**6. Simulation and validation of the model**

Now, the relevance of the Petri net model represented in Figure 6 and described in the previous sections is demonstrated through several simulations made for different interesting configurations of the system. They are defined according to the functions to be activated (or not) in the Petri net model of the PBS system.


## **6.1. Simulation of the PBS model under the configuration (a)**

Let us consider the configuration (a) with the parameters given in the Table 6. The first part of the Table 6 gives the initial marking of the Petri net where we consider that initially there are 15 bicycles available in each station *Si*. In the second part of the table, we read the parameters *Ri* and *Ci* of the control function of the system. It is assumed that the reorder point *Ri* (resp. the capacity *Ci*) of each station is the same and fixed to 10 (resp. 20) bicycles. Finally, the table gives the parameters of the transitions of the Petri net model. For each transition, are indicated its nature which can be an immediate transition (I); a deterministic transition (D); or an exponential transition (E); its rule policy namely continue process (C) or restart process (R); and also its firing delay given in minutes (constant firing delay for deterministic transitions; zero firing delay for immediate transitions, and mean firing delay for stochastic transitions).


**Table 6.** Configuration data of the Petri net model

484 Petri Nets – Manufacturing and Computer Science

capacity of the place *P*). The performance indices are particularly interesting to measure the rates of two uncomfortable situations that can occur in the system. That is the case

Now, the relevance of the Petri net model represented in Figure 6 and described in the previous sections is demonstrated through several simulations made for different interesting configurations of the system. They are defined according to the functions to be

1. *Configuration (a).*— *The dynamic model with the regulation system:* In this case, we consider the general functioning of the system. That is, stations are available for users and the system is under control to rebalance bicycles between stations that are emptying out and those that are filling up. The Petri net model represented in Figure 6 reproduces this case when all of its subnets are not disabled as it is the case in the following

2. *Configuration (b).*— *The dynamic model without the regulation system*: Unlike the first case, here, we consider that the regulation system is unavailable. So, the stations remain operational for users but without any control of the bicycle flows. According to the Petri net model, this configuration is obtained when the initial marking of all the places *PRi* is equal to zero. This configuration deactivates the redistribution vehicle path and the

3. *Configuration (c).*— *The static model:* This case represents the behavior of the system when the frequentation of the stations is very low (functioning of the system during night, for example). In terms of the Petri net model, this situation can be simulated by increasing considerably the transition firing delays of the transitions *TSij* which represent the displacements of bicycles between different stations. Similarly to the case B, the stations remain operational for users but without any control because (for the

Let us consider the configuration (a) with the parameters given in the Table 6. The first part of the Table 6 gives the initial marking of the Petri net where we consider that initially there are 15 bicycles available in each station *Si*. In the second part of the table, we read the parameters *Ri* and *Ci* of the control function of the system. It is assumed that the reorder point *Ri* (resp. the capacity *Ci*) of each station is the same and fixed to 10 (resp. 20) bicycles. Finally, the table gives the parameters of the transitions of the Petri net model. For each transition, are indicated its nature which can be an immediate transition (I); a deterministic transition (D); or an exponential transition (E); its rule policy namely continue process (C) or restart process (R); and also its firing delay given in minutes (constant firing delay for deterministic transitions; zero firing delay for immediate transitions, and mean firing delay

of empty and full stations which have necessary to be avoided.

**6. Simulation and validation of the model** 

activated (or not) in the Petri net model of the PBS system.

configurations (case *b* and case *c*).

whole control function of each station.

case *c*) of the limited bicycle flows.

for stochastic transitions).

**6.1. Simulation of the PBS model under the configuration (a)** 

Considering all the parameters of this configuration, evolution graphs and some performances of the system can be established thanks to our simulation tool. We focus our attention on the behavior of the time evolution of the number of available bicycles in the stations which can be observed in Figure 11. According to the control function integrated in the model, it can be observed that the number of bicycles (marking the places *PSi*) in the three stations "oscillates" around the reorder point *Ri* (=10) and the capacity *Ci* (= 20) of each station is respected. Obviously, the dynamic behavior of this part of the system is similar to the one of inventory control systems [17].

As shown in Figure 12, the dynamic behavior of the redistribution vehicle can also be observed through the time evolution of the marking of the places *PRi* together. For the chosen parameters of the system, we see that there are always enough bicycles in the redistribution vehicle.

## **6.2. Simulation of the PBS model under the configuration (b)**

As defined previously, in this configuration, we consider the dynamic model without the regulation system. Here, the initial markings of all the places *M(PRi)* are equal to zero. That is the only modification to be made in the Table 6 to obtain this situation. Thanks to this change, the redistribution vehicle path and all of the control function of each station are not available in the Petri net model (see Figure 6). Formally, for *M(PRi) = 0* and *M(PCi) = 0*, the transitions *TEi, TSi, TOi, TRi* (control function) and *TRij* (displacement of the redistribution vehicle) are never enabled for this configuration. Consequently, the stations remain operationnal for users but whitout any control.

Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 487

Thanks to our simulator, the evolution time of the number of bicycles in the different stations is given in Figure 13. With regard to the first configuration (a), here, we see clearly the absence of the control functions of the system. It is observed that the stations are very frequently full. In contrast, empty stations are not observed in this configuration. This is only due to the parameters chosen for this case. Indeed, recall that we have 15 bicycles in each station (there are a total of 45 bicycles in the system) and the capacity of each station is fixed to 20. Thus, the minimal number of bicycles which can be found in a given station is 5 bicycles (Indeed, 45 - 20\*2 (two full stations) = 5) which can be observed in the graphs.

Finally, the Table 7 gives some performances of the system for the two configurations obtained by computing the equations (5-9). The average number of available bicycles in each station is represented by the average marking of the places *PSi*. For the configuration A, *Mavr(PSi)* is approximately equal to 10 which is coherent with the reorder point *Ri* fixed to 10 for this configuration. In contrast, for the configuration B where the system is without the regulation function, the average number of bicycles in the stations is over 15. The rates of empty and full stations are also given in the table. The rate of empty and full stations (together), *RF/F(PSi)=RFull(PSi)+REmpty(PSi)*, is estimated to 6.80% in the case of the

> Perf. evaluation of the configuration (a) Mavr(PS1) Mavr(PS2) Mavr(PS3) 10,18 10,14 10,21 REmpty(PS1) REmpty(PS2) REmpty(PS3) 3.33% 3.30% 3.31% RFull(PS1) RFull(PS1) RFull(PS1) 3.52% 3.50% 3.51% RE/F(PS1) RF/E(PS1) RF/E(PS1) 6.85% 6.80% 6.82% Perf. evaluation of the configuration (b) Mavr(PS1) Mavr(PS2) Mavr(PS3) 15,72 15,57 15,53 REmpty(PS1) REmpty(PS2) REmpty(PS3) 0% 0% 0% RFull(PS1) RFull(PS1) RFull(PS1) 51.0% 51.1% 50.8% RF/E(PS1) RF/E(PS1) RF/E(PS1) 51.0% 51.1% 50.8%

configuration (a), and 51% in the case of the configuration (b) for each station.

**Table 7.** Some performances indices of the model (for the considered parameters)

**Figure 11.** Evolution of the number of available bicycles in the stations (configuration a)

**Figure 12.** Evolution of the number of bicycles in the redistribution vehicle (configuration a).

Thanks to our simulator, the evolution time of the number of bicycles in the different stations is given in Figure 13. With regard to the first configuration (a), here, we see clearly the absence of the control functions of the system. It is observed that the stations are very frequently full. In contrast, empty stations are not observed in this configuration. This is only due to the parameters chosen for this case. Indeed, recall that we have 15 bicycles in each station (there are a total of 45 bicycles in the system) and the capacity of each station is fixed to 20. Thus, the minimal number of bicycles which can be found in a given station is 5 bicycles (Indeed, 45 - 20\*2 (two full stations) = 5) which can be observed in the graphs.

486 Petri Nets – Manufacturing and Computer Science

**Figure 11.** Evolution of the number of available bicycles in the stations (configuration a)

**Figure 12.** Evolution of the number of bicycles in the redistribution vehicle (configuration a).

Finally, the Table 7 gives some performances of the system for the two configurations obtained by computing the equations (5-9). The average number of available bicycles in each station is represented by the average marking of the places *PSi*. For the configuration A, *Mavr(PSi)* is approximately equal to 10 which is coherent with the reorder point *Ri* fixed to 10 for this configuration. In contrast, for the configuration B where the system is without the regulation function, the average number of bicycles in the stations is over 15. The rates of empty and full stations are also given in the table. The rate of empty and full stations (together), *RF/F(PSi)=RFull(PSi)+REmpty(PSi)*, is estimated to 6.80% in the case of the configuration (a), and 51% in the case of the configuration (b) for each station.


**Table 7.** Some performances indices of the model (for the considered parameters)

Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 489

As noted previously, to obtain the configuration c, the Petri net model shown in Figure 6 must be parameterized as in the configuration b but by increasing considerably the transition firing delays of the transitions *TSij*. Contrary to the behavior of the system shown in Figure 13, in this situation, we obtain a static model with very low (negligible) evolution

In this chapter, we have presented an original Petri net approach for public bicycle sharing systems modelling and performance evaluation for control purposes. A modular dynamic model based on Petri nets with marking dependent weights is proposed and a simulation approach is developed and used to simulate and validate models described in this chapter. Very likely, this approach is the first one in the literature dedicated to this urban transportation mode by using Petri nets. The authors believe that this new area of research has significant promise for the future to help planners and decision makers in determining how to implement, and operate successfully these complex dynamical systems. Now, we are working in the following directions: (1) Development of some natural extensions of the models including more complex modelling features such as the application of other control strategies used for the latest systems which operate with smart technologies and provide users and controllers with real-time bike availability information; (2) Development of optimization methods for optimal control purposes. For example, the objective is how to search optimal values of the decision parameters *Ri, Ci* of the model in order to minimize the two uncomfortable situations *M(PSi) = 0* (empty station) and *M(PSi) = Ci* (full station) that

of the distribution of the bicycles in the network.

may occur in the different stations of the system.

*science*, Vol. 35, No. 12, pp. 693-706.

September 5-9 2011, Toulouse, France.

*Advances in Complex Systems*, Vol. 14, No. 3, pp. 415-438.

Karim Labadi, Taha Benarbia, Samir Hamaci and A-Moumen Darcherif *EPMI, Ecole d'Electricité, de Production et Management Industriel, France* 

[1] Abbas-Turki A.; Bouyekhf R.; Grunder O.; El Moudni A.; (2004) "On the line planning problems of the hub public-transportation networks", *International journal of systems* 

[2] Benarbia, T., Labadi, K., and Darcherif, M., (2011) "A Petri Net Approach for Modelling, Performance Evaluation and Control of Self-Service Public Bicycle Systems", *16th International IEEE conference on Emerging Technologies on Factory Automation,* ETFA'2011,

[3] Borgnat, P., Abry, P., Flandrin, P., Robardet, C., Rouquier, J-B., Fleury, E., (2011), "Shared Bicycles in a City: a Signal Processing and Data Analysis Perspective",

**7. Conclusion** 

**Author details** 

**8. References** 

**Figure 13.** Evolution of the number of available bicycles in the station 3 (configuration b)

As noted previously, to obtain the configuration c, the Petri net model shown in Figure 6 must be parameterized as in the configuration b but by increasing considerably the transition firing delays of the transitions *TSij*. Contrary to the behavior of the system shown in Figure 13, in this situation, we obtain a static model with very low (negligible) evolution of the distribution of the bicycles in the network.

### **7. Conclusion**

488 Petri Nets – Manufacturing and Computer Science

**Figure 13.** Evolution of the number of available bicycles in the station 3 (configuration b)

In this chapter, we have presented an original Petri net approach for public bicycle sharing systems modelling and performance evaluation for control purposes. A modular dynamic model based on Petri nets with marking dependent weights is proposed and a simulation approach is developed and used to simulate and validate models described in this chapter. Very likely, this approach is the first one in the literature dedicated to this urban transportation mode by using Petri nets. The authors believe that this new area of research has significant promise for the future to help planners and decision makers in determining how to implement, and operate successfully these complex dynamical systems. Now, we are working in the following directions: (1) Development of some natural extensions of the models including more complex modelling features such as the application of other control strategies used for the latest systems which operate with smart technologies and provide users and controllers with real-time bike availability information; (2) Development of optimization methods for optimal control purposes. For example, the objective is how to search optimal values of the decision parameters *Ri, Ci* of the model in order to minimize the two uncomfortable situations *M(PSi) = 0* (empty station) and *M(PSi) = Ci* (full station) that may occur in the different stations of the system.
