**1. Introduction**

Public Bicycle-Sharing Systems (PBSS), also known as self-service public bicycle systems, are available in numerous big cities in the world (Vélib' in Paris, Bicing in Barcelona, Call-a-Bicycle in Munich, OyBicycle in London, etc.). Since its inception, Bicycle-sharing programs have grown worldwide. There are now programs in Europe, North America, South America, Asia, and Australia. A still growing list of cities which provides such green public transportation mode can be found at the Bicycle-sharing world map (http://Bikesharing.blogspot.com) as shown in Figure 1. As a good complementary to other urban transportation modes, bicycle use entails a number of benefits including environmental, mobility and economic benefits. The public bicycle sharing systems are especially useful for short-distance city transport trips and to face many public transport problems, including growing traffic congestion, pollution, greater car dependency, buses caught in city congestion, and ageing transport infrastructure.

A PBS system can be described as a bank of bicycles that can be picked up and dropped off at numerous stations (service points) across an urban area. The bicycle stations are usually located 300 meters apart, consisting of terminals and stands for fastening the bicycles. Every station is equipped with roughly twenty bicycle stands (the number can be estimated depending on the location of the service point and the estimated level of use). A customer uses a bicycle to travel from one station to another. A bicycle can be taken out from any station and returned to the same or any other station, provided that there is an available locking berth. A PBS system requires more than just bicycles and stations; a variety of other equipment is needed to keep the bicycles and stations functioning at adequate level of service. Particularly, this includes a fleet of vehicles for redistribution of bicycles between stations in order to balance the network (see Figures 2 to 4).

© 2012 Labadi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Labadi et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 467

**Figure 4.** Redistribution vehicle (http://www.velib.paris.fr/)

other users from collecting or delivering bicycles.

in two different modes [30]:

Amsterdam.

Over the recent years, public bicycle-sharing schemes have developed from being interesting experiments in urban mobility to mainstream public transport options in cities as large and complex as Paris and London (http://www.velib.paris.fr/). PBS schemes have evolved dramatically since their introduction in the 1960s and undergone various changes. These changes can be categorized into three key phases, known as Bicycle sharing generations [9]-[8]. These include the rst generation, called white Bicycles (or free Bicycles); the second generation, coin-deposit systems; and the third generation, or information technology based systems. Potential "fourth generation" design innovations are already under development including electric bicycles, movable docking stations, solar-powered docking stations, and mobile phone and iPhone real time availability applications. Of these innovations, the introduction of electric bicycles is likely to be the most significant in terms of attractiveness. The Table 1 gives a survey of some significant public bicycle-sharing programs over the world since the first generation schemes that were introduced in

A crucial question for the success of a PBS system is its ability to meet the fluctuating demand for bicycles at each station and to provide enough vacant lockers to allow the renters to return the bicycles at their destinations. Indeed, some stations have more demand than others, especially during peak hours. In addition, not surprisingly stations located at the top of hills are chronically empty of bicycles, as the customers ride down the hill but do not wish to make the return trip uphill. Bicycles also tend to collect in stations in the city centres and stay there. In some cases, the *imbalance is temporary*, e.g., high return rate in a suburban train station in the morning and high renting rate in the afternoon. In other cases, the *imbalance is persistent*, e.g., relatively low return rate in stations located on top of hills [30]. If no action is taken by the service provider they rapidly fill or empty, thus preventing

Thus, the system requires constant monitoring to balance the network. The monitoring system dispatches motorized redistribution vehicles (trucks) to rebalance bicycles between stations that are emptying out and those that are filling up. This operation can be carried out

 *Static mode* — The bicycle redistribution operation can be carried out during the night when the usage rate of the PBS system is very low. The bicycle repositioning is

**Figure 1.** Bike-Sharing world map (source http://Bike-sharing.blogspot.com).

**Figure 2.** Full station, (http://www.velib.paris.fr/)

**Figure 3.** Empty station, (http://www.velib.paris.fr/)

**Figure 4.** Redistribution vehicle (http://www.velib.paris.fr/)

466 Petri Nets – Manufacturing and Computer Science

**Figure 1.** Bike-Sharing world map (source http://Bike-sharing.blogspot.com).

**Figure 2.** Full station, (http://www.velib.paris.fr/)

**Figure 3.** Empty station, (http://www.velib.paris.fr/)

Over the recent years, public bicycle-sharing schemes have developed from being interesting experiments in urban mobility to mainstream public transport options in cities as large and complex as Paris and London (http://www.velib.paris.fr/). PBS schemes have evolved dramatically since their introduction in the 1960s and undergone various changes. These changes can be categorized into three key phases, known as Bicycle sharing generations [9]-[8]. These include the rst generation, called white Bicycles (or free Bicycles); the second generation, coin-deposit systems; and the third generation, or information technology based systems. Potential "fourth generation" design innovations are already under development including electric bicycles, movable docking stations, solar-powered docking stations, and mobile phone and iPhone real time availability applications. Of these innovations, the introduction of electric bicycles is likely to be the most significant in terms of attractiveness. The Table 1 gives a survey of some significant public bicycle-sharing programs over the world since the first generation schemes that were introduced in Amsterdam.

A crucial question for the success of a PBS system is its ability to meet the fluctuating demand for bicycles at each station and to provide enough vacant lockers to allow the renters to return the bicycles at their destinations. Indeed, some stations have more demand than others, especially during peak hours. In addition, not surprisingly stations located at the top of hills are chronically empty of bicycles, as the customers ride down the hill but do not wish to make the return trip uphill. Bicycles also tend to collect in stations in the city centres and stay there. In some cases, the *imbalance is temporary*, e.g., high return rate in a suburban train station in the morning and high renting rate in the afternoon. In other cases, the *imbalance is persistent*, e.g., relatively low return rate in stations located on top of hills [30]. If no action is taken by the service provider they rapidly fill or empty, thus preventing other users from collecting or delivering bicycles.

Thus, the system requires constant monitoring to balance the network. The monitoring system dispatches motorized redistribution vehicles (trucks) to rebalance bicycles between stations that are emptying out and those that are filling up. This operation can be carried out in two different modes [30]:

 *Static mode* — The bicycle redistribution operation can be carried out during the night when the usage rate of the PBS system is very low. The bicycle repositioning is performed based on the status of the system at that time and the demand forecast for the next day.

Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 469

comprising decisions about the location, number and size of stations, *(2) Tactical (mid-term)* incentives for customer based distribution of bicycles. For example, the Vélib in Paris grants some extra minutes for returning bicycles at uphill stations, *(3) Operational (short-term)* provider based repositioning of bicycles. The most important issues for the success of a PBS

*How many bicycles and stations?* 

*How often should the bicycles be* 

 *Is the current number of redistribution vehicles and projections sufficient? Should more bicycles be purchased?* 

*Regular or preventative maintenance* 

To help planners and decision makers answer these crucial questions, modelling and performance analysis and optimization methods and tools for PBS systems are needed. A literature review describing some existing works developed in this research area is dressed in the section 2. After, the rest of this chapter deals with an original Petri net approach dedicated for PBS systems modelling for control purposes [2]-[19]-[20]-[21]. The section 3 provides an introduction for Petri nets with the "marking dependent weights" concept [17] as it is used throughout this chapter to model PBS systems for control purposes. In the sections 4 to 6, a modular framework based on Petri nets with marking dependent weights is developed for modelling and performance evaluation of PBS systems. The concluding section 7 gives some remarks and perspectives of this work. Our approach is intended to help planners and decision makers in determining how to implement, and operate

Public Bicycle-sharing systems have attracted a great deal of attention in recent years. Although the growth of the system has been rapid following the development of better

 *Where to locate stations? What is the size of each station? How should the bicycles be distributed? How many vehicles (and what sizes) are needed for bicycle redistribution?* 

**Stage Questions**

*redistributed?* 

*required?* 

*...* 

*More stations needed?* 

*...*

system are summarized in Table 2.

Before deployment (Strategic / Tactical)

After deployment

(Operational)

**Table 2.** Management and design measures of a PBS system

successfully these complex dynamical systems.

**2. A review of the literature** 

 *Dynamic mode* — The bicycle redistribution operation can be carried out during the day when the usage rate of the PBS system is significant. The bicycle repositioning is performed based on the current state of the station as well as aggregate statistics of the station's usage patterns [16].


**Table 1.** Summary of some PBS systems over the world

Vogel et al., (2011) [38] identify three management and design measures alleviating these imbalances divided into different planning horizons: *(1) Strategic (long-term)* network design comprising decisions about the location, number and size of stations, *(2) Tactical (mid-term)* incentives for customer based distribution of bicycles. For example, the Vélib in Paris grants some extra minutes for returning bicycles at uphill stations, *(3) Operational (short-term)* provider based repositioning of bicycles. The most important issues for the success of a PBS system are summarized in Table 2.


**Table 2.** Management and design measures of a PBS system

To help planners and decision makers answer these crucial questions, modelling and performance analysis and optimization methods and tools for PBS systems are needed. A literature review describing some existing works developed in this research area is dressed in the section 2. After, the rest of this chapter deals with an original Petri net approach dedicated for PBS systems modelling for control purposes [2]-[19]-[20]-[21]. The section 3 provides an introduction for Petri nets with the "marking dependent weights" concept [17] as it is used throughout this chapter to model PBS systems for control purposes. In the sections 4 to 6, a modular framework based on Petri nets with marking dependent weights is developed for modelling and performance evaluation of PBS systems. The concluding section 7 gives some remarks and perspectives of this work. Our approach is intended to help planners and decision makers in determining how to implement, and operate successfully these complex dynamical systems.

## **2. A review of the literature**

468 Petri Nets – Manufacturing and Computer Science

station's usage patterns [16].

the next day.

**1960** Amsterdam

**1995** Copenhagen

**2005** Lyon

**2008** Hangzhou

**2008** Washington

**2009** Canada

(Netherlands)

(Denmark)

(France)

(China)

(USA)

(Quebec)

deposit.

first 30 minutes free.

America.

**Table 1.** Summary of some PBS systems over the world

performed based on the status of the system at that time and the demand forecast for

The first Bicycle sharing system in the world appeared in Amsterdam, the Netherlands on July 28, 1965. Bicycles were painted white and offered to the public who would like to use them. Due to theft and abuse, unfortunately the

In 1995 in Copenhagen, Bycyklen or City Bicycles was operated as the first large-scale second generation Bicycle sharing program. The system had improved in many aspects. The bicycles are specifically designed for intensive urban use and stations were set up with each equipped with a coin

system in the world and the first one in France, and was also operated by ClearChannel, a private company. In 2009, the operator changed to EFFIA

Lyon started its Bicycle sharing system Vélo'V with an unusually large scale in 2005. The city of Lyon and JC Decaux funds the system together through an advertisement contract, which the latter one operates the scheme. Vélo'V is based on stations and has long and short term subscription available with

Lyon, city of Paris and Cycocity fund the system together and the latter one operates it. Velib' has fixed stations and requires registration beforehand.

In May, 2008, the first Bicycle sharing program in China started its operation

SmartBicycle DC was a bicycle sharing system implemented in August 2008 with 120 bicycles and 10 automated rental locations in the central business district of Washington, D.C. The network was the first of its kind in North

Bixi is a public bicycle sharing system serving Montreal, Quebec, Canada. The system was launched on May 12 2009, with 3000 bicycles and 300 stations located around Montreal's central core, and it expanded to 5,000

in Hangzhou, a city 180 km southwest of Shanghai, with 4.2 million

 *Dynamic mode* — The bicycle redistribution operation can be carried out during the day when the usage rate of the PBS system is significant. The bicycle repositioning is performed based on the current state of the station as well as aggregate statistics of the

**Year City Bicycle-sharing program**

program only survived for several days.

**1998** Rennes (France) Rennes launched "Vélo à la Carte" in 1998, which was the first computerized

**2007** Paris (France) Velib' was introduced to Paris in 2007. With the similar operation scheme as

inhabitants in the metropolitan area, 8 districts included.

bicycles and 400 stations later that summer.

Vogel et al., (2011) [38] identify three management and design measures alleviating these imbalances divided into different planning horizons: *(1) Strategic (long-term)* network design

and shifted to a new system called "VéloStar".

Public Bicycle-sharing systems have attracted a great deal of attention in recent years. Although the growth of the system has been rapid following the development of better

#### 470 Petri Nets – Manufacturing and Computer Science

tracking technology, most of the studies related to PBS systems in the literature have focused on their history, development and some practical advises [9]-[31]-[39]. There are, however, relatively few studies addressing strategic and operational issues that arise in such systems.

Petri Nets Models for Analysis and Control of Public Bicycle-Sharing Systems 471

mathematical optimization models. According to our knowledge, unlike our work [2]-[19]- [20]-[21], no other studies has been undertaken on the dynamics modelling and performance evaluation of such dynamical systems. In addition to their self-service mode, PBS systems are dynamic, stochastic, and complex systems, this makes their modelling and analysis very complicated. Among the formalisms used to model the dynamic systems, Petri nets are one of the graphical and formal specification techniques for the description of the operational behavior of the systems. They are widely used in a number of different disciplines including engineering, manufacturing, business, chemistry, mathematics, and even within the judicial system [17]-[18]-[40]-[32]-[42]-[41]. They have been accepted as a powerful formal specification tool for a variety of systems including concurrent, distributed, asynchronous,

However, although Petri nets have been widely used in various domains, they played a relatively minor role in modelling and analysis of urban transportation systems. According to some existing works, the modelling of the systems by using Petri nets formalisms can be considered from either a discrete and/or a continuous point of view. Continuous Petri nets for the macroscopic and microscopic traffic flow control are used in [15]-[34], while hybrid Petri nets are used in [11] to provide a valuable model of urban networks of signalized intersections. Recently, batch Petri nets with controllable batch speed are used [10] to study a portion of the A12 highway in The Netherlands. From a discrete point of view, generalized stochastic Petri nets are used [1]-[4] for modelling and planning of public transportation systems. The two complementary tools, Petri nets and (max, +) algebra, have been used [28]- [29] to deal with the modelling and the performance evaluation of a public transportation system. For the modelling of passenger flows at a transport interchange, as shown in [33] colored Petri nets are able to incorporate some specific parameters and data in the model such as the variation of walking speeds between passengers and the restricted capacity of

These are a few works that demonstrate the high potential of Petri nets as a tool for modelling and performance analysis of urban transportation systems, but also on the other hand, it is shown that the application of Petri nets is still in its early stage and particularly limited to intersection traffic control [11]-[14]-[15]-[25]-[35]-[34] and to some studies dealing with the modelling of urban transportation systems for planning purposes [28]-[29]-[33]. In addition, according to our knowledge, unlike traditional urban transportation systems, no work has been undertaken on PBS systems modelling and performances analysis by using Petri net models. Our contribution in this context is the first one in the literature [18]-[19]-

In its basic form, a Petri net may be defined as a particular bipartite directed graph consisting of places, transitions, and arcs. Input arcs are ones connecting a place to a transition, whereas output arcs are ones connecting a transition to a place. A positive weight may be assigned to each arc. A place may contain tokens and the current state (the marking)

parallel, deterministic and non-deterministic.

features of the interchange infrastructure.

**3. Petri nets with variable arc weights** 

[20]-[21].

About strategic issues, Lin and Yang (2011) [22] and Lin et al., (2011) [23] address the strategic problem of finding optimal stations using mathematical programming techniques. The problem is formulated as a hub location inventory model. The key design decisions considered are: the number and locations of bicycle stations in the system, the creation of bicycle lanes between bicycle stations, the selection of paths of users between origins and destinations, and the inventory levels of sharing bicycles to be held at the bicycle stations. The design decisions are made with consideration for both total cost and service levels. Dell'Olio et al. (2011) [7] present a complete methodology for the design and implementing of bicycle sharing systems based on demand estimates considering the stations and the fares. Vogel et al. (2011) [38] develop a methodology for strategic and operational planning using data mining. A case study shows how Data Mining applied to operational data offers insight into typical usage patterns of bike-sharing systems and is used to forecast bike demand with the aim of supporting and improving strategic and operational planning.

Regarding operational issues, besides the work presented in [38], the static balancing problem studying the repositioning of bicycles among bicycle stations where the customer demand is assumed to be negligible is addressed in [5]-[30]. Several mathematical formulations of the problem can be found in [30] and an exact algorithm based on column generation and a suitable pricing algorithm based on dynamic programming are given by Chemla et al.; (2011) [5]. From an OR perspective, the bicycle repositioning problem bears great similarities to some other routing problems which have been largely studied in the literature. As an example from this point of view, Forma et al. (2010) [12] consider the bicycle repositioning problem as a variation of the Pickup and Delivery problem (PDP). Naturally, some similarities between bicycle-sharing and car-sharing systems can be explored in order to adapt some existing results in this field (see, for example [26]-[36]).

Besides OR approaches developed, particularly by using mathematical programming techniques [5]-[12]-[22]-[23]-[30] to support decision making in the design and management of PBS systems, Data Mining techniques receives attention in academia as well as in practice. Data Mining is particularly suitable to analyze and to predict the dynamics of a PBS system. By exploring and analyzing the temporal and geographic human mobility data in an urban area using the amount of available bicycles in the stations of a PBS system [3], statistical and prediction models can be developed [13]-[16] for tactical and operational management of such systems.

As noted in [37], although extensive analysis of bicycle data or customer surveys can be applied to predict future bicycle demand at stations, the demand still has to be considered stochastic and not deterministic. Moreover various points in time have to be incorporated in a suitable mathematical optimization models. Such a stochastic and dynamic model can be computational intractable. In addition, customer behavior cannot be modelled in these mathematical optimization models. According to our knowledge, unlike our work [2]-[19]- [20]-[21], no other studies has been undertaken on the dynamics modelling and performance evaluation of such dynamical systems. In addition to their self-service mode, PBS systems are dynamic, stochastic, and complex systems, this makes their modelling and analysis very complicated. Among the formalisms used to model the dynamic systems, Petri nets are one of the graphical and formal specification techniques for the description of the operational behavior of the systems. They are widely used in a number of different disciplines including engineering, manufacturing, business, chemistry, mathematics, and even within the judicial system [17]-[18]-[40]-[32]-[42]-[41]. They have been accepted as a powerful formal specification tool for a variety of systems including concurrent, distributed, asynchronous, parallel, deterministic and non-deterministic.

However, although Petri nets have been widely used in various domains, they played a relatively minor role in modelling and analysis of urban transportation systems. According to some existing works, the modelling of the systems by using Petri nets formalisms can be considered from either a discrete and/or a continuous point of view. Continuous Petri nets for the macroscopic and microscopic traffic flow control are used in [15]-[34], while hybrid Petri nets are used in [11] to provide a valuable model of urban networks of signalized intersections. Recently, batch Petri nets with controllable batch speed are used [10] to study a portion of the A12 highway in The Netherlands. From a discrete point of view, generalized stochastic Petri nets are used [1]-[4] for modelling and planning of public transportation systems. The two complementary tools, Petri nets and (max, +) algebra, have been used [28]- [29] to deal with the modelling and the performance evaluation of a public transportation system. For the modelling of passenger flows at a transport interchange, as shown in [33] colored Petri nets are able to incorporate some specific parameters and data in the model such as the variation of walking speeds between passengers and the restricted capacity of features of the interchange infrastructure.

These are a few works that demonstrate the high potential of Petri nets as a tool for modelling and performance analysis of urban transportation systems, but also on the other hand, it is shown that the application of Petri nets is still in its early stage and particularly limited to intersection traffic control [11]-[14]-[15]-[25]-[35]-[34] and to some studies dealing with the modelling of urban transportation systems for planning purposes [28]-[29]-[33]. In addition, according to our knowledge, unlike traditional urban transportation systems, no work has been undertaken on PBS systems modelling and performances analysis by using Petri net models. Our contribution in this context is the first one in the literature [18]-[19]- [20]-[21].
