**4.2. Transition synchronization and place synchronization**

People would think, based on experiences in daily life, that synchronization is something participated by different parties like hand clapping, or some events occurring at the same time like a live casting on TV with an on-going game. Such synchronization is characterized by distance 0, since one cannot see one hand clapping or see the TV show without the game.

We have said that hand clapping is a single transition consisting of actions of two hands; A live show and the on-going game would appear in a net as one transition as well since if they were separated as different transitions in parallel, ordered firings would produce the same effect , but this cannot be true. Synchronization characterized by distance 0 is "transition synchronization". Synchronization with σ(T1,T2) > 0 is "place synchronization" since such synchronization is achieved via places and it can be observed by taking an added place as an observation window.

System Σ5 in Figure 5 (a) is the same system as shown in Figure 3 (b) with an added place p denoted by a dotted circle and connected to transitions by dotted arrows. This added place does not belong to Σ5, it is to be used for observations.

(a) Σ5 (b) Σ<sup>6</sup>

114 Petri Nets – Manufacturing and Computer Science

the physical meaning of σ(a,b) =2 for Σ2.

the initial marking. Figure 4 (b) is still another case of σ(a,b) =2.

The synchronous distance defined by Definition 8 satisfies distance axioms:

when σ(T1,T2)> 0, since σ(T1,T2) = σ(T1-T2, T2 - T1) by definition.

It is easy to prove the above two theorems, so omitted here.

**4.2. Transition synchronization and place synchronization** 

σ(T1,T2) = 0 if and only if T1 = T2 ; σ(T1,T2)≥ 0; σ(T1,T2) = σ(T2,T1); σ(T1,T2) + σ(T2,T3)≥ σ(T1,T3).

This theorem explains why the concept is called distance. It is assumed that T1∩T2= ∅

σ(T1,T2)<∞ if and on ly if for any repeatable portion δ of any sequence in ρ, #(δ, T1,T2) = 0. ◆

People would think, based on experiences in daily life, that synchronization is something participated by different parties like hand clapping, or some events occurring at the same time like a live casting on TV with an on-going game. Such synchronization is characterized by distance 0, since one cannot see one hand clapping or see the TV show without the game.

hand clapping.

(a) Σ3 (b) Σ<sup>4</sup>

Theorem 2

Theorem 3

◆

**Figure 4.** σ(a,b) =2: a and b in conflict

number of firings of transition b and transition a. This is why σ(a,b) is finite. The consecutive firings of transition a count at most to 2, so do the consecutive firings of transition b. This is

The distance (i.e. synchronous distance from now on) between left hand action and right hand action is 0, since it is impossible to see only an action of either hand in the course of

Figure 4 (a) shows another situation of σ(a,b) =2, where transitions a and b are in conflict at

**Figure 5.** Added Place as Observation Window

An observer records what he finds through the window by putting a token into p when transition a fires and removing a token from p when transition b fires. It is assumed that place p has enough tokens, say n tokens, to start with, so the recording would not be interrupted. The maximum number of tokens in p is n+1 while the minimum is n-1. So the difference is 2 between transition b and transition a. The same observation applies to Σ6 to get the same distance.

The observation window p for disjoint transition sets T1 and T2 is an added place whose input arcs are from T1 and output arcs are pointing to T2. Place p gets a token whenever a transition in T1 fires and loses a token whenever a transition in T2 fires. Place p has enough tokens to start with to ensure a smooth observation. In case the number of tokens in p is not bounded, the distance between T1 and T2 is ∞. Otherwise, the difference between the maximum and the minimum is the distance.

It is easy to find that σ(a,b) = ∞ for the P/T-system in Figure 6 (a), since the repeatable sequence t1abat2 contains two firings of a and only one firing of b. The added place p as shown in Figure 6 (b) has weight 2 on the arc from p to transition b, it would have n+1 tokens after the first firing of a and n-1 tokens after the firing of b. The difference is 2.This is a weighted distance between b and a: σ(a,2b) = 2, the weight for a is 1 while the weight for b is 2. The concept of weighted synchronous distances makes a distinction when infinite distance is encountered. For simplicity, the formal definition of weighted synchronous distances is omitted here.

Workflow Modelling Based on Synchrony 117

We go no further on synchronous distances in this chapter since what concerns us is workflow modelling and what has been said about synchronous distances is already

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

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

A business process is a pre-designed process in an enterprise or an organization for conducting the manipulation of individual cases of a business. It existed even before the birth of computers. The manipulation of individual business cases consists of business tasks and management tasks. The concept of "workflow" aims at "computerized facilitation or automation of a business process, in whole or part" (WFMC). To this end, the separation of business tasks and management tasks is of first importance. In a way, this is similar to the separation of data processing from a program, leading to the concepts of databases and

A single business task may be carried out by a computer program. But "computerized automation of a business process" focuses on management automation rather than task automation. Management automation relies on clearly specified management rules. Most of the rules apply to all cases while some of the rules apply to individual cases. The former is "workflow logic" and the latter is "case semantics". The purpose of workflow modelling is nothing but to establish a formal specification of management rules to serve as a guide in the design, implementation and execution of a computer system called "workflow engine". It is the execution of the engine that conducts the processing of business cases. Figure 7

The workflow logic specifies how business tasks are ordered (for causally dependent tasks) and/or synchronized. But, ordering is also a way of synchronization. Thus, workflow logic specifies how business tasks are synchronized for all conceivable cases in a business. In other words, workflow logic specifies all possible routes that a business case may take when being processed. What workflow logic cares about a single business task is not what data it requires or what data it will produce, let along what exact values are inputted or outputted.

In this sense, workflow logic is concerned with abstract business tasks only.

sufficient.

problem.

**5. Business process** 

"workflow" and "business process".

database management systems.

illustrates how workflow is related to business process.

**5.1. Business process vs. workflow** 

(a) Σ7 (b)Σ<sup>8</sup>

**Figure 6.** Weighted Synchronous Distance

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 smallest leads to uniqueness of the distance.

### Theorem 4

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 transition t in T1.◆

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

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