**5. Business process**

116 Petri Nets – Manufacturing and Computer Science

**Figure 6.** Weighted Synchronous Distance

smallest leads to uniqueness of the distance.

**4.3. Computing synchronous distance** 

computing weighted distances based on this algorithm.

distances is omitted here.

(a) Σ7 (b)Σ<sup>8</sup>

Theorem 4

transition t in T1.◆

asynchronous.

distance is encountered. For simplicity, the formal definition of weighted synchronous

Generally speaking, T1 and T2 may have more than one transition. It is possible that all transitions in the same set share the same weight, it is also possible that different transitions need different weights. It is assumed that the weights, if exist, would take the smallest possible values. For simplicity, we write σ'(T1,T2) for weighted distance between T1 and T2 when the weights for individual transitions are known. The restriction on weights to be

For an arbitrary place s in S in P/T-systemΣ, as long as s has disjoint pre-set and post-set, σ'(T1,T2) = max{ M(s)∣M∈[M0>} – min{ M(s)∣M∈[M0>} if exists, otherwise σ′(T1,T2) = ∞, where T1 and T2 are respectively the pre-set and post-set of s and σ'(T1,T2) denotes a weighted distance with W(s,t) as the weight of transition t in T2 and W(t,s) as the weight of

In case there are no weights that yield a finite distance between T1 and T2, T1 and T2 are

All transition sequences can be found on the graph G(Σ). Synchronous distances, weighted or not, are defined in terms of transition sequences. Thus, it is possible to design algorithms for the computing of distances. Firstly, if a ring exists on G(Σ) that contains different numbers of firings of T1 transitions and T2 transitions, the distance is ∞. All algorithms are designed for computing finite distances only. The book in Chinese entitled *Petri Net Applications* by the author will soon appear, in which an algorithm for the computing of synchronous distances in P/T-systems can be found. It is easy to design an algorithm for

This theorem is true since what the added place records is exactly what happens in s.

A successful application of Petri nets requires a full understanding of the application problem.

The term "workflow" was used as a synonym for "business process" in the book *Workflow Management* by W. Aalst and K. Hee. The terminology was developed by the Workflow Management Coalition (WFMC), an organization dedicated to develop standard terminology and standard interfaces for workflow management systems components.

As the first step towards workflow modelling, we make a clear distinction between "workflow" and "business process".
