The statistical independence of time of every two adjacent bus links plays a crucial role in deciding the feasibility of using many mathematical models to analyze urban transit networks. Traditional research generally ignores the time independence that acts as the ground of their models. Assumption is usually made that time independence of every two adjacent links is sound. This is, however, actually groundless and probably causes problematic conclusions reached by corresponding models. Many transit assignment models such as multinomial probit-based models lose their effects when the time independence is not valid. In this paper, a simple method to predetermine the time independence is proposed. Based on the predetermination method, a modified capacity-restraint transit assignment method aimed at engineering practice is put forward and tested through a small contrived network and a case study in Nanjing city, China, respectively. It is found that the slope of regression equation between the mean and standard deviation of normal distribution acts as the indicator of time independence at the same time. Besides, our modified assignment method performs better than the traditional one with more reasonable results while keeping the property of simplicity well.
Urban transit network is becoming a hot issue especially in developing countries like China where transit priority has risen to become a national policy. Developed countries in Europe witness this trend as well. Even in the US where car traffic dominates, as a major focus of transit network, transit accessibility is widely researched (e.g., [
The issue “independence” acts as one of the prerequisites that make a great difference to the validity of many models targeted at analyzing transit network. Traditional works, however, hardly paid enough attention to time independence of every two adjacent bus links. Substantially, this independence can never be neglected. MNP model, which is short for multinomial probit model, was first used in stochastic traffic assignment [
This work is targeted at engineering practice of transit systems planning and management while two simple methods are proposed. To start with, we put forward a method to predetermine time independence of every two adjacent bus links. A characteristic quantity,
Most bus drivers are technical in the network. The There is only a little interference to transit operations caused by pedestrians and bicycles. Transit priority control is advanced, so intersections have limited effect on transit operations.
These four basic assumptions are not impractical. Assumptions
As for time independence predetermination, the “time” here is to be defined strictly. The two sides of a bus line have a bus stop. When a bus begins to pull over at the upstream stop, the time is
We fit the mean and std, which is short for standard deviation in this paper, of the normal distribution mentioned above. Bus links that satisfy all the basic assumptions to the greatest degree can be selected in a transit network. Accordingly, we could get several normal distributions:
The regression model is the fundamental of time independence predetermination and the modified capacity-restraint model. According to the data in Nanjing, when all chosen links meet the assumptions well, the goodness of fitting is perfect with high
After the regression analysis, we could predetermine time independence of every two adjacent bus links in a network. Time independence predetermination is to predetermine the independence of SST at every two adjacent links. Note that all bus links in the network conform to normal distribution and the goodness of fitting of regression equation is high on the basis of our analysis above. In this part, we put forward an indicator of independence whose value can simply predetermine time independence with practicability.
Considering normal distribution
Then the variance is given by the following expression:
A transit route is given in Figure
A transit route with its stops and SST.
Supposing that SST at every two adjacent links are mutually independent, then
If there are
By subtracting
Function
The analytical results could be obtained. When
As is analyzed before, When When When
Under the premise of a fixed mean of
When
When
Therefore, the parameter
If time independence is predetermined to be not valid, many transit assignment methods such as MNP-based models should never be used. Under our basic assumptions, methods based on logit models are untenable due to the normal distribution for SST. As a result, many assignment methods prove ineffective. Traditional capacity-restraint transit assignment method is not affected, though it could not take time uncertainty into consideration. So this part proposes a modified capacity-restraint transit assignment method on the basis of normal distribution for SST and regression equation between the distribution mean and std. When the invalidity of time independence has been predetermined, unlike many assignment methods, the proposed one keeps its effectiveness. It remains useful even though time independence is proved to be solid.
Compared with traditional capacity-restraint method, the modified one considers time uncertainty by using 95% quantile of normal distribution while keeping the advantages of traditional method like the simplicity of calculation and so on. It is believed that traditional heuristic assignment methods are faced with the “common lines” problem, and many modifications have been made to overcome the obstacle [
Flow chart of the modified capacity-restraint assignment method.
Comparison of the proposed and traditional abstraction of networks.
The rationale of the modified method is similar to the traditional one, though the coefficient of assignment
Note that the generalized cost of bus travel changes after loading one part of OD into transit network, but the std of SST is to remain the same unless some changes about the mean of SST take place. Then the remaining parts of the modified method are the same as that of the traditional one. To make this revised method more useful in engineering practice, efficiency should be paid special attention to. As a result, we recommend dividing the whole OD into three shares by 50%, 30%, and 20%, respectively.
Data has been collected through transit survey in metropolitan area of Nanjing, China. Considering all the basic assumptions of our methods and the traffic environment in real world, the bus lines are selected according to the following points: All the bus lines pass the metropolitan area of Nanjing, which are widely distributed in the city center. Bicycles and pedestrians make a little difference to bus operations, especially where the side median exists. Buses have priority on signalized intersections. Survey had better be conducted on a sunny day to avoid small probability events.
Taking all these factors into consideration, 7 groups of adjacent bus stops were randomly picked out, and corresponding bus lines were selected in the metropolitan area of Nanjing. Note that there may exist several choices of bus lines when a pair of bus stops is determined. So it is rationally assumed that there exists no difference between these lines concerning their operations between the pair of bus stops. To avoid waiting for a long time at bus stops, generally bus lines with too low frequency were never selected in this survey.
The transit survey was conducted from 4:00 p.m. to 7:00 p.m. on weekdays to collect data at peak hours. The investigators must record the time when their targeted buses stop for the first time at the bus stop. Then they should go on their buses and record the time when the buses pull over at the next stop. In theory, investigators should repeat the process for 16 times in single direction. In this way, however, it is hard for the survey to get finished in 3 hours. So investigators were permitted to repeat the process for 8 times in dual directions if the bus frequency was rather low. The error is typically rather limited if the bus stop on the other side of road is very close to the stop on this side in terms of the distance along the road.
From the recorded time when buses pull over at stops, SST samples are figured out and displayed in Table
Result of the SST survey in metropolitan area of Nanjing.
Number | 201 | 100 | 792 | 405 | 126 | 160 | 16 |
---|---|---|---|---|---|---|---|
1 | 159 | 32 | 42 | 61 | 55 | 35 | 49 |
2 | 127 | 37 | 49 | 52 | 56 | 36 | 46 |
3 | 166 | 32 | 42 | 59 | 54 | 37 | 51 |
4 | 158 | 28 | 46 | 56 | 65 | 43 | 50 |
5 | 122 | 31 | 41 | 67 | 67 | 41 | 58 |
6 | 134 | 33 | 51 | 60 | 56 | 35 | 60 |
7 | 138 | 35 | 41 | 62 | 59 | 35 | 47 |
8 | 148 | 33 | 45 | 53 | 51 | 36 | 52 |
9 | 166 | 29 | 42 | 58 | 52 | 39 | 51 |
10 | 140 | 31 | 47 | 70 | 52 | 37 | 53 |
11 | 155 | 35 | 42 | 71 | 62 | 35 | 51 |
12 | 138 | 35 | 40 | 67 | 55 | 40 | 57 |
13 | 151 | 30 | 45 | 58 | 48 | 36 | 52 |
14 | 144 | 33 | 49 | 64 | 55 | 37 | 62 |
15 | 138 | 36 | 45 | 57 | 54 | 35 | 48 |
16 | 132 | 33 | 48 | 57 | 52 | 38 | 51 |
|
|||||||
Mean | 144.8 | 32.7 | 44.7 | 60.8 | 55.8 | 37.2 | 52.4 |
std | 13.4 | 2.5 | 3.4 | 5.7 | 5.1 | 2.4 | 4.6 |
Frequency histogram of SST of each bus line.
Although it has been assumed that SST at bus links conforms to normal distribution, distribution test could be made to give a preliminary proof. We conduct Jarque-Bera test on the basis of the data in Nanjing using MATLAB. According to the results displayed in Table
Result of the distribution test of SST.
Bus lines |
|
CV | Mean | Variance |
---|---|---|---|---|
201 | 0 | 3.4140 | 144.75 | 180.47 |
100 | 0 | 3.4140 | 32.69 | 6.36 |
792 | 0 | 3.4140 | 44.69 | 11.56 |
405 | 0 | 3.4140 | 60.75 | 32.47 |
126 | 0 | 3.4140 | 55.81 | 26.30 |
160 | 0 | 3.4140 | 37.19 | 5.90 |
16 | 0 | 3.4140 | 52.38 | 21.18 |
Note:
Fitting curve of the SST of number 201 bus line.
Data from 6 lines is used for fitting the relationship between the mean and std, and the rest is for examining the fitting result simply. According to Section
Result of the regression analysis.
|
1.1919 | −3.2673 | ||
---|---|---|---|---|
bint | [0.9169, 1.4669] | [−4.3756, −2.1590] | ||
stats | 0.9731 | 144.8071 | 0.0003 | 0.0139 |
Note:
Regression analysis between the mean and std of normal distributions.
Judging from the results, the coefficient of determination is close to 1 and the probability of significance values 0.0003 that is smaller than 0.05 under the level of significance of 0.05. Furthermore, the
On the basis of our regression equation (
A contrived transit network is established to compare our proposed assignment method with the traditional one. Impact from other traffic modes on transit operations is not considered in this example. There are three transit lines in Figure
A contrived transit network.
Generalized cost function is commonly used in transportation studies [
As is recommended in [
Cost update of transit routes.
OD pair | Transit route | Our cost during loading 50%/30%/20% of OD (95% quantile SST if necessary) |
---|---|---|
Origin 1 to destination 3 | 1 → 2 → 3 | 4.9 (5.52)/8.1 (9.216)/10.02 (11.4336) |
1 → 5 → 6 → 3 | 28.5/32.7/34.5 | |
1 → 9 → 10 → 6 → 3 | 11.7/13.1/14.66 | |
|
||
Origin 1 to destination 6 | 1 → 3 → 6 | 12.9/17.7/20.58 |
1 → 5 → 6 | 8.7 (9.63)/10.8/ 11.7 (13.095) | |
1 → 9 → 10 → 6 | 8.9/10.3 (11.757)/11.86 | |
|
||
Origin 1 to destination 12 | 1 → 2 → 3 → 4 → 12 | 10.9 (11.55)/17.1 (19.611)/20.82 |
1 → 9 → 10 → 6 → 3 → 4 → 12 | 27.9/34.5/39.54 | |
1 → 5 → 6 → 7 → 12 | 17.7/19.8/20.7 (23.49) | |
1 → 9 → 10 → 6 → 7 → 12 | 23.7/25.8/28.14 |
Process of passenger flow assignment.
Transit route | 50% OD share | 30% OD share | 20% OD share |
---|---|---|---|
1 → 2 → 3 | 400 | 240 | 85 |
1 → 5 → 6 | 150 | 30 | 60 |
1 → 9 → 10 → 6 | 50 | 90 | 20 |
1 → 2 → 3 → 4 → 12 | 250 | 150 | 25 |
1 → 5 → 6 → 7 → 12 | 0 | 0 | 75 |
Note: routes in Table
Results of transit assignment.
Passenger volume obtained by the modified method
Passenger volume obtained by traditional method
As for the independence predetermination, it is hoped that the indicator of time independence
The results of metropolitan area of Nanjing indicate the key point before modeling urban transit network. It is the basic assumptions that researchers should lay great emphasis on. Instead of asserting that all model assumptions could be satisfied with qualitative analysis, some quantitative analysis is also indispensable. The proposed indicator of time independence along with some statistics of probabilistic tests exactly plays the role. Some uncomplicated transit survey mentioned in this paper could be conducted to figure out these key indicators before selecting reasonable models.
Some basic assumptions could be strengthened with the development of urban transportation. For assumption
Although it is assumed that SST conforms to normal distribution when bus links satisfy all the four basic assumptions, we could conduct distribution test to confirm the normal distribution assumption just like the case study in Nanjing, China. Only in a few cases may such SST at bus links not conform to normal distribution though there is no exception in Nanjing. When the assumption of normal distribution does not pass the test, the modified assignment method will lose its effect, and the method to predetermine time independence cannot be used in the whole urban transit network. However, we can focus on the subnetworks that SST at all bus links are tested to conform to normal distribution, so the predetermination method is able to be adopted in these subnetworks. In this way, the results of predetermination could also act as part of evidence to decide the validity of some models.
Another assumption is made in which the regression equation between the mean and std of normal distribution in the Double Logarithmic Coordinate is significant. Nevertheless, this cannot be guaranteed even though all bus links satisfy those basic assumptions. To obtain high goodness of fitting, samples could probably be optimized by selecting locations. This measure is quite flexible, which can make successive adjustments in the whole network, and there is a high chance that a significant regression equation exists characterizing a transit network that satisfies the basic assumptions. Even if poor goodness of fitting remains unsettled, which is not likely to happen, our proposed assignment method will work all the same if all std of SST at bus links could be worked out through transit survey. Then, it is evident the transit network cannot be too large. But in this case, anyhow, our method to predetermine time independence will become invalid.
A simple method has been proposed to predetermine time independence of every two adjacent bus links, which can prejudge the validity of many stochastic network models for public transit. Then a modified capacity-restraint transit assignment method is put forward and aimed at engineering practice when such independence is predetermined to be invalid.
Some basic assumptions have been made to elicit two direct assumptions to the proposed methods: First, SST at bus links conform to normal distribution. Second, the regression equation between the mean and standard deviation has high goodness of fitting. These two assumptions are tenable according to the results in Nanjing. Then the indicator of time independence
We take good advantage of the data acquired in Nanjing to conduct a case study. From the result of distribution test, all the seven bus links conform to normal distribution with a good significance level of 0.05. Then regression analysis is made in the Double Logarithmic Coordinate with a large
Although the findings in this paper are some of engineering practices, this work is limited by some issues. Firstly, it is very difficult to completely satisfy basic assumption
Further research could focus on how to loosen the proposed assumptions. Even if normal distribution is invalid, others like Negative Binomial Distribution and others may replace it. Different distributions can be introduced to a network simultaneously, and accordingly the methods of predetermination and assignment might exist as well, which is probably complicated to be put into practice yet highly prospective. In addition, existing approaches could be introduced to have our assignment model well consider the “common lines” problem as with the development of previous heuristic capacity-restraint models. Related details are worth exploring. Moreover, the validity of the modified assignment method could be verified using real data of transit passenger flow before getting adopted by traffic engineers. Last but not least, work could be continued in the case
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is funded by the National Key Basic Research Program of China (no. 2012CB725402), the Fundamental Research Funds for the Central Universities (no. KYLX_0171), and the Scientific Research Foundation of the Graduate School of the Southeast University.