**4. PN as a general system representation framework**

As pointed in the beginning of this work, Petri Nets has developed for the last fifty years to become a general schema for systems modeling.

**Figure 7.** Time diagram for the web user latency.

*V* = *vi* is a finite set of state values;

following.

of the gas station, the total time interval spent to serve one or nine users is the summation of

Timed Petri Nets 373

Thus, the complete process would be to elicit the requirements using UML diagrams including time diagrams - synthesize a Timed Petri Net from this model, and them perform the requirement analysis and final synthesis of a model for the problem. In fact, the final

Formally, the timelines are drawn according the behavior of *state variables*, defined in the

A timeline is the tracking of all changes in state value in a interval [0, *τ*) where *τ* is the observed time horizon. A timeline is said to be completely closed if the union of its not superposed values is exactly *τ*. In that case the transitions occur in deterministic time.

the not superposed time intervals required for each dependent action.

results for the example of the gas station were obtained following this approach.

Definition 15.[State Variables] A state variable is a triple (*V*, *T*, *D*) where:

*T* : *V* → *V* specify each atomic transition or change in value. *D* : *V* → **N** × **N** is the time duration for each state value.

In the previous section, we showed that a modeling discipline should be followed to achieve good results with Petri Nets formal representation, specially when time is an important variable to consider, either by deterministic time or using continuous dense time intervals. However, in the example above time does not appear explicitly at the beginning, since we started with the class diagram where there was no reference to duration time of the processes (supplying or payment). The problem them begins with a demand to a proper representation of time duration in UML that could later be transformed in a timed net.

To fit this demand UML 2.0 specification inserted an interaction diagram derived from the sequence diagram where time intervals or time duration are very important issues. Thus, once identified the actors and sub-systems in the model, their interaction could be viewed and modeled taking in account that it occurs during a running time where specific events can cause a change in the status of that interaction. Thus, the full relation can be described in what is called a state lifetime where several timelines show the evolution of the interacting components.

OMG (www.omg.org) shows a very appealing example of time diagram to model

In Figure 7 we can see a hierarchical superposition of levels and the action derived from the interaction between a user and a web system. Sub-systems invoked by this action and the time they spent to provide a proper action are explicitly depicted. As in the previous problem

**Figure 7.** Time diagram for the web user latency.

14 Will-be-set-by-IN-TECH

The introduction of deterministic time (transition) would add more detail about the process, with the characteristic time for processing the payment or to fill a car. An organized queue would fail (even if works quite fine in the model) since this time can be modified depending of the user or to unpredictable events during the payment or during the supply process. However if specific (and deterministic)intervals such as 3 min for the payment and 5 min for the filling of gas are stablished, the system could handle 9 drivers in 25 min with a waiting

More convincing examples can be found in business, manufacturing or computer networks. More challenge problems emerged in the spatial applications or satellite control, but what is important is that even deterministic time approach can be used to solve a diversified set of problems. However, it could be stressed that the timed approach should be supported for tools and environments that rely in a sound and complementary approach to Timed Nets including Time Petri Nets. The approach shown here, inserted in the GHENeSys environment is exactly one of this cases. Besides, GHENeSys is an implementation of a unified net, that

In the next section we go further in the discussion of using Petri Nets and specifically Timed Petri Nets to fit requirements that come in the new version of UML, which includes time

As pointed in the beginning of this work, Petri Nets has developed for the last fifty years to

In the previous section, we showed that a modeling discipline should be followed to achieve good results with Petri Nets formal representation, specially when time is an important variable to consider, either by deterministic time or using continuous dense time intervals. However, in the example above time does not appear explicitly at the beginning, since we started with the class diagram where there was no reference to duration time of the processes (supplying or payment). The problem them begins with a demand to a proper representation

To fit this demand UML 2.0 specification inserted an interaction diagram derived from the sequence diagram where time intervals or time duration are very important issues. Thus, once identified the actors and sub-systems in the model, their interaction could be viewed and modeled taking in account that it occurs during a running time where specific events can cause a change in the status of that interaction. Thus, the full relation can be described in what is called a state lifetime where several timelines show the evolution of the interacting

In Figure 7 we can see a hierarchical superposition of levels and the action derived from the interaction between a user and a web system. Sub-systems invoked by this action and the time they spent to provide a proper action are explicitly depicted. As in the previous problem

OMG (www.omg.org) shows a very appealing example of time diagram to model

time of at most 2 min for some drivers.

diagrams and timelines.

components.

follows the specifications in ISO/IEC 15909 standard.

become a general schema for systems modeling.

**4. PN as a general system representation framework**

of time duration in UML that could later be transformed in a timed net.

of the gas station, the total time interval spent to serve one or nine users is the summation of the not superposed time intervals required for each dependent action.

Thus, the complete process would be to elicit the requirements using UML diagrams including time diagrams - synthesize a Timed Petri Net from this model, and them perform the requirement analysis and final synthesis of a model for the problem. In fact, the final results for the example of the gas station were obtained following this approach.

Formally, the timelines are drawn according the behavior of *state variables*, defined in the following.

Definition 15.[State Variables] A state variable is a triple (*V*, *T*, *D*) where:

*V* = *vi* is a finite set of state values;

*T* : *V* → *V* specify each atomic transition or change in value.

*D* : *V* → **N** × **N** is the time duration for each state value.

A timeline is the tracking of all changes in state value in a interval [0, *τ*) where *τ* is the observed time horizon. A timeline is said to be completely closed if the union of its not superposed values is exactly *τ*. In that case the transitions occur in deterministic time.

If the transitions occur in a time interval [*tmin*, *tmax*] the timeline is said to be flexible. In this case we can represent the transition in a Timed Petri Net by an interval, as proposed in the first section. If we want to deal with deterministic time transition it is enough to make *tmin* = *tmax* and the same net framework could be used.

Timeline models can be very useful in some critical problem applications such as intelligent planning and scheduling. Some of those applications could be used in spatial projects [14]5. In other applications Petri Nets were used to perform requirements analysis including deterministic time, as in the one proposed by Vaquero et al.[40][41]. In that case the idea of solving real life planning problems starts with the elicitation and specification of requirements using UML, goes through the analysis of this requirements using Timed Petri Nets, synthesizes a model also in Petri Nets and finally uses a specific language, PDDL, to transfer the model to software planners which will provide the final result. Also, a modeling design environment were developed to perform this process[41][40].

**Figure 9.** Time diagram in the itSIMPLE system.

In conclusion it is important to remark that the evolution of Petri Nets towards a formal representation, capable to treat complex systems should be based in two basis: the extension to model timed systems; and the development of a unified net that includes all extensions besides the timed approach - hierarchy, gates, not controled elements, always respecting the recent published ISO/IEC standard and its next release to appear in 2013. This is the fundamental concepts to have new environments that could support a complementary treatment of timed systems, that is, that could deal with deterministic timed net as well as

Timed Petri Nets 375

Besides, it would be advisable that the same environment could deal, also in a complementary way, with classic P/T nets as well as with high level (HLPN) nets or even with simetric nets, which is also part of the ISO/IEC standard. The novelty would be to use a unified net system as a platform to reach the further challenge which would be the introduction of abstract nets. In what concerns the unified net to treat time intervals and dense time, we achieve a good point with the system GHENeSys where the present work focus most in the first part. However, in [17] a more detailed description of the state class algorithm is given and the basic concepts that lead to a modeling and simulation approach to dense time nets. Therefore the unification with timed PN is a promising result in the near future. Also the system GHENeSys

with time PN in the same environment. That is the focus of the present work.

**5. Conclusions**

**Figure 8.** Language structure in itSIMPLE 3.1

A specific state lifetime were developed to model and analyze the timelines for the agents and objects that would compose the plan, as shown in Figure 8.

Based on this time diagram Petri Nets could be synthesized to make the proper validation of the model. It is important to notice that there are a large number of approaches and tools that claim to perform a good analysis of models directly associated to a planer software with good results. However, most of this systems address only model problems which are well behaved and/or have a limited size and complexity. When the challenge is to model a large system, such as the space project mentioned before or a port to get and deliver petroleum, the challenge could be too big to be faced by these proposals.

Therefore the combination UML/Timed Petri Nets could be successful in the modeling of large and complex problems also in the planning area, with the possibility to be applied in practice to real systems.

<sup>5</sup> See also the Mexar 2 Project and the use of intelligent software application in the link mexar.istc.cnr.it/mexar2.

**Figure 9.** Time diagram in the itSIMPLE system.
