# Perceived Trip Time Reliability and Its Cost in a Rail Transit Network

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Passenger’s Clock-Based Trip Time and PTT

#### 3.2. Probability Distributions of Clock-Based Trip Time Components

#### 3.2.1. Probability Distributions of Walking Time

#### 3.2.2. Probability Distributions of Waiting Time

- (1)
- Probability Distribution of Waiting Time on Frequency-based Lines

- (2)
- Probability Distribution of Waiting Time on Schedule-based Lines

#### 3.2.3. Probability Distribution of In-Vehicle Time

#### 3.3. PTR and PTRC Metrics

#### 3.3.1. PTR and PTRC on a Path

#### 3.3.2. PTR and PTRC among OD Pairs

#### 3.3.3. PTR and PTRC on Lines

#### 3.3.4. PTR and PTRC in an RTN

#### 3.4. PTR and PTRC Estimation Based on Passenger Trip Assignment and Monte Carlo Simulation

#### 3.4.1. The Length-Based C-Logit Stochastic User Equilibrium Model

#### 3.4.2. The Method of Successive Weighted Averages

- Step 1:
- Set the iteration’s number $h=1$, the algorithm variable ${\gamma}_{0}=1$, the algorithm parameter $a\ge 0$, and the stop iteration criterion $\phi $. The effective path sets among OD pairs are determined using Yen’s algorithm. The PTTs of paths in the effective travel path sets are computed without considering passenger flow.
- Step 2:
- Passenger trips are assigned to the RTN with Equations (23) and (24) according to the PTTs of effective travel paths to compute passenger flow on each link, which is represented as ${f}_{e}^{h},\forall e\in E$.
- Step 3:
- The PTTs of effective travel paths among OD pairs are computed according to the passenger flows on links ${f}_{e}^{h},\forall e\in E$. The passenger trips among OD pairs are assigned to the RTN again. The passenger flow on each link, ${z}_{e}^{h},\forall e\in E,$ is recomputed.
- Step 4:
- Let ${\gamma}_{h}={\gamma}_{h-1}+{h}^{a}$, ${\theta}_{h}=\raisebox{1ex}{${h}^{a}$}\!\left/ \!\raisebox{-1ex}{${\gamma}_{h}$}\right.$ and $h=h+1$. Passenger flows on links are updated with Equation (28):$${f}_{e}^{h+1}={f}_{e}^{h}+{\theta}_{h}\xb7\left({z}_{e}^{h}-{f}_{e}^{h}\right)$$
- Step 5:
- Convergence assessment. If $\raisebox{1ex}{$\sqrt{{\left({f}_{e}^{h+1}-{f}_{e}^{h}\right)}^{2}}$}\!\left/ \!\raisebox{-1ex}{${{\displaystyle \sum}}_{e\in E}{f}_{e}^{h}$}\right.\le \phi $, then stop iterating and ${f}_{e}^{h+1}$ is the passenger flow after passenger trip assignment; otherwise, go to step 3.

#### 3.4.3. The Method of Successive Weighted Averages

## 4. Case Study

#### 4.1. Chengdu’s RTN

#### 4.2. Chengdu’s RTN Operation Data and Surveyed Data

#### 4.3. PTR and PTRC Estimation

#### 4.3.1. PTR and PTRC among OD Pairs

#### 4.3.2. PTRs and the PTRCs on Lines and on the RTN

#### 4.4. Regression Analysis Relating PTRs and PTRCs to Influential Factors

#### 4.4.1. Regression Models Relating PTR and PTRC to Influential Factors

#### 4.4.2. Multiple Linear Regression Analysis for PTRs and Influential Factors

^{−6}, which is below 0.05 (0.05 is a commonly used value for judging whether the significance test is passed). This demonstrates that the linear relation between PTR and influential factors is significant. The coefficient of determination (i.e., R_square in Figure 10b) that quantifies the degree of linear correlation is 0.82433. It indicates that Equation (31) has a high goodness-of-fit.

#### 4.4.3. Multiple Linear Regression Analysis for PTRC and Influential Factors

^{−6}(i.e., the value of $p$ in Figure 11b) is the significance value of the model, which is below 0.05 (0.05 is a commonly used value for judging whether the significance test is passed). Therefore, the linear relation between PTRC and influential factors is significant. R_square in Figure 11b is 0.9014. It is close to 1 and thus demonstrates that Equation (32) has a high goodness-of-fit. The multiple linear regression model between PTRC and influential factors is shown in Equation (32).

#### 4.4.4. PTR Enhancement and PTRC Reduction Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**(

**a**) Cumulative distribution of generated 1000 passengers’ PTTs on path 1; (

**b**) Cumulative distribution of generated 1000 passengers’ PTTs on path 1.

**Figure 10.**(

**a**) The value of RMSE when introducing influential factors into the model; (

**b**) The result of applying the stepwise regression method.

**Figure 11.**(

**a**) The value of RMSE when introducing influential factors into the model; (

**b**) The result of applying the stepwise regression method.

**Figure 12.**The percent decreases in ${Y}_{1}$ and ${Y}_{2}$ when the values of ${X}_{4}$ and ${X}_{6}$ are decreased.

Indicators | Description |
---|---|

Coefficient of variation [23] | The ratio of the standard deviation to the mean. |

Skewness of time [24] | The ratio of the difference between the 90th percentile trip time and 50th percentile trip time to the difference between the 50th percentile trip time and 10th percentile trip time. |

90th or 95th percentile trip time [25] | 90th or 95th percentile trip time used as the reliable trip time |

Buffer time [26] | The difference between the average trip time and 95th percentile trip time. |

Buffer time index [26] | The percentage of buffer time with respect to the average trip time. |

On-time arrival [27] | The probability that a trip arrives within the trip time budget. |

Time unreliability [25] | The fraction of late arriving trips. |

Total time budget [27] | The minimum trip time threshold that satisfies a certain reliability requirement given by decision-makers at a certain confidence level. |

Mean-excess total time [28] | The conditional expectation of trip times exceeding the corresponding total trip time budget at a given confidence level. |

Load Factor (%) | Sitting | Standing |
---|---|---|

0–75 | 0.86 | ---- |

75–100 | 0.95 | ---- |

100–125 | 1.05 | 1.62 |

125–150 | 1.16 | 1.79 |

150–175 | 1.27 | 1.99 |

175–200 | 1.40 | 2.20 |

>200 | 1.55 | 2.44 |

**Table 3.**Values of $\mathsf{\mu}$ and $\mathsf{\sigma}$ for walking speed distribution $\mathrm{N}\text{}\left(\mu ,{\mathsf{\sigma}}^{2}\right)$ [38].

Walking Place | $\mathit{\mu}$ (m/s) | $\mathit{\sigma}$ |
---|---|---|

passageways | 1.39 | 0.463 |

upstairs | 0.79 | 0.236 |

downstairs | 0.81 | 0.174 |

Headways (min) | ζ | ω | θ |
---|---|---|---|

5 | 0.43 | 0.41 | 2.85 |

10 | 0.52 | 0.36 | 3.39 |

15 | 0.58 | 0.32 | 3.98 |

20 | 0.64 | 0.27 | 4.57 |

30 | 0.90 | 0.24 | 6.52 |

Line | Operation Type | Headway (min) | Seats (per Hour) | Capacity (Passenger Trips per Hour) |
---|---|---|---|---|

1 | frequency-based | 2.00 | 348 × 30 | 1460 × 30 |

2 | frequency-based | 2.73 | 348 × 22 | 1460 × 22 |

3 | frequency-based | 3.00 | 348 × 20 | 1460 × 20 |

4 | frequency-based | 3.00 | 348 × 20 | 1460 × 20 |

5 | frequency-based | 4.00 | 348 × 15 | 1460 × 15 |

6 | frequency-based | 6.00 | 348 × 10 | 1460 × 10 |

7 | schedule-based | 10.00 | 250 × 6 | 680 × 6 |

8 | schedule-based | 15.00 | 250 × 4 | 680 × 4 |

9 | schedule-based | 15.00 | 250 × 4 | 680 × 4 |

10 | schedule-based | 15.00 | 610 × 4 | 1280 × 4 |

11 | schedule-based | 10.00 | 610 × 6 | 1280 × 6 |

12 | schedule-based | 10.00 | 610 × 6 | 1280 × 6 |

Transfer Station | Transfer Direction | Walking Distance (m) | Transfer Direction | Walking Distance (m) |
---|---|---|---|---|

3 | Line 1 to line 5 | 178 | Line 5 to line 1 | 155 |

3 | Line 1 to line 7 | 237 | Line 7 to line 1 | 207 |

6 | Line 1 to line 4 | 252 | Line 4 to line 1 | 155 |

7 | Line 1 to line 2 | 215 | Line 2 to line 1 | 200 |

10 | Line 1 to line 3 | 200 | Line 3 to line 1 | 104 |

13 | Line 1 to line 5 | 252 | Line 5 to line 1 | 126 |

Link (Station–Station) | Running Time (min) | Link (Station–Station) | Running Time (min) | ||
---|---|---|---|---|---|

Upstream | Downstream | Upstream | Downstream | ||

1–2 | 1.87 | 1.88 | 18–19 | 1.35 | 1.33 |

2–3 | 2.08 | 2.07 | 19–20 | 1.38 | 1.37 |

3–4 | 1.47 | 1.47 | 20–21 | 1.57 | 1.57 |

4–5 | 1.57 | 1.58 | 21–22 | 1.95 | 1.92 |

5–6 | 1.22 | 1.25 | 22–23 | 1.50 | 1.50 |

6–7 | 1.35 | 1.33 | 23–24 | 1.83 | 1.83 |

7–8 | 1.15 | 1.20 | 24–25 | 1.70 | 1.78 |

8–9 | 1.15 | 1.17 | 25–26 | 1.78 | 1.77 |

9–10 | 1.32 | 1.32 | 26–27 | 1.93 | 1.93 |

10–11 | 1.27 | 1.28 | 27–28 | 2.00 | 2.25 |

11–12 | 1.35 | 1.33 | 28–29 | 1.75 | 1.50 |

12–13 | 1.40 | 1.45 | 29–30 | 1.77 | 1.77 |

13–14 | 1.63 | 1.63 | 30–31 | 1.58 | 1.58 |

14–15 | 1.50 | 1.50 | 31–32 | 1.50 | 1.50 |

15–16 | 1.17 | 1.17 | 32–33 | 1.33 | 1.33 |

16–17 | 1.20 | 1.20 | 33–34 | 1.87 | 1.87 |

17–18 | 1.78 | 1.78 | 34–35 | 2.00 | 2.00 |

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## Share and Cite

**MDPI and ACS Style**

Liu, J.; Schonfeld, P.; Chen, J.; Yin, Y.; Peng, Q.
Perceived Trip Time Reliability and Its Cost in a Rail Transit Network. *Sustainability* **2021**, *13*, 7504.
https://doi.org/10.3390/su13137504

**AMA Style**

Liu J, Schonfeld P, Chen J, Yin Y, Peng Q.
Perceived Trip Time Reliability and Its Cost in a Rail Transit Network. *Sustainability*. 2021; 13(13):7504.
https://doi.org/10.3390/su13137504

**Chicago/Turabian Style**

Liu, Jie, Paul Schonfeld, Jinqu Chen, Yong Yin, and Qiyuan Peng.
2021. "Perceived Trip Time Reliability and Its Cost in a Rail Transit Network" *Sustainability* 13, no. 13: 7504.
https://doi.org/10.3390/su13137504