# Public Transport Network Vulnerability and Delay Distribution among Travelers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodological Framework

## 3. Model

#### 3.1. Public Transport System Model

_{y}, s

_{w}) may be connected by: (i) a line link e

^{l}

_{y,w}if a line service l exists between these two stops; (ii) a pedestrian link g

_{y,w}, otherwise. The capacity c

_{e}of the road link e is defined as the maximum number of vehicles that can use e in the time period τ (veh/τ). Finally, the transport service is realized by vehicles characterized by seat capacity and operating speed.

_{OD}is modelled as a time-dependent origin-destination (OD) matrix at stop level, where each element d

_{od}represents the flow of travelers starting their trip at stop o and arriving at stop d in a given time period. For each o-d pair, a passenger n uses a path k

_{n}from the set of all the available paths K

^{od}, k

_{n}∊ K

^{od}, where k

_{n}is defined as a sequence of links e. Each path may include one or more transfers, trans

_{k}, occurring when two consecutive links e of k

_{n}belong to different lines.

_{n}, p

_{n}(k

_{n}), is modelled by a multinomial logit model [63]:

_{n}for passenger n defined as [17]

_{n}, which includes running times on links and dwell times at stops; $E\left[{t}_{{k}_{n}}^{g}\right]$ is the expected walking time on pedestrian links; $c{m}_{{k}_{n}}$ is the monetary cost of k

_{n}; and ${m}_{{k}_{n}}^{trans}$ is the number of transfers along k

_{n}. β

_{s}are the respective parameters. In-vehicle times are the same for passengers following the same path between o and d, while monetary costs may depend on fare policies (e.g., discounts for elderly or teenagers/children, special fares for regular working commuters, agreements with firms for discounts to their employees, etc.). Walking times depend on walking speed, which may vary among passengers following the same path. Finally, for a given o-d pair, the number of transfers may depend on the user’s choices if several alternative lines share some stops.

#### 3.2. Simulation Model

_{OD}.

#### 3.3. Disrupted Scenario and Vulnerability Evaluation

_{e}, is defined as

_{n}is the path chosen by passenger n.

_{e}for a given period t

^{δ}, c

_{e}

^{δ}being the capacity of link e in disrupted conditions, which is lower than the capacity c

_{e}

^{b}in baseline conditions. Reduction in link capacity generally causes travel time increase and then delays at successive stops with cascading effects on vehicle scheduling and waiting times at stops, which are likely to be longer. Then, passenger travel times in disrupted scenarios are generally higher than travel times in baseline conditions.

_{n}

^{δ}, is higher than his/her travel time in the baseline condition, TT

_{n}

^{b}. For a given scenario—included the baseline one—the total travel time for passenger n from o to d is given by the sum of waiting time at stops ${t}_{{k}_{n}}^{wait}$, in-vehicle times ${t}_{{k}_{n}}^{ivt}$, and walking times on pedestrian links ${t}_{{k}_{n}}^{g}$. Delay of passenger n in the disrupted scenario δ is computed as

^{δ}is the number of passengers experiencing a delay in scenario δ. The higher the average delay, the higher the vulnerability of the network in the disrupted scenario.

^{δ}is built by evaluating the share of delay DEL

_{n}

^{δ}for each delayed traveler n. Each point in the curve is identified by the set of coordinates (X

_{n}

^{δ}; Y

_{n}

^{δ}) where X

_{n}

^{δ}is the cumulated proportion of passengers (X

_{n}

^{δ}< X

_{n+1}

^{δ}) and Y

_{n}

^{δ}is the corresponding cumulated proportion of delays. Finally, the Gini index in scenario δ, ${V}_{GINI}^{\delta}$ is computed as

_{0}

^{δ}= Y

_{0}= 0, X

_{n}

_{δ}

^{δ}= Y

_{n}

_{δ}= 1, and V

_{GINI}

^{δ}ranges in [0, 1].

## 4. Case Study

#### 4.1. Public Transport System Description and Implementation

^{wait}= β

^{g}= −0.07, β

^{ivt}= −0.04, and β

^{trans}= −0.334. As for monetary costs, in the case study, the ticket fare is the same for all the lines, and then for all paths, i.e., no single ticket discounts for users’ categories exist, and fare zones are not applied so that the hypothesis that the user’s path choices are not affected by monetary costs is reasonable. In the considered case study, information regarding expected remaining time until next vehicle arrival for all lines serving all connected stops is provided to passengers at stops. Finally, when using connection links between stops, passengers are assumed to walk at a constant speed v

_{g}= 1.2 m/s (~4.5 km/h) with a maximum walking threshold of 500 m.

#### 4.2. Disrupted Scenario Definition: Central Links Identification

_{e}of each link has been computed in the baseline scenario, and links have been ranked according to it. The first five links have been selected for modelling five disrupted scenarios δ. More specifically, the first scenario corresponds to the disruption of the first-ranked link, the second to the second-ranked, and so on. In addition, for a more realistic representation, adjacent links to the most central ones having similar LC

_{e}values (up to 10% difference) have also been disrupted to assure line closures of a relevant length—which have been identified as “segment”. Table 1 and Figure 4 show the links with the highest LC

_{e}values that were selected for modelling the disrupted Scenarios (δ1–δ5). The simulated disruption corresponds to the complete closure of the segment due to unexpected events (such as road accidents or infrastructure issues), and then the capacity c

_{e}of such links has been set to zero for the morning peak-hour between 8:00 and 9:00 a.m. (t

_{δ}= 1 h). Moreover, no service replacement has been planned, due to both the unexpected nature of the event and the short duration of the disruption. For such scenarios, vulnerability has been quantified in terms of average passenger delays (see Section 3, Equations (6) and (7)). Then, the Lorenz curves LO

^{δ1−5}and Gini indices V

_{GINI}

^{δ1–5}have been computed, in order to refine the evaluation of the disruption impacts according to the proposed approach.

## 5. Results and Discussion

^{δ}are depicted in Figure 6 for the five scenarios, while the Gini coefficients V

_{GINI}

^{δ}are provided in Table 3. Overall, the curves suggest relatively low symmetry in terms of impacts on passengers, as showed by the difference between the symmetrical distribution line (in red in Figure 6) and the Lorenz curves, and only a small fraction of users experiences the majority of delays. Similarly, the Gini indices V

_{GINI}

^{δ}vary in the five scenarios ranging from 0.34 to 0.44, confirming that delay distributions are far from perfectly symmetrical.

_{GINI}

^{δ}= 0.44), with passengers experiencing delays also greater than 1 h. In this scenario, 80% of delayed passengers experience only 45% of the total amount of delay, while the remaining 20% of passengers experience the majority of delays. Referring to cases δ1 and δ3, delays are slightly more symmetrical: in these cases, 80% of the passengers share approximately 50% of delays. Scenarios δ4 and δ5, in which the disrupted links are the less central ones among the considered cases, have similar and lower Gini indices (V

_{GINI}

^{δ}= 0.34), suggesting a slightly more symmetric delay distribution.

_{e}are generally the ones whose closure causes the highest average delay (see Table 3). In the scenarios where the most central links are closed (δ1 and δ2), the magnitude of the impact is greater, and the network is more vulnerable. Conversely, for Scenarios δ3 and δ5, delays are lower and the network less vulnerable. These results seem to suggest the existence of some correlation between the centrality measure and the network vulnerability, although the centrality of a link does not necessarily mean that its closure leads to the most critical situation, particularly in terms of asymmetry of impacts (compare, for example, Scenarios δ1 and δ2 in Table 3).

_{GINI}

^{δ5}= 0.344 vs. V

_{GINI}

^{δ1}= 0.402), with δ5 being less vulnerable than δ1. Even if average delays are similar, the disrupted conditions of the system in the two cases are not the same, as delays distribute differently, as do the effects perceived by users.

## 6. Conclusions and Further Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**(

**a**) Average passenger delay and (

**b**) delay distribution among passengers for scenarios (δ1–δ5).

Scenario | Disrupted Lines | Disrupted Segment Length (m) | LC_{e} |
---|---|---|---|

δ1 | 14, 11, 13, 19, 20, 25, 27 | 900 | 25,382 |

δ2 | 13, 19, 21, 25, 35, 36 | 1300 | 19,772 |

δ3 | 11, 20, 21, 25 | 1300 | 19,603 |

δ4 | 13, 19, 36 | 850 | 13,246 |

δ5 | 14, 20, 21 | 1000 | 13,200 |

Scenario | Average Travel Time (min) | Percentage of Delayed Passengers (%) |
---|---|---|

b | 34.19 | 0 |

δ1 | 36.80 | 15.38 |

δ2 | 37.80 | 20.22 |

δ3 | 35.46 | 11.51 |

δ4 | 35.28 | 18.14 |

δ5 | 36.55 | 14.79 |

**Table 3.**Vulnerability Gini indices, load centrality, and average passenger delay for Scenarios (δ1–δ5).

Scenario | V_{GINI}^{δ} | LC_{e} | $\overline{\mathit{D}\mathit{E}{\mathit{L}}^{\mathit{\delta}}}$ (min) |
---|---|---|---|

δ1 | 0.40238 | 25,382 | 15.74 |

δ2 | 0.44415 | 19,772 | 16.88 |

δ3 | 0.38586 | 19,603 | 13.79 |

δ4 | 0.34230 | 13,246 | 12.99 |

δ5 | 0.34366 | 13,200 | 14.51 |

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**MDPI and ACS Style**

Malandri, C.; Mantecchini, L.; Paganelli, F.; Postorino, M.N. Public Transport Network Vulnerability and Delay Distribution among Travelers. *Sustainability* **2021**, *13*, 8737.
https://doi.org/10.3390/su13168737

**AMA Style**

Malandri C, Mantecchini L, Paganelli F, Postorino MN. Public Transport Network Vulnerability and Delay Distribution among Travelers. *Sustainability*. 2021; 13(16):8737.
https://doi.org/10.3390/su13168737

**Chicago/Turabian Style**

Malandri, Caterina, Luca Mantecchini, Filippo Paganelli, and Maria Nadia Postorino. 2021. "Public Transport Network Vulnerability and Delay Distribution among Travelers" *Sustainability* 13, no. 16: 8737.
https://doi.org/10.3390/su13168737