Techniques of petri net reduction

Petri Nets have the capability to analyze large and complex concurrent systems. However, there is one constraint. The number of reachability states of the concurrent systems outweighs the capability of Petri Nets. Previous Petri Net reduction techniques focussed on reducing a subnet to a single transition and hence not powerful enough to reduce a Petri Net. This paper presents six reduction rules and discusses their drawbacks. A new reduction technique called Knitting Technique to delete paths of a Petri Net while retaining all the properties of the original net is presented. Further Structural matrix which facilitates reduction is presented

An efficient scheduling method for grid systems based on a hierarchical stochastic petri net

This paper addresses the problem of resource scheduling in a grid computing environment. One of the main goals of grid computing is to share system resources among geographically dispersed users, and schedule resource requests in an efficient manner. Grid computing resources are distributed, heterogeneous, dynamic, and autonomous, which makes resource scheduling a complex problem. This paper proposes a new approach to resource scheduling in grid computing environments, the hierarchical stochastic Petri net (HSPN). The HSPN optimizes grid resource sharing, by categorizing resource requests in three layers, where each layer has special functions for receiving subtasks from, and delivering data to, the layer above or below. We compare the HSPN performance with the Min-min and Max-min resource scheduling algorithms. Our results show that the HSPN performs better than Max-min, but slightly underperforms Min-min

Petri net controllers for Generalized Mutual Exclusion Constraints with floor operators

In this paper a special type of nonlinear marking specifications called stair generalized mutual exclusion constraints (stair-GMECs) is defined. A stair-GMEC can be represented by an inequality whose left-hand is a linear combination of floor functions. Stair-GMECs have higher modeling power than classical GMECs and can model legal marking sets that cannot be defined by OR–AND GMECs. We propose two algorithms to enforce a stair-GMEC as a closed-loop net, in which the control structure is composed by a residue counter, remainder counters, and duplicate transitions. We also show that the proposed control structure is maximally permissive since it prevents all and only the illegal trajectories of a plant net. This approach can be applied to both bounded and unbounded nets. Several examples are proposed to illustrate the approach

Modeling and Simulation of Task Allocation with Colored Petri Nets

The task allocation problem is a key element in the solution of several applications from different engineering fields. With the explosion of the amount of information produced by the today Internet-connected solutions, scheduling techniques for the allocation of tasks relying on grids, clusters of computers, or in the cloud computing, is at the core of efficient solutions. The task allocation is an important problem within some branch of the computer sciences and operations research, where it is usually modeled as an optimization of a combinatorial problem with the inconvenience of a state explosion problem. This chapter proposes the modeling of the task allocation problem by the use of Colored Petri nets. The proposed methodology allows the construction of compact models for task scheduling problems. Moreover, a simulation process is possible within the constructed model, which allows the study of some performance aspects of the task allocation problem before any implementation stage

Аппроксимация ресурсных эквивалентностей в сетях Петри с невидимыми переходами

Two resources (submarkings) are called similar if in any marking any one of them can be replaced by another one without affecting the observable behavior of the net (regarding marking bisimulation). It is known that resource similarity is undecidable for general labelled Petri nets. In this paper we study the properties of the resource similarity and resource bisimulation (a subset of complete similarity relation closed under transition firing) in Petri nets with invisible transitions (where some transitions may be labelled with an invisible label (τ) that makes their firings unobservable for an external observer). It is shown that for a proper subclass (p-saturated nets) the resource bisimlation can be effectively checked. For a general class of Petri net with invisible transitions it is possible to construct a sequence of so-called (n, m)-equivalences approximating the largest τ-bisimulation of resources.Два ресурса (подразметки) называются подобными, если в любой разметке любой из них может быть заменен другим, и при этом наблюдаемое поведение сети не изменится (относительно бисимуляции разметок). Известно, что подобие ресурсов неразрешимо для обыкновенных сетей Петри. В этой статье мы изучаем свойства подобия ресурсов и бисимуляции ресурсов (подмножество отношения подобия, замкнутое по срабатыванию переходов) в сетях Петри с невидимыми переходами (где некоторые переходы могут быть помечены специальной меткой (τ), что делает их срабатывания невидимыми для внешнего наблюдателя). Показано, что для собственного подкласса (p-насыщенных сетей) бисимуляция ресурсов может быть эффективно проверена. Для общего класса сетей Петри с невидимыми переходами можно построить последовательность так называемых (n, m)-эквивалентностей, аппроксимирующую наибольшую τ-бисимиляцию ресурсов

Resource Bisimilarity in Petri Nets is Decidable

Petri nets are a popular formalism for modeling and analyzing distributed systems. Tokens in Petri net models can represent the control flow state or resources produced/consumed by transition firings. We define a resource as a part (submultiset) of the Petri net marking and call two resources equivalent iff replacing one of them with another in any marking does not change the observable Petri net behavior. We investigate the resource similarity and the resource bisimilarity -- congruent restrictions of the bisimulation equivalence on Petri net markings and prove that the resource bisimilarity is decidable in contrast to the resource similarity.Comment: New version for submission to the journa

Subclasses of Formalized Data Flow Diagrams: Monogeneous, Linear & Topologically Free Choice RDFD\u27s

Formalized Data Flow Diagrams (FDFD\u27s) and, especially, Reduced Data Flow Diagrams (RDFD\u27s) are Turing equivalent (Symanzik and Baker, 1996). Therefore, no decidability problem can be solved for FDFD\u27s in general. However, it is possible to define subclasses of FDFD\u27s for which decidability problems can be answered. In this paper we will define certain subclasses of FDFD\u27s, which we call Monogeneous RDFD\u27s, Linear RDFD\u27s, and Topologically Free Choice RDFD\u27s. We will show that two of these three subclasses of FDFD\u27s can be simulated via isomorphism by the correspondingly named subclasses of FIFO Petri Nets. It is known that isomorphisms between computation systems guarantee the same answers to corresponding decidability problems (e. g., reachability, deadlock, liveness) in the two systems (Kasai and Miller, 1982). This means that problems where it is known that they can (not) be solved for a subclass of FIFO Petri Nets it follows immediately that the same problems can (not) be solved for the correspondingly named subclass of FDFD\u27s

Составные редукции моделей Крипке и автоморфизмы

Kripke factor-model concept is investigated. It is shown, that every factor-model is representexl as a decomposition of several spexdal facctor-models, which groups of automorphisms are primes. Moreover, we show, that every finite group is isomorphic for a group of automorphisms of a certain Kripke model.Показано, что с помощью понятия фактор-модели, предложенного в [1], произвольные модели Крипке могут быть представлены в виде композиции моделей с простыми группами автоморфизмов. Доказано также, что любая конечная группа изоморфна группе автоморфизмов некоторой подходящей мо¬дели Крипке