se presenta una clase de redes de Petri, llamada PC2R, que cumple dicha lista de requerimientos, manteniéndose a su vez respetuosa con la filosofía de diseño de anteriores subclases enfocadas a otros contextos de aplicación. Junto con la revisión e integración de anteriores resultados en el nuevo marco conceptual, se aborda el estudio de propiedades inherentes a los sistemas resultantes y su relación profunda con otros tipos de modelos, la confección de resultados y algoritmos eficientes para el análisis estructural de vivacidad en la nueva clase, así como la revisión y propuesta de métodos de resolución de los problemas de bloqueo adaptadas a las particularidades físicas del dominio de aplicación. Asimismo, se estudia la complejidad computacional de ciertas vertientes relacionadas con el problema de asignación de recursos en el nuevo contexto, así como la traslación de los resultados anteriormente mencionados sobre el dominio de la ingeniería de software multihilo, donde la nueva clase de redes permite afrontar problemas inabordables considerando el marco teórico y las herramientas suministradas para subclases anteriormente explotadas

Analytics of Condition-Effect Rules

This thesis studies properties such as confluence and termination for a rule model with condition-effect rules. A rule model is first defined and the complexity of solving these problems is analysed. Analysis of both confluence and termination shows that they are PSPACE-complete for our rule model. We give algorithms for testing these properties. We also study certain syntactic and structural restrictions under which these problems become easier and can be solved in polynomial time for practical purposes

Recent advances in petri nets and concurrency

CEUR Workshop Proceeding

Verifying temporal properties of systems with applications to petri nets

This thesis provides a powerful general-purpose proof technique for the verification of systems, whether finite or infinite. It extends the idea of finite local model-checking, which was introduced by Stirling and Walker: rather than traversing the entire state space of a model, as is done for model-checking in the sense of Emerson, Clarke et al. (checking whether a (finite) model satisfies a formula), local model-checking asks whether a particular state satisfies a formula, and only explores the nearby states far enough to answer that question. The technique used was a tableau method, constructing a tableau according to the formula and the local structure of the model. This tableau technique is here generalized to the infinite case by considering sets of states, rather than single states; because the logic used, the propositional modal mu-calculus, separates simple modal and boolean connectives from powerful fix-point operators (which make the logic more expressive than many other temporal logics), it is possible to give a relatively straightforward set of rules for constructing a tableau. Much of the subtlety is removed from the tableau itself, and put into a relation on the state space defined by the tableau-the success of the tableau then depends on the well-foundedness of this relation. This development occupies the second and third chapters: the second considers the modal mu-calculus, and explains its power, while the third develops the tableau technique itself The generalized tableau technique is exhibited on Petri nets, and various standard notions from net theory are shown to play a part in the use of the technique on nets-in particular, the invariant calculus has a major role. The requirement for a finite presentation of tableaux for infinite systems raises the question of the expressive power of the mu-calculus. This is studied in some detail, and it is shown that on reasonably powerful models of computation, such as Petri nets, the mu-calculus can express properties that are not merely undecidable, but not even arithmetical. The concluding chapter discusses some of the many questions still to be answered, such as the incorporation of formal reasoning within the tableau system, and the power required of such reasoning