Dynamic Optimization of Exclusive Bus Lane Location Considering Reliability: A Case Study of Beijing

Abstract

For metropolises like Beijing, heavy congestions cause transit passengers’ unreliable travel time, including in vehicle time and waiting time. Comparing with other managerial measures, designing a lane for bus use only is an effective method to improve travel reliability, for it can eliminate the influence on bus-driving conditions. This paper proposes a reliable and practical method to determine exclusive bus lanes (EBL). A reliability-based optimization model is established, in which the tradeoff among bus and private car passengers’ travel time, reliability, and EBL construction cost are considered. Based on the actual network, a user equilibrium demand assignment model is applied to estimate the dynamic bus flow distribution. Since the model is nonlinear, a two-step method is proposed where tangent lines are introduced to constitute an envelope curve to linearize the model. This work conducts the statistical modeling and fitting analysis with actual bus trajectory data, collected on EBL in Beijing during peak hours. Passenger travel time distributions are fitted to estimate the statistical passenger travel time; Lognormal distribution and Gaussian distribution are the best fit. The optimization results indicate that the passenger travel time reliability can be improved by 5.5% by the optimized EBL location scheme. This study will provide a theoretical basis and methodological support for improving the service level of the public transportation system in large cities through the scientific planning of exclusive bus lanes.

1. Introduction

Travel time reliability is defined as the consistency or dependability in travel times. Due to heavily congested traffic in metropolises like Beijing, the operation of transit systems always becomes unreliable. Although bus services have timetables, the unpredictable and frequent road congestion makes commuters’ travel time fluctuate greatly. As a result, low-level travel reliability has significantly influenced passengers to choose the public transportation system. It has become the core issue for transit operation managers and researchers to improve reliability conditions of bus networks and attract more passengers to use the bus system. Recently, the exclusive bus lane (EBL) has been implemented as an effective solution to the above problem in a lot of cities. An EBL is a lane restricted to buses. Compared to the existing transit priority strategies, such as the transit signal priority for buses to pass through intersections first [1,2,3], the queue-jumper lanes that allow buses to easily enter traffic flow in a priority position [4], and the intermittent bus lanes that give buses temporary running priority [5], an EBL can provide an isolated running environment and prevent various disturbances on the road, which is attractive to travel time and reliability, and supports a shift of demand from private cars to public transportation [6].

This paper proposes a practical method for EBL implementation to enhance reliability for transit-commuting passengers. A bi-level optimization model is established, incorporating dynamic demand considerations in the lower-level model. In addition, travel time reliability indexes are defined for bus passengers by considering their in-vehicle and waiting process during the whole trip. Statical regression analysis is used for evaluating the key parameters when estimating the passengers travel time reliability. Finally, a real-world case study is conducted to assess the applicability of the proposed model. The research will provide theoretical support for improving the reliability of bus systems through scientific EBL planning.

The work is organized as follows: in Section 2, the literatures are reviewed; in Section 3, the method for optimizing EBL location and a bi-level optimization model are established; in Section 4, a case study with actual data is discussed; Section 5 and Section 6 give the results, discussions, and conclusions.

2. Literature Review

EBLs are widely applied in many big cities. Due to the provision of dedicated road rights for buses, the use of dedicated bus routes will significantly improve the speed, stability, and convenience of buses for passengers. For example, in the practice of Shanghai [7] and Los Angeles [8], the use of bus lanes has reduced delays by 10% and 14.2%, respectively.

EBL location optimization is a complex optimization problem considering the tradeoff between car and bus passengers [9,10]. As an NP-hard problem, efficiency maximization or cost minimization is usually set as the optimization goal [11,12]. Yu proposed a bi-level optimization model of EBL on the multi-modal network [13]. The upper model considered passenger travel time and comfort, and the lower model was a multi-modal distribution model. Yao analyzed the design method of EBL in random traffic networks from the perspective of the competition relationship between cars and buses, and focused on the uncertainty of bus waiting time [14]. Khoo established a bi-level programming model to study the temporal and spatial characteristics of EBL [15]. Truong compared the application effect of continuous EBL under different scenarios, and proposed the corresponding planning method [16]. Sun analyzed the EBL optimization by three levels including EBL’s structure, layout, and service frequency [17]. The upper model was established from the perspective of urban planners, the middle model considered the optimal operation cost of enterprise, and the lower model aimed to maximize the satisfaction of passengers. Several researchers also combined the EBL location optimization with intersection signal optimization [18,19]. Considering the condition that EBL location optimization was always a complex programming problem under the result of transit assignment [20], different solution algorithms were proposed, such as genetic algorithms [21], branch and bound [22], and benders decomposition [23].

Since EBLs have been widely applied by many transit agencies, a lot of research focused on the evaluation of the operation stability and service reliability after the implementation of EBL [24]. For the EBL location optimization studies, quantified travel time and reliability of both transit and car passengers are key factors. Travel time is usually measured by the Bureau of Public Roads (BPR) function or related functions [25,26,27,28]. For travel reliability, with the development of multi-source data, more data-driving methods are proposed, and AVL data, GIS data, and GPS data are used to estimate reliability conditions. McLeod [29] analyzed headway variance using incomplete data. The research focused on the missing buses or discarded spurious bus headways. Chen [30] analyzed service reliability at the levels of stop, route, and network. Barabino proposed an offline framework for the outputting of time reliability by analyzing AVL data [31,32].

The implementation of EBL considers real-time response to dynamically changing traffic demands, making the design of efficient service processes and solution algorithms the core issue of this research. Similar problems, including studies on the integration of different modes of transport, are generally based on the extension of the vehicle routing problem with time windows [33]. For solving the optimization model, the two main types of algorithms currently are analytical algorithms and heuristic algorithms. For analytical algorithms, since they can obtain the precise solution to the model through a complex mathematical iterative process, they are suitable for solving small-scale problems. In contrast, heuristic algorithms can obtain relatively optimal solutions through a limited number of iterations, and are more widely applied in practice [34,35,36].

Although much research has been conducted to support improving reliability of transit systems by EBL, the following critical issues are deserving of further investigations:

(1)

Travel time is concentrated more in EBL location research. However, in the real-world case, traffic congestions and crowded waiting environments may cause transit systems to become more and more unreliable, which could be the most important factor to dynamic demand.

(2)

With different EBL location plans, travel time and reliability of different bus lines can be changed when part of the trip operates on an EBL. The number of passengers taking related bus lines will be changed correspondingly. This dynamic demand should be analyzed in detail.

(3)

Previous studies were always conducted under numerical grid networks, and different EBL locations and key parameters were tested. However, in big cities, the traffic conditions are complex, and traffic is heavily congested during peak hours. It is indicated that most quantitative analyses resulting from the numerical studies have a larger gap with the actual ones. Therefore, with more detailed travel data from advanced IC card systems and on-vehicle GPS equipment, case studies should be conducted to examine the effect of the EBL location optimization model.

3. Materials and Methods

3.1. Descriptions

In recent years, advancements in data acquisition technology have allowed for more precise descriptions of transit passenger travel patterns using mobile GPS data, which can capture bus trajectories down to the second. In this paper, GPS data collected from onboard devices in Beijing are utilized to estimate parameters related to bus operating reliability.

A framework to quantify passenger travel time reliability is proposed and an optimization model to determine EBL location is established considering reliability. Firstly, formulation of passengers’ travel reliability is quantified. Then, optimization formulation is proposed and a corresponding linearization algorithm is designed. Finally, a case study is conducted to validate the model. The optimization model can provide EBL locations in which the real-time passenger demand is considered. The overall flowchart is described in Figure 1.

Parameters and variables in this paper are described in

Table 1. Parameters and variables.

Parameters	Descriptions
$t_j$$, t_i$	Bus arrival time samples of stop i and j.
$t_{ij}$	Sample time from i to j.
$t_{ij}^{in-vehicle}$	Reliability-based in-vehicle time.
E($t_{ij}$)	Expectation of IVT.
${\sigma (t}_{ij})$	Standard deviation (SD) of ij.
$\rho$	Reliability preference parameter.
N	Number of samples.
w	Sample waiting time.
$t^{waiting}$	Reliability-based waiting time.
E($w$)	Expectation of WT for all the samples.
D(w)	Deviation of sample waiting time
${\sigma }_w$	Standard deviation of w.
zw	Expectation of passenger arrival time/bus arrival interval.
s	Travel arc.
Cs	General cost of Travel arc s.
u	Unit travel time cost (yuan/h).
$S_{in-vehicle}$	Set of in-vehicle travel arc.
$S_{waiting}$	Set of waiting arc.
$d_s$	Extra delay time.
$y_s$	Number of waiting passengers.
${\delta }_{od}^{ps}$	Parameter that determines whether travel arc s passes on path p from node o to node d
$y_{od}^p$	Number of passengers on available path p from nodes ij
K	Vehicle capacity.
$a,b, \beta , { \beta }_1$, n	Parameters to be estimated.
$S_{transfer}$	Set of transfer arc.
$u^{\prime }$	Unit construction cost.
$l_s$	the length of EBL.
$t_s^0$	Unit travel time of car under free flow velocity
$y^{car}$	Volume of car.
H	Passenger arrival headway
${\phi }_s$	whether the EBL is set on the arc s.

. The decision variables and corresponding parameters to reflect EBL implementation and passenger travel time and reliability on EBLs are proposed in the model. The other definitions of variables that reflect the transit travel time reliability follows the Ref. [37].

3.2. Estimation of Travel Time Reliability

“Buffer time” is used to reflect reliability which is composed by expected travel time and the potential variation. Travel time reliability will decrease with the increasing of the buffer time [37]. Passenger travel time includes reliability-based in-vehicle time (IVT) and reliability-based waiting time (WT). As Figure 2 shows, IVT and WT are estimated respectively.

$$t_{ij}=t_j-t_i$$

(1)

$$t_{ij}^{in-vehicle}=\mathrm{E}\left[t_{ij}\right]+\rho {\sigma (t}_{ij})$$

(2)

$$\mathrm{E}[t_{ij}]={\displaystyle \frac{{\sum }_{n=1}^N{t}_{ij}\left(n\right)}{N}}$$

(3)

$${\sigma (t}_{ij})=\sqrt{{{\displaystyle \frac{1}{N-1}}{\sum }_{n=1}^N{(t}_{ij}\left(n\right)-E(t_{ij}))}^2}$$

(4)

$$w=t_i-t_0$$

(5)

$$t^{waiting}=\mathrm{E}\left[w\right]+\rho {\sigma }_w$$

(6)

$$\mathrm{E}\left(w\right)=\mathrm{E}(t_i-t_0)=\mathrm{E}\left(t_i\right)-\mathrm{E}\left(t_0\right)$$

(7)

$${\sigma }_w=\sqrt{D\left(w\right)}$$

(8)

$$P(T=t)={\displaystyle \frac{(z_{w}H{)}^t{e}^{-z_{w}H}}{t!}}$$

(9)

$$E(t)={\int }_0^{H}tP\left(T\right)dT=z_{w}H$$

(10)

$$\sigma \left(t\right)=\sqrt{z_w}$$

(11)

Equations (1)–(4) reflect the estimation of IVT, and Equations (5)–(8) reflect the estimation of WT. It is mentioned that Equation (2) reflects that the reliability-based IVT of stop i and j is composed of expected time and “buffer time”. Equation (3) reflects the expectation of IVT. Equation (4) reflects standard deviation. Equation (6) reflects the reliability-based WT. The variation of WT is caused by bus arrival time and passenger arrival time. The bus interval h is assumed to follow the certain probability distribution. E(h) is estimated by field GPS data. In public transportation systems, transit passengers’ arrival at bus stops can be regarded as a stochastic process that follows Poisson distribution [0, H] [38].

3.3. Model Formulation and Analysis

3.3.1. General Cost

(1)

Formulation cost of a transit passenger

The general costs include the cost of IVT, WT, extra waiting delay, and transfer time. For the passenger travel arc, the formulations are determined as follows:

Cost of IVT

$$C_s=u{t}_s^{in-vehicle}\hspace{1em}\hspace{1em}\forall s\in S_{in-vehicle}$$

(12)

Cost of WT

$$C_s=u{t}_s^{waiting}\hspace{1em}\hspace{1em}\forall s\in S_{waiting}$$

(13)

Cost of extra waiting delay time

Considering the scenario that the boarding and alighting process will be slowed down with the increasing number of passengers [39,40], an extra delay will be caused, which is determined as follows:

$$d_s={\beta }_1({\displaystyle \frac{y_s}{K}}{)}^n\hspace{1em}\forall s\in S_{waiting}$$

(14)

$$C_s=u{d}_s\hspace{1em}\hspace{1em}\forall s\in S_{waiting}$$

(15)

Equation (14) reflects extra delay time, which is determined by the actual passengers in the bus. The more boarding passengers there are, the longer the delay will be.

Cost of transfer time

Transfer time equals to the additional WT, which is expressed as follows:

$$C_s=u{t}_s^{transfer}=u{t}_s^{waiting}\hspace{1em}\forall s\in S_{transfer}$$

(16)

(2)

Formulation of the private car passenger cost

On the basis of the BPR model [25,26,27,28], travel time cost of a private car passenger can be calculated as Equation (17), in which the implementation of an EBL is considered. $y^{car}$ means the volume of the car.

$$C_s=u{t}_s^0\left(1+a{\left({\displaystyle \frac{y^{car}}{b\left(1-{\phi }_s\right)+\frac{N-1}{N}b{\phi }_s}}\right)}^{\beta }\right)\hspace{1em}\forall s\in S_{in-vehicle}$$

(17)

(3)

Formulation of EBL construction cost

The construction cost of an EBL is composed of the unit construction cost and the length of the EBL, which can be calculated as follows:

$$C_s={u^{\prime }l}_s{\phi }_s\mathrm{}\forall s\in S_{in-vehicle}$$

(18)

3.3.2. Model Formulation

The EBL location optimization model can provide the optimal scheme to determine the distribution of an EBL on the road network. Dynamic demand distribution is considered. For transit passengers, an EBL can significantly improve time reliability. However, for private vehicle passengers, after the implementation of an EBL, the road capacity will decrease and the traffic will be more congested. The EBL location optimization model is expressed as follows:

$$\mathit{min}{ f}_1=\sum _s{u}^{\prime }{l}_s{\phi }_s+\sum _{s}u{(t}_s^{in-vehicle}(1-{\phi }_s)+t_s^{EBL}{\phi }_s{)y}_s+\sum _{s}u{t}_s^{car}{y}_s^{car}$$

(19)

$$\mathit{min}{ f}_2=\sum _{\mathrm{s}}{\int }_0^{{\mathrm{y}}_{\mathrm{s}}}{\mathrm{c}}_{\mathrm{s}}\left(\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }$$

(20)

s.t.

$$\sum _s{l}_s{\phi }_s\le L_{max}\hspace{1em}\hspace{1em}\forall s\in S$$

(21)

$${\phi }_s\in \left\{\mathrm{0,1}\right\}\hspace{1em}\hspace{1em}\forall s\in S$$

(22)

$$y_s=\sum _o\sum _d\sum _p{y}_{od}^p{\delta }_{od}^{ps}$$

(23)

$$\sum _{p\in P}{y}_{od}^p=q_{od} \forall o\in V,\forall d\in V$$

(24)

$$y_{od}^p\ge 0 \forall o\in V,\forall d\in V,\forall p\in P$$

(25)

f₁ considers the tradeoff among the transit passenger’s travel time cost, the private car passenger’s travel time cost, and the construction cost of an EBL. In f₂, dynamic transit demand is assigned. The objective function of f₂ can reflect the sum of the results of integrating the general cost. As Equation (14) shows, extra waiting delay is determined by the real-time volume of waiting passengers The Beckman model is used, as f₂ shows, which can demonstrate that the optimal solution is equivalent with user equilibrium [41].

Equation (20) can be expressed as follows:

$${\int }_0^{y_s}{c}_s\left(\xi \right)d\xi =u{t}_s^{in-vehicle}+u{(t}_s^{waiting}+d_s)+u^{\prime }{l}_s$$

(26)

Equation (21) reflects the constraint of the maximum length of alternative EBL schemes. Equation (24) reflects that the sum of passengers on all paths should equal the demand of each OD pair. Equation (25) means that the decision variable is non-negative.

3.3.3. Solution Algorithm

The multi-objective model is a bi-level optimization model in which the upper model decides the locations of EBLs, and the lower model describes the demand assignment. In addition, the lower function f₂ is a nonlinear optimization model, which is hard to solve. Therefore, a two-stage algorithm based on a feedback-loop process is proposed to solve the bi-level optimization model first. In addition, a linearization algorithm is proposed to transform the nonlinear model into a linear function, which can significantly decrease the difficulty of solving the model [41,42]. For the two-stage algorithm, the first step is searching, which determines the alternative solutions set in function f₁. The second step is a feedback-loop process in which each the EBL scheme resulting from the function f₁ is applied into function f₂, in which the transit demand distribution will be calculated. Then, the result will be input back into function f₁. With the continuously cycling iteration, the optimal EBL scheme will be estimated.

Step 1: Searching. Determine the range of decision variables P{P₁, P₂, P₃,…, P_n} in function f₁. Delete the unfeasible solutions from the alternative solution set. For roads which can be the candidates to set an EBL, the following conditions of the traffic environment should be met: (1) The number of bus lines running on an EBL is greater than N₁. (2) The number of road lanes is greater than N₂. (3) The ratio of private car volume and road capacity should be greater than a₁, which indicates that only congested roads are suitable for EBL. (4) The maximum total length of an EBL should be less than L_max.

Step 2: Feedback-loop step. According to feasible EBL schemes P^*{P₁^*, P₂^*, P₃^*,…, P_n^*}, the results are input into function f₂. Then, the passenger-flow distribution results F^*{F₁^*, F₂^*, F₃^*,…, F_n^*} can be calculated and input back into the upper model again. Using iterations, the best scheme P** is selected. The algorithm is terminated. The solution framework is shown in Figure 3.

The objective function f₂ is combined with the costs of an in-vehicle arc, waiting arc, and transfer arc [37]. As Equations (14) and (15) show, f₂ includes a part of the nonlinear functions, which are hard to solve and is presented in Equation (27) as follows:

$${\int }_0^{y_s}{c}_s\left(\xi \right)d\xi ={\left.C_s\left(\xi \right)\right|}_0^{y_s}={\displaystyle \frac{u{\rho \beta }_1}{\left(n+1\right)K^n}}(y_s{)}^{n+1}\hspace{1em}s\in S_{waiting}$$

(27)

Referring to Ref. [35], a series of tangent lines are introduced to linearize the nonlinear function, which is described as follows:

Step 1. Take the second derivative of Equation (27) and obtain the following

$$C(y_s{)}^{″}={\displaystyle \frac{d(\frac{dC\left(y_s\right)}{d{y}_s})}{d{y}_s}}={\displaystyle \frac{u\rho {\beta }_1}{n{k}^n}}(y_s{)}^{n-1}\hspace{1em}s\in S_{waiting}$$

(28)

Usually, $n\ge 1$,so $C(y_s{)}^{″}\ge 0$. It is concluded that $C\left(y_s\right)$ is the concave function within the range, which is shown as Figure 4.

Step 2. N nodes are inserted on the curve with space $\Delta h$ and a tangent line is added on each node, as Figure 4 shows. It is easily concluded that when N is large enough, the original curve is equivalent to the envelope curve. On any node h, the envelope curve can be expressed as follows:

$$\mathrm{L}:\left\{\begin{array}{c}{y}_1\left(h\right)y_s+y_2\left(h\right) h=n\Delta h, n=\mathrm{1,2},\dots ,N\\ y_1\left(h\right)=C^{\prime }\left(h\right)\\ y_2\left(h\right)=C\left(h\right)-y_1\left(h\right)h\end{array}\right.$$

(29)

Step 3. The assistant decision variable $z_s$ is introduced and formulated as follows:

$$y_1\left(h\right)y_s+y_2\left(h\right) \le z_s h=n\Delta h, n=\mathrm{1,2},...,N$$

(30)

For Equation (30), it is indicated that the value of z_s is greater than the value of the ordinate node on the blue grid. For any y_s, z_s will be restricted in the blank area off the blue grid in Figure 4. The optimization model f₂ aims to select the minimum solution, so $\min { f}_2={\sum }_s{\int }_0^{y_s}{c}_s\left(\xi \right)d\xi$ can be replaced by ${\sum }_s{z}_s$. As a result, for any y_s, the nonlinear function ${\sum }_s{\int }_0^{y_s}{c}_s\left(\xi \right)d\xi$ can be replaced by the linear form.

After the linearization processing of f₂, the optimization model is transferred into the linear optimization model, which is solved by Python+Gurobi in this paper.

4. Case Study

4.1. Local Bus Network

The local transit network of the Zhongguancun Software Park zone is a busy commercial area in Beijing. The heavy traffic congestions occur frequently during peak hours. As a result, when passengers choose the bus for commuting, the travel time becomes unreliable. Therefore, reliability is important for commuters. The transit network of the study area is presented in Figure 5. The travel distance between each stop is shown in

Table 2. Direct distance between bus stops.

Bus Stop	Bus Stop	Distance (km)	Bus Stop	Bus Stop	Distance (km)	Bus Stop	Bus Stop	Distance (km)
1	4	1.5	2	3	0.5	10	11	0.3
4	8	1.9	1	9	1	11	12	0.4
1	3	1	9	12	0.5	2	9	0.6
3	4	0.5	12	13	0.3	9	25	1.5
4	6	1	13	14	1.8	1	26	1.3
6	8	0.9	14	15	0.7	26	27	0.7
4	5	0.7	15	16	0.6	1	22	1.3
5	7	0.8	16	17	0.7	22	23	0.6
1	2	0.3	1	10	1.1	23	24	0.8

.

Table 3. Travel demand in case study.

OD (Person)	S2	S3	S5	S8	S9	S10	S12	S13	S14	S15	S16	S17	S18	S22	S23	S24	S26
S1	2	6		96	209	371	2	270	2	29	149		2	327	139	91	3
S3			3	10
S9								59		23	46
S11										19	4
S13										13	23
S15											6	6
S22															15	23
S23																14

shows the transit demand, which is extracted from the IC card system. The volume of basic data is over five hundred thousand pieces. It reflects the number of passengers that will travel from an origin stop to a destination stop. There are a total of 1962 passengers using buses during the peak hour.

The average ticket for transit costs 0.3 yuan/km. Unit travel time is 11.34 yuan/h, and travel time reliability is 19.27 yuan/h [43]. Reliability preference is determined as 2, in which the potentially unreliable traffic needs extra buffer time [37].

Table 4. Key parameters in case study.

Parameter	IVT		WT		K	n	${\mathit{\beta }}_1$
	E	SD	E	SD
Value	3.38 min/km	2.35 min/km	4.18 min	6.18 min	80 person	3	0.5
Parameter	u		$u^{\prime }$	C
Value	Travel time yuan/h	Travel time reliability yuan/h	14.7 yuan/km	600 pcu/lane
	11.34	19.27
Road Name	Xierqi	Houchang Cun	Malianwa N	Xinxi	Dongbeiwang M	Shangdiqi	Shangdisan
Traffic Volume (pcu/h/lane)	549	1116	692	160	130	634	420
Road Name	Shangdixi	Dongbeiwang W
Traffic Volume (pcu/h/lane)	880	201

gives the key parameters for the proposed optimization model. Other parameters for the optimization model are described in Table 4.

4.2. Travel Time Reliability Estimation

GPS data of 11 bus lines in the case area are collected for fitting analysis. The period of the data is from 6:00 to 11:00 a.m. on Monday. Considering the condition that there is still no EBL in the local area of the case study, to simulate passenger reliability value when the bus is running on an EBL, three bus lines (Line 300 N, Line 300 W, and Line 300 KN) are selected for the case study, which run on the Third Ring Road, completely covered by an EBL during peak hours in Beijing. The sample GPS data are shown on Figure 6.

Passenger travel time distributions are fitted to estimate the statistical passenger travel time. Four typical fits for passenger travel time are applied to select the best statistical distribution. As Figure 7 and

Table 5. Fitting result.

Line		Gaussian	Weibull	Laplace	Lognormal	Parameters of Best Fit
300 N	R-square	0.900	0.882	0.909	0.973	a = 0.799 b = 0.356
	RMSE	0.055	0.060	0.052	0.028
300 W	R-square	0.898	0.873	0.903	0.967	a = 0.728 b = 0.385
	RMSE	0.053	0.059	0.051	0.030
300 KN	R-square	0.958	0.897	0.936	0.953	a = 0.915 b = 1.877 c = 0.506
	RMSE	0.050	0.078	0.061	0.052

show, results of a Gaussian fit, Weibull fit, Laplace fit, and Lognormal fit are presented. In addition, the fitting results are validated using Root Mean Square Error (RMSE) and R-squared. It is indicated that for Lines 300 N and 300 KN, Lognormal distributions are the best fit. For Line 300 KN, a Gaussian distribution is the best fit.

On the basis of the best fit distributions, the IVT of bus lines on an EBL and the IVT and WT of conventional lines are estimated, and the results are shown in

Table 6. Result of travel time fluctuation.

Line	IVT (min/km)
	E	SD
330 N	1.96	1.27
300 W	1.92	1.25
300 KN	0.88	0.92
Average	1.59	1.15

and

Table 7. Result of expectation and standard deviation.

Line	IVT (min/km)		WT (min)
	E	SD	E	SD
902	3.7	3.3	2.4	5.7
320	3.7	2.4	5.1	5.6
333	3.7	2.4	3.6	5.6
362	2.9	2	3.6	5.6
509	3.3	2.2	5.1	6.8
521	3.7	2.7	4.1	6.4
570	3	1.9	5.1	6.8
963	3.8	2.6	2.4	5.7
636	3.2	2.2	4.4	6.2
85	3	1.9	5.1	6.8
82	3.2	2.2	5.1	6.8
Average	3.38	2.35	4.18	6.18

. On the EBL, passengers’ average expectation and standard deviation of the IVT are 1.59 min/km and 1.15 km, respectively. Off of the EBL, 11 conventional bus lines in the case study area are selected, and results of the IVT and WT are estimated. For IVT, the expectation and standard deviation are distributed from 2.9 min/km to 3.8 min/km, and from 1.9 min/km to 3.3 min/km. For WT, the distributions are from 2.4 min to 5.1 min, and from 5.6 min to 6.8 min. It is indicated that the variation of WT is larger for the waiting process. After the estimation of bus passengers’ IVT and WT, reliability can be estimated as Equations (2) and (6) described. Then facing different plans of EBL locations, dynamic demand is estimated considering the reliability conditions in the optimization model.

5. Result and Discussions

CR_network is used to determine the overall reliability. which can be calculated as follows:

$$C{R}_{network}=\sum _{s}u\rho {(\sigma }_s{+D_s)y}_s\hspace{1em}\hspace{1em}\forall s\in S_{in-vehicle}\cup S_{waiting}$$

(31)

Equation (31) reflects the total buffer time, including passengers’ in-vehicle time and waiting time [43]. There are a total of nine roads that could be implemented with an EBL. If any of them are implemented with an EBL, corresponding travel time and reliability parameters will be changed. As is shown in

Table 8. Optimized EBL scheme.

Road Name	Xierqi	Houchang Cun	Dongbeiwangzhong	Shangdiqi	Dongbeiwangxi
φ	1	0	0	0	0
Road name	Malianwabei	Xinxi	Shangdisan	Shangdixi
φ	1	1	0	1

, for the optimized EBL location scheme, the following roads are suggested to implement an EBL: Shangdixi, Xierqi, Xinxi, and Malianwabei. According to the calculation results in Figure 8, total reliability cost CR_n under the background of an EBL is 14,785.2 yuan, which is 5.5% lower than the actual value of 15,650.8 yuan. This means that when an EBL is implemented, all passengers can save 5.5% of time cost due to various uncertain influencing factors. It is indicated that a reasonable EBL location scheme will significantly improve reliability and reduce the passengers’ extra time cost caused by various unreliable factors.

In addition, comparing with other similar research, the results of this paper are more suitable for the commercial area. Although these areas are small, bus passengers often have extremely high requirements for travel time reliability. At the same time, these areas often experience severe traffic congestion during peak hours. Therefore, implementing an EBL will have a better optimization result.

6. Conclusions

Planning an EBL in big cities can effectively improve the travel time reliability of bus passengers. This paper proposes a reliable and practical method to optimize EBL locations in order to improve travel time reliability on a network, considering the realistic transit demand dynamics. Conclusions are summarized as follows:

(1)

The indicator based on buffer time is defined to quantify the transit passenger travel time reliability, which is composed by passenger reliability preference and time volatility. Comparing the traditional methods of traffic models, the methodology proposed in this paper uses statistical modeling and fitting analysis to evaluate travel time and reliability, based on actual AVL data. Data are collected from bus lines that are operating on the EBL of the Third Ring Road in Beijing during peak hours. Calculation results show that unit expectation and standard deviations of travel time are 1.59 min/km and 1.15 min/km, respectively, which can be important parameters to reflect the fluctuation of passenger travel time reliability in the optimization model.

(2)

A relatively complete framework of a reliability-based EBL location optimization model is established, and is composed of the EBL location and demand assignment optimization. The transit passengers’ dynamic choices are considered under different EBL location schemes. A two-stage solution algorithm on the basis of iteration is designed to solve the nonlinear programming model. To solve the complex nonlinear problem, a linearization method is proposed by cutting tangent lines on the concave curves.

(3)

To validate the model’s quality and robustness, the case study is analyzed in Beijing. Results show that the best-performing bus lane locations mainly depend on travel demand, road structure, and traffic conditions. For the network of bus passengers and private car passengers, the best EBL location scheme resulting from the proposed model can evidently improve the travel time reliability by 5.5% in the case study. In addition, comparing with other cities that have applied the EBL like Los Angeles, Beijing has more business districts and office areas, and commuters have higher requirements for the reliability of bus systems. Therefore, the implementation of an EBL can better improve the service level of the public transportation system within the area of highly concentrated demand. It will attract more passengers to choose the bus for commuting.

There are two main limitations in this paper. First, the impacts from other transportation modes, such as shared bicycles, electric bicycles, and pedestrians, are not considered. Second, the case study in the paper needs to be extended to transit networks with a larger scale. Those limitations need to be improved in future research.