Simulation Modelling Petri Net

With the aid of FTA, the reliability of the system of interest can be obtained analytically. However, when the system is large and complex, or the mission to be performed is made up of many phases, FTA would become inaccurate and computationally expensive. To overcome this issue, some alternative reliability evaluation methods are developed by the approach of simulation modelling, one of which is Petri Net (PN) developed by Petri [18]. Simulation modelling is a computer- based method, which uses mathematical algorithms and equations to solve real-life problems efficiently and cost-effectively. In comparison of analytical analysis, simulation modelling method allows us to deal with more complex issues and moreover is easier to be verified and understood.

Similar to the FTA, PNs provide an intuitive graphical representation of the reliability problem of interest. But by contrast, the PN method is more suited to dealing with the reliability issues in complex systems that involve more components, functions, and more complex system configurations than in a simple system. PNs have shown many advantages in performing system simulation and modelling [81]. For example, it removes redundant information in the model of the system so that the problem can be simplified. Also, the PN model can be easily adapted to modelling different problems by simply modifying the network settings.

The PN method is, in essence, a direct bipartite graph. As shown in Figure 2.4, it basically consists of the following four types of symbols:

⚫ Circles – represent the places, which are conditions or states such as mission failure, phase failure, or component failure depending on the issue being considered. ⚫ Rectangles – represent the transitions, more abstractly actions or events. It is

worthy to note that in the case of timed transitions, a solid rectangular bar can be used when the time spent for completing the transition is zero. Otherwise, the rectangular bar is hollow.

⚫ Arcs – represent connections between places and transitions. It should be noted that arcs with a slash on and a number, n, next to the slash represent a combination of

n single arcs and each arc has a weight n. The weight will be 1 when there is no

slash.

⚫ Small marks – represent tokens that carry the information in the PN.

To ease understanding, an example of the movement of tokens through a net is illustrated in Figure 2.4.

Figure 2.4 Enabling and switching of transition, (a) before enabling transition, (b) after enabling transition

From Figure 2.4(a), it is seen that there are two inputs and one output place connected to a timed transition with a time delay t. The input places have arcs with weights 2 and 3, respectively. The transition is enabled when the number of tokens contained in every input place is not less than the corresponding arc weights. Once the

transition is enabled, the number of tokens corresponding to the arc weight will be taken out from the corresponding input place to fulfil the transition after the time delay t associated with the transition. If the transition time t is greater than zero, the PN is known as a timed Petri Net. For example, as shown in Figure 2.4, two and three tokens are respectively taken out of the upper and lower input places, and one more token will be present in the output place. But it is necessary to note that after completing the transition, the number of tokens that are increased in the output place is also dependent on the corresponding arc weight. For example, if the arc weight connected to the output place is ‘n’, then n more tokens will appear in the output place after enabling the transition. After the transition, there will be zero, one, and two tokens in the corresponding places, as shown in Figure 2.4(b). This forms a new distribution of tokens in the places of the PN. This is called marking, which represents a PN configuration with the distribution of tokens in the places.

The movement of tokens through a PN can be transformed into matrix form as shown in Equation (2.16).

𝑀_𝑟 = 𝑀₀ + 𝐵𝑇𝐸 (2.16)

where 𝑀𝑟 is the final marking after 𝑟 transitions. 𝐸 represents a finite set of

transitions, which forms a column matrix, (𝑚, 1). Here, 𝑚 indicates the number of transitions in the net, representing how many times each transition has fired after r transitions. 𝑀₀ refers to the initial marking of the net. It is a 𝑛 × 1 column matrix, where 𝑛 is the number of places. 𝐵 is known as the incidence matrix, which is a 𝑚 × 𝑛 matrix. Each element in matrix 𝐵 , 𝑏_𝑖𝑗, corresponds to the effect that the transition 𝑖 has on the place 𝑗. It should be noted that 𝐵𝑇 is the transpose of matrix 𝐵. In order to explain this method more clearly, the matrix expression for the example shown in Figure 2.4 is presented. Since there are three places in total and there are 2, 4 and 1 tokens respectively in them, the initial marking of the net is expressed as:

𝑀0 = [

2 4 1

] (2.17)

According to the numbers of the places and the transitions, as well as the connections and the weight of each arc in the figure, the matrix, 𝐵, is denoted as:

𝐵 = [−2 −3 1] (2.18)

In matrix B, the values -2 and -3 represent the input places respectively lose two and three tokens after the transition according to the weights of the arcs connecting to them. The value 1 means the output place will obtain one more token after the transition.

Since there is only one transition only firing once, then has

𝐸 = [1] (2.19)

After the associated time delay t, the resultant marking, 𝑀₁, can be calculated by using Equation (2.20), i.e.

𝑀₁ = 𝑀₀+ 𝐵𝑇𝐸 = 𝑀_𝑟 = [ 2 4 1 ] + [ −2 −3 1 ] = [ 0 1 2 ] (2.20)

where, 𝑀1 is the matrix expression of the resultant PN after the transition in Figure

2.4(b).

Despite the outstanding merit of flexibility, it is found that conventional PN has difficulty in describing complex systems or a system that is designed to carry out complex tasks and missions [82]. To further improve the ability and capability of PN, many extension forms of PN have been developed. For example, as shown in Figure 2.5, a type of arc that is terminated with a circle was developed, which is called inhibit arc. This kind of arc prevents the firing of transitions when the input place is marked, thereby enhancing the decision power of PN. From Figure 2.5, it is noticed that the top

input place is connected with the transition by an inhibitor arc. Since there is a token in the place, this transition cannot fire.

Figure 2.5 PN with an inhibitor arc

Apart from the modifications of arcs, an enhanced PN, namely Coloured Petri Net (CPN), was introduced in [83]. Different from the conventional PN, the individual tokens in the CPN model are characterised by different colours, which represent different identities or different information. For example, these coloured tokens could represent components with different functions or workers undertaking different jobs. Therefore, the CPN is also known as a high-level PNs which is more informative than the conventional PNs. In addition, the colours of the tokens in the CPN model are also associated with transitions, so that the transitions can be activated if and only if the tokens with the same colour enable the transition. To facilitate understanding, two CPN transition examples are illustrated in Figure 2.6. In order to ensure that black and white figures can also clearly express the information, different filling patterns are also adopted to characterise the tokens that are characterised using different colours. Moreover, a key has been included in the figure, where token 1 represents green, token 2 red and token 3 blue. As the transition in Figure 2.6(a) is green, only the green token (token 1) can enable the transition. It should be noted that the coloured tokens can also activate non-coloured transitions and the tokens in the output places will still carry the colour information, as shown in Figure 2.6(b). However, as observed, only tokens with the same colour are able to enable the transition. No further firing of the transition will occur in Figure 2.6(b) as there are not enough tokens of the same colour to activate the transition.

(a) CPN with a coloured transition

(b) CPN with a normal transition Figure 2.6 Diagrams of the CPN

To date, the PN has become a popular tool used for evaluating the reliability of a system or a mission. For example, an extended object-oriented PN model was proposed by Wu in 2015 to analyse the reliability of a phased mission with common cause failures [84]. Considering industrial applications, a PN-based wind turbine asset model was developed by Le and Andrews to study the degradation, maintenance and inspection processes of different wind turbine components [85]. In the area of AGVs and AGV systems, many PN-based models have been developed for eliminating deadlock in the AGV systems through performing system design and analysis. For example, a coloured resource-oriented Petri Net (CROPN) method was developed by Wu and Zhou by addressing the resources in automated manufacturing systems, and then the CROPN was used to find the shortest routes for the AGVs while avoiding both deadlock and blocking in the AGV systems [13]. One of the CROPNs developed by them is shown in Figure 2.7.

Figure 2.7 An example of the CROPN in [13]

In the figure above, the places represent the resource zones, target destinations, or joint junctions in the AGV system. The arcs represent the guidepath connecting the zones. In addition, the tokens represent the four AGVs in places P1, P2, P3, and P8, respectively. Also, the PNs were used by Luo et al. to design a programmable logical controller (PLC) for preventing the collisions of vehicles in an AGV system [86]. In their study, two different ordinary PNs, namely a control-hardware PN and a closed- loop PN, were constructed. The former was used to model the control elements including sensors, up-down counters, coils, and wiring loops. The latter, designed based on the former, was used to describe the control specification of collision-prevention in an AGV system. In addition, the PN based method was developed by Nishi and Maeno to optimise the routing planning for the AGVs in semiconductor fabrication bays [87]. Despite so many applications, so far, all existing PN based models were developed mainly for investigating route planning and control strategies of the AGV systems. Its application in studying the reliability of AGVs and AGV systems has not been reported in open literature.