**4. Modelling of a PBS system**

As mentioned in the second section of this chapter, modelling, analysis and performance evaluation of PBS systems is crucial not only for their successful implementation and performance improvement but also for ensuring an effective regulation of bicycle traffic flows. This section deals with our original Petri net approach dedicated for PBS system modelling and analysis for control purposes. We consider a PBS system with *N* stations noted by *S = {S1, S2, …, SN}*. Each station *Si S* is equipped with *Ci* bicycle stands (the capacity of a station *Si*). In practice, the number *Ci* depends on the location of the service point and the estimated level of use. The system requires a constant control which consists in transporting bicycles from stations having excess of bicycles to stations that may run out of bicycles soon. In the general way, the main objective of the control system, performed by using redistribution vehicles as shown in Figure 4, is to maintain *Ri* (reorder point) bicycles per station *Si* to ensure bicycles are available for pick up and thus (*Ci – Ri*) vacant berths available for bicycle drop off at every station.

#### **4.1. The Petri net model**

Firstly, for the sake of clarity, we consider a PBS system with only three stations and then we design a Petri model of the system as shown in Figure 6. The designation of the elements and parameters of the Petri net model are given in Tables 3, 4 and 5. Thereafter, as will be shown in this section, according to the modular structure of the resulting Petri net model, its generalization to represent PBS systems with *N* stations will be straightforward and intuitive.

Before a formal description and analysis of the model in the next sub-sections, the Figure 6 and the Tables 3, 4, and 5 allow readers to quickly gain an understanding of the Petri net model of the system. A closer look at the Petri net model shows three subnets (modules) representing three different functions named as follows: (*1*) the "station control" subnet; (*2*) the "bicycle flows" subnet; and (*3*) the "redistribution circuit" subnet. The main function of each subnet is described in the following:



**Table 3.** Interpretation of places of the PN model

474 Petri Nets – Manufacturing and Computer Science

since the following condition is satisfied:

marking *Mf* = (*2, 10, 8*) as the following:

**4. Modelling of a PBS system** 

available for bicycle drop off at every station.

**4.1. The Petri net model** 

The model represents an inventory control system with continue review (*s, S*) policy (here *s* = *4* and *S* = *10*) [17]. The different operations of the system are modelled by using a set of transitions: generation of replenishment orders (*t3*); inventory replenishment (*t2*); and order delivery (*t1*). In the model, the weights of the arcs (*t3, p2*), (*t3, p3*) are variable and depend on

According to the current marking *Mi* = (*2, 2, 0*) of the net, the transition *t3* is enabled,

<sup>2</sup> 2 3 *M p Inhib p t* ( ) 2 ( ,) 4 After the firing of *t3*, 10 - *M(p2) =* 10–2 = 8 tokens are added into the places *p2* and *p3*. In other words, the firing of the transition *t3* from the initial marking *Mi* leads to a new

22 2 ( ) ( ) 10 ( ) 2 10 2 10. *M p Mp Mp fi i*

33 3 ( ) ( ) 10 ( ) 0 10 2 8. *M p Mp Mp fi i*

As mentioned in the second section of this chapter, modelling, analysis and performance evaluation of PBS systems is crucial not only for their successful implementation and performance improvement but also for ensuring an effective regulation of bicycle traffic flows. This section deals with our original Petri net approach dedicated for PBS system modelling and analysis for control purposes. We consider a PBS system with *N* stations noted by *S = {S1, S2, …, SN}*. Each station *Si S* is equipped with *Ci* bicycle stands (the capacity of a station *Si*). In practice, the number *Ci* depends on the location of the service point and the estimated level of use. The system requires a constant control which consists in transporting bicycles from stations having excess of bicycles to stations that may run out of bicycles soon. In the general way, the main objective of the control system, performed by using redistribution vehicles as shown in Figure 4, is to maintain *Ri* (reorder point) bicycles per station *Si* to ensure bicycles are available for pick up and thus (*Ci – Ri*) vacant berths

Firstly, for the sake of clarity, we consider a PBS system with only three stations and then we design a Petri model of the system as shown in Figure 6. The designation of the elements and parameters of the Petri net model are given in Tables 3, 4 and 5. Thereafter, as will be shown in this section, according to the modular structure of the resulting Petri net model, its generalization to represent PBS systems with *N* stations will be straightforward and intuitive. Before a formal description and analysis of the model in the next sub-sections, the Figure 6 and the Tables 3, 4, and 5 allow readers to quickly gain an understanding of the Petri net

the parameters s and *S* of the system and on the marking of the model (*S - M(p2); s*).


**Table 4.** Interpretation of transitions of the PN model


Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 477

**Figure 6.** The Petri net model of a self-service public bicycles system (with three stations)

**Table 5.** Decision parameters of the PN model

Thanks to the modularity of the developed model, its generalization for a system with *N*  stations is simple to make according to the different functions cited previously. For example, to model *N* stations *Si (*i = 1, 2, …, *N*), we need to *N* places denoted by *PSi* and the control subnet is duplicated for each station similarly to the model represented for three stations (see Figure 6). Finally, by considering all the modules, the Petri net model representing a PBS system with *N* stations should contain:

<sup>2</sup> *TN N* 5 \* transitions; *P N* <sup>4</sup> places; <sup>2</sup> 2 \* 21 \* *Ad N N* directed arcs, and <sup>2</sup> 7 \* *Ai N N* inhibitors arcs.
