NETWORK MODELS (V-SCHEMES), Basic relationships. - Modeling of systems

NETWORK MODELS (V-SCHEMES)

In the practice of object modeling, it is often necessary to solve problems associated with a formalized description and analysis of cause-effect relationships in complex systems, where several processes simultaneously occur in parallel. The most common formalism currently describing the structure and interaction of parallel systems and processes is the petri nets proposed by Petri [28, 30].

Basic relationships.

The theory of Petri nets develops in several directions: the development of mathematical foundations, the structural theory of networks, various applications (parallel programming, discrete dynamical systems, etc.).

Formally, the Petri net (N-scheme) is given by a four-type

where B is a finite set of characters, called positions, B ^ 0; In - a finite set of symbols called transitions, BΦ0, B (1) Φ0 I is the input function (direct incidence function), 1: ВхВ - + { 0, 1}; O is the output function (the inverse function of incidence), 0 : B x 2? - * {0, 1}. Thus, the input function I maps the transition 4 to the set of input positions 6, e/(4), and the output function O maps the transition 4 to the set of output positions 6, e/)(4). For each transition 4 7), it is possible to determine the set of input positions of the transition 7 (4) and the output transition positions O (4) as

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Similarly, for each transition b ( eB , the definitions of the set of input transitions of the position 1 (b) and the set of output transitions of the 0 (b)

Graphically, the M-scheme is depicted as a bipartite oriented multigraph, representing a set of positions and transitions (Figure 2.8). As can be seen from this figure, the Scheme graph has two types of nodes: positions and transitions, represented by 0 and 1, respectively. Approximate arcs connect positions and transitions, each arc directed from an element of one set (position or transition) to an element of another set (transition or position). The graph The schema #/i> is a multigraph, since it allows the existence of multiple arcs from one vertex to another.

Fig. 2.8. Graphic image of the N-cxeme

Example 2.7. We formally represent the H-scheme, shown in the form of a graph in Fig. 2.7:

Possible applications. The above representation Schema #/i> can only be used to reflect the statics of the simulated system (the relationship of events and conditions), but it does not allow to reflect the dynamics of the functioning of the modeled system in the model. To represent the dynamic properties of an object, the marking (marking) function M is introduced: & pound; - + {0, 1, 2, ...}. Marking M is the assignment of some abstract objects called tags (chips), positions M-scheme, and the number of labels corresponding to each position may vary. With the graphical N-scheme graphical markup, the markup is displayed by placing inside the top-positions of the corresponding number of points (when the number of points is large, numbers are placed).

A labeled (marked) N-scheme can be described in the form of a five # m = <2 ?, D, /, 0, N) and is a collection Petri net and labeling M [28, 30].

The functioning of schema #/i> is reflected by moving from the markup to the markup. The initial markup is denoted as L/ 0 : # - * {0, 1, 2, ...}. Change of markings occurs as a result of operation

Fig. 2.9. Example of functioning of the marked H-scheme

one of the transitions ^ 6 0 of the network. A necessary condition for triggering the transition & lt; 1 L is 6, e/D) {N/(6 & lt;) & gt; 1}, where M (b) is the marking of the position b {. The transition (1 P /i> for which the specified condition is met, is defined as being in the ready state for triggering or as an excited transition.

Triggering the transition alters the network layout A/(6) = (A/(L 1 ), N (6 2 ), A/(6 ") ) on the marking M '(b) according to the following rule:

t. ie the transition takes one label from each of its input positions and adds one label to each of the output positions. To display the change of marking M on LH, use the notation m1m

Example 2L. Consider the marked M-scheme with the initial markup A/ 0 = {1, 0, 0, 0, 1, 0, 1}, which is shown in Fig. 2.9, a. With this initial layout of the N-scheme , the only ready-to-trigger is the transition 2 , triggering

which leads to a change in the markup M 0 and M and where M x ~ {0, 1, 1, 0, 1,0, 1} (Figure 2.9, b). When marking M x , the transitions <& , and 5 are possible. Depending on which transition worked first, one of three possible new markings is obtained (Figure 2.9, c, b, e). The operation of the X-scheme continues as long as there is at least one possible transition.

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Thus, Scheme is executed by running transitions under the control of the number of labels and their distribution in the network. The transition is started by removing the labels from its input positions and creating new labels placed in the output positions. A transition can only be triggered when it is allowed. A transition is called allowed if each of its input positions has a number of labels at least equal to the number of arcs from the position to the transition.

Example 2.9. For some given labeled M-scheme (Figure 2.8) with the initial

with the A/ 0 mark (1, 2, 0, 0, 1} (section 2.10, a) only the transition is allowed, and the remaining transitions < i> & lt; 1 2 , and - forbidden. As a result of this transition, we get a new labeled N-scheme (Fig.

2.10, b). Now the transitions c1 2 and & lt; 1 b are enabled as a result of their launching we get a new marked scheme. The transitions c1 2 and 3 are in conflict, since only one of them can be started. For example, if you start & lt; 1 b we get the network shown in Figure 2.10, in. Now only the c1 A and get a new scoped network (Figure 2.10, d). Now two transitions are allowed: a a and (in the conflict). Run the transition ^ (Figure 2.10, q). No transition can now be started and the network is terminated.

An important feature of the models of the process of functioning of systems using the typical L schemes is the simplicity of constructing the hierarchical constructions of the model. On the one hand, each Schema can be considered as a macro transition or a macroposition of a higher-level model. On the other hand, the transition or position # - of the scheme can be detailed in the form of a separate subnet for more in-depth study of the processes in the simulated system 5. This implies the possibility of efficient use of N-cxem

Figure 2.10. Example of functioning of a marked pre-defined H-scheme

for modeling parallel and competing processes in different systems.

The typical N-schemes based on conventional labeled Petri nets are suitable for describing events of arbitrary duration in the S modeled system. In this case, the model constructed using such N-schemes reflects only the order of occurrence of events in the system S. To reflect the time parameters of the functioning process of the simulated system S on the basis of N-schemes the extension of the device of Petri nets is used: temporary networks, "networks", Merlin networks, etc. [19]. In detail, the issues related to simulation using N- schemes will be discussed further below.

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