Optimal Reservation Volume of Urban Roads Based on Travel Reservation Strategy

Improving the spatial-temporal balance between the supply and demand of urban transportation and alleviating trafc congestion are important ways to build sustainable cities. Te travel reservation strategy (TRS) is more fexible and refned than traditional trafc demand management methods. Tis study aims to determine the optimal reservation volume (ORV) for urban roads and verify the efectiveness of the TRS. First, we employed the sustained fow index to estimate the ORV from the degree of trade-of between the road breakdown probability and capacity. Ten, a bilevel programming model based on ORV constraints was established to analyse the efectiveness of the TRS. Te results indicated that the ORV range is 0.79–0.89 times the road capacity. Te TRS can achieve the best steady beneft when the demand for reservation travel reaches at least 40%. Selecting the most congested critical roads in the network to implement TRS is more efective than on a large area. Te driver default behavior will increase the V/C ratios and travel costs of all roads in the network. It has been proven that the reservation transportation mode will promote the spatial-temporal balance between supply and demand to alleviate trafc congestion.


Introduction
Te increasing imbalance between supply and demand in urban transportation systems aggravates trafc congestion.It signifcantly afects residents' daily lives, causing economic losses and environmental pollution and aggravating the hidden dangers of travel safety [1,2].Te core cause of trafc congestion is the mismatch between transportation infrastructure supply and residents' travel demand, manifested by the lack of efective trafc demand management measures to adjust to the growing travel demand.With the development of emerging technologies such as instant messaging and artifcial intelligence, the travel reservation strategy (TRS) has become a new method for solving urban trafc congestion [3].
TRS is a more fexible and refned trafc supply and demand balance strategy.It refers to the regulation of the total number of vehicles that can pass through the key roads in recurrent congested areas during specifc peak hours.Te implementation of TRS should frst determine the recurrent congested roads and periods that require reservations by monitoring the trafc state of the urban road network.Second, it is crucial to determine the optimal reservation volume (ORV) for the corresponding roads and periods.A small ORV wastes road resources, whereas a large ORV causes road congestion.Terefore, a reasonable ORV is the basis for ensuring the efective implementation of the TRS.Finally, by creating a travel reservation timetable for private cars, travellers can choose the appropriate route and departure time on the reservation platform.Private cars without reservations are prohibited from driving on the reservation road during the reservation period, except buses and special vehicles (such as police cars, fre trucks, and ambulances).TRS can transform queues at roadway bottlenecks into virtual online queues.Travellers can reserve passage roads and time in advance on an open reservation platform, thus realising precise regulation of regional trafc travel demand.Akahane and Kuwahara [4] frst introduced a travel reservation method for managing urban trafc demand.Tey explored the possibility of using reservations to adjust the departure time of travellers to alleviate trafc congestion on holidays.Since then, several studies have applied travel reservations to various scenarios, such as highways [3], urban roads [5], and urban perimeters [6].Te complex network structure and diverse trafc behavior in urban transportation make it more challenging to establish an urban road travel reservation system.
Te urban reserved transportation mode is further divided into the regional reservation, road reservation, and lane reservation, according to the scope of the reservation space.A regional reservation strategy means that the trafc demand of an area is limited to below the critical capacity of the area.Only when the area's trafc volume is below the critical capacity can vehicles enter the area.Zhao et al. [6] built a downtown space reservation system to mitigate trafc congestion in a cordon-based downtown area by requiring people who want to drive to this area to make reservations in advance.Menelaou et al. [7] assumed that a heterogeneous urban area is partitioned into multiple homogeneous regions.Te vehicle is allowed to enter a region only if it is anticipated that the region will stay within its critical capacity during the time interval that the vehicle will be required to traverse an urban region.Te apparent advantage of the regional reservation strategy is that it is easy to implement and only needs to control the boundaries of each region.However, the trafc fow within the region needs to be more delicately managed; otherwise, congestion may still occur, and queuing may occur when entering the region boundary.Te lane reservation strategy refers to establishing a high-priority lane on the road for reserved vehicles.Bai et al. [8] showed that the lane reservation strategy could reduce the travel time and air pollution of reserved lanes based on real trafc data.Zhang et al. [9] investigated a bilevel optimisation model for the hazmat transportation problem with lane reservation.Tey found that the bilevel model can efectively reduce the total risk of hazmat transportation while considering the interests of hazmat carriers and ordinary travellers.Te lane reservation strategy will inevitably aggravate trafc congestion in nonreserved lanes; however, the convenience of reserved lanes will gradually change residents' travel concepts.TRS implies that all lanes of a road are only available for reserved vehicles.Menelaou et al. [10] proposed a time-dependent route reservation method for relieving trafc congestion.Some studies have increased the fexibility of travellers' choice of appointment time by introducing evolutionary [11] or pricing [12] mechanisms that combine a dynamic auction with capacity control rules.TRS combines the advantages of road charging and a tradable credit scheme strategy.Te core idea of TRS is to encourage travellers to choose a reasonable travel route and period.Trough peak load shifting management, travel demands are reasonably arranged in diferent periods, and congestion queues on the road are transformed into virtual online queues.TRS can alleviate trafc congestion by guiding travel behavior, controlling trafc fow within a reasonable range, and ensuring travel time and speed from departure to destination.
Te premise of implementing TRS is to determine ORV.However, previous studies primarily focused on the design of vehicle scheduling models and algorithms for TRS and less on calculating the ORV of reserved roads and analysing the trafc state of the road network under the ORV constraint.Some studies have analysed specifc problems with travel reservations by directly setting the reservation volume, such as considering road capacity or the sum of road and parking capacities as the reservation volume [6,13].It does not consider the trafc state of reserved roads, which deviates from the essence of TRS.Terefore, in recent years, scholars have preferred to mine trafc fow data to determine reservation volumes [14].Menelaou et al. [15] replaced the capacity constraint with the density constraint and used infnitesimal perturbation analysis to capture the dynamic change in the critical density value.By comparing the estimated capacity and critical density distribution function, Geistefeldt and Shojaat [16] showed that the relative variability of the capacity is lower than that of the critical density.It revealed that the trafc volume is more appropriate than density to represent the trigger of trafc congestion.Traditionally, road capacity has been treated as a constant value.However, various studies demonstrate that road capacity is not a fxed number but a random variable [17].Brilon et al. [18] proposed a method to estimate the stochastic capacity function, which can evaluate the breakdown probability for any given volume representing the capacity.Shojaat et al. [19] used the sustained fow index (SFI) to analyse freeway fow performance.Te SFI represents the theoretical volume that can maintain fuid trafc under capacity uncertainty.Although stochastic capacity is generally accepted, only a few studies have applied it to quantify road performance in practical trafc management.Furthermore, the complex network structure and various trafc behaviors in urban transportation make it more challenging.
While the literature extensively explores the TRS and its applications, there is a notable dearth of data-driven research specifcally targeting its implementation on urban roads.Predominantly, prior studies emphasize vehicle scheduling models and TRS algorithms, sidelining the pivotal task of determining the ORV and assessing the road network's trafc state under ORV constraints.Tis study seeks to bridge these identifed gaps and contribute a novel perspective.To achieve this goal, this study frst used the SFI to calculate the ORV of the reservation roads.Ten, a bilevel programming model based on ORV constraints was established to explore the infuences of the number of reservation roads and reservation tendency on road network capacity, V/C ratio, and travel cost under the implementation of TRS.It can provide managers and decision makers with new ideas for alleviating trafc congestion and determining the critical parameters for implementing TRS.
Te rest of this study is organized as follows.Section 2 introduces the method used in this paper in detail.Section 3 conducts a case study with real trafc data and discusses the results.Section 4 concludes this paper.

Methodology
2.1.Preliminaries.Setting a scientifc and reasonable ORV is the premise of implementing the TRS.It can maximise the reserved road's trafc volume to ensure a smooth road and avoid wasting road resources.Based on the traditional threephase trafc fow theory (as shown in Figure 1), as the speed increases, the trafc state of the road experiences the evolution process from the "stop-and-go" congested fow to the smooth fow.Terefore, when the road reaches the critical speed, the trafc volume is the theoretical maximum volume.When the road speed exceeds the critical speed, the road remains smooth, but the road resources are not fully utilised.Trafc congestion occurs when the road speed is lower than the critical speed.In the process of trafc fow transition from smooth fow to congestion fow, there is an oscillating region, and trafc fow in this region may lead to congestion once it is subjected to signifcant external interference.Terefore, determining the critical speed is the key to calculating the optimal total reservation.Te speed threshold can be determined by mining the road trafc data.Brilon et al. [18] used the boundary between the upper and lower branches of the speed-fow scatter plot to distinguish noncongested and congested fows.Shojaat et al. [20] determined the speed threshold as the boundary between the fuid and congested trafc by analysing the speed and fow time series.Given this, from the perspective of road network resilience, we used the previous research results to determine the critical threshold of trafc congestion difusion by using the percolation theory to determine the critical speed of the road [21,22].

ORV Calculation
Based on SFI Method.Tis section describes the ORV calculation method in detail.First, the capacity distribution function of the road is estimated based on the product limit method.Te SFI approach is then used to determine the optimal road capacity.

Estimation of the Road Capacity Distribution Function.
Te capacity distribution function of the road was estimated using the product limit method (PLM) [18].Capacity is defned as the maximum fow that sustains fuid trafc conditions [23].Te road performance was acceptable when the trafc volume was below the maximum fow.Beyond this fow, the average travel speed drops sharply, causing trafc congestion.Te term "breakdown" of the fow has been used to describe the transition from relatively free-fowing trafc to congestion, which is usually called stop-and-go trafc [24].Generally, the trafc fow observed before the breakdown can be regarded as capacity (momentary).Te breakdown can be detected by a sudden and sharp reduction in the trafc speed; therefore, the speed threshold must be determined frst.After determining the speed threshold, the period was divided into several time intervals (typically 5 min), and the interval set was defned according to the collected data.
Set B: Te average speed in interval i is above the speed threshold and falls below the threshold in the next interval i + 1, and thus i ∈ B. In this case, the fow in interval i is the trafc volume observed before trafc breakdown, which can be regarded as the momentary capacity of the road.
After determining the set of breakdown intervals, the PLM was used to estimate the road capacity distribution function, and the calculation formula is as follows: where F(q) is the capacity distribution function, S(q) is the capacity survival function, q is the trafc volume, q i is the trafc volume in interval i, k i is the number of intervals with trafc volume q i ≤ q, and b i is the number of breakdowns at volume q i .B is the set of breakdown intervals.

Determination of ORV Based on SFI Method. Te
ORV is determined to fnd an optimal trafc volume in the capacity distribution function to sustain fuid trafc conditions.Shojaat et al. [19] introduced the sustained fow index (SFI) method.Te SFI is defned as the product of the trafc volume and probability of survival at that volume, which calculates the theoretical average volume that can be sustained in fuid trafc under capacity uncertainty.Te SFI provides a joint performance measure, and the maximum SFI is the best compromise between maximising the volume and minimising the probability of trafc breakdown.Before the maximum SFI, the risk of trafc breakdown was low, but road resources were underutilised.After the maximum SFI is reached, the trafc volume becomes larger and more vulnerable to trafc breakdown.Te SFI was calculated using the following formula: where SFI is the sustained fow index, S(q i ) is the probability of survival at volume q i , and F(q i ) is the probability of breakdown at volume q i .Te PLM is a nonparametric estimation method that depends on a large number of samples.Obtaining an accurate estimation of the entire capacity distribution function is difcult, and we cannot obtain a relatively accurate Journal of Advanced Transportation maximum SFI.Terefore, it is necessary to know the mathematical distribution functions for parameter estimation.First, we compare the ft of diferent distributions through experiments and select the most suitable function as the capacity distribution function.Ten, we used the SFI to calculate the optimal capacity of the road, as shown in the following equation: where f(q) is the density function of capacity.

Bilevel Programming Model Based on ORV Constraints.
Tis study used the weekday morning rush hour as an example to verify the efectiveness of TRS.Travellers who enter reservation roads during this period must make reservations in advance.Te traveller must detour if the road has reached the reservation volume.Tis study assumes that all travellers who have obtained the right-of-way through reservations arrive evenly within the reserved period.Te parameters and variables used in this section are listed in Table 1.
Te optimal capacity calculated in Section 2.1 is set as the ORV constraint, denoted by q res a .Terefore, the admissible state of the request of the vehicle j to enter link a in a certain period is denoted by res j a .If the request of vehicle j is admissible, res j a � 1; otherwise, res j a � 0. Terefore, res j a can be calculated as follows: Tis study divides vehicles into two types: those with reservation tendencies and ordinary vehicles.Vehicles with reservation tendencies will make reservations on the reservation platform before travel and can drive on the reservation road after success.By contrast, ordinary vehicles cannot drive on reservation roads.To simplify the problem, we assume that vehicles with reservation tendencies follow the travel route provided by the reservation system, which is based on the system optimization.In contrast, ordinary vehicle travel is based on the user equilibrium.A bilevel programming (BP) model based on reserved capacity was established to explore the infuence of TRS on the trafc state and travel cost of the road network.Te upper-level optimisation objective is to maximise the trafc demand, and the constraint condition is the capacity limit of the reservation road.Te lower-level optimisation objectives are the system optimal for reservation travellers and user equilibrium for ordinary travellers.
Te upper-level model is as follows: Equation ( 5) shows the objective function of the upperlevel model.w is a set of OD pairs and w ∈ W; d w represents the original travel demand for the OD pair w. μ w is the OD pair w and the value of μ indicates whether the current network capacity has spare capacity [25].If μ > 1, the network has a spare capacity; otherwise, the network is overloaded. w μ w d w represents the total capacity of the road network.Equation ( 6) represents the capacity constraint on the reservation road, where q a is the trafc fow on link a.
Under the TRS, two types of travellers exist in the road network that pursue optimisation under the infuence of each other.Te lower-level model is a mixed trafc equilibrium model, which can be expressed as follows: Min Z so down �  a q so a × t a q so a + q ue a , Equation ( 7) is the objective function of the system optimal model, which is constrained by equations ( 8)- (11).q so a and q ue a represent trafc fow on link a under the system optimal and user equilibrium model, respectively.t a (q) is represents the travel demand of the reservation-tendency vehicle and λ is the reservation tendency.Equation (9) ensures that the sum of the fows assigned to paths equals the travel demand, where R represents the set of paths, r ∈ R, and u so,w r is the trafc fow on path r on the OD pair w.Equation (10) ensures that the sum of all path fows through link a is equal to q so a .δ so,w a,r are path-link relations.If link a is on path r, then δ so,w a,r � 1; otherwise, δ so,w a,r � 0. Equation (11) indicates that the variables are positive.
Equation ( 12) represents the objective function of the user equilibrium model, which is constrained by equations ( 13)- (16).In equation (13), d ue w represents the travel demand of an ordinary vehicle.Equation (14) ensures that the sum of the fows assigned to paths equals the travel demand, where u ue,w r is the trafc fow on the path r on the OD pair w. ε ue,w r is the reservation section-path relation, which ensures that ordinary vehicles cannot enter the exclusive section of the reservation travel.If no reservation section exists on path r, then ε ue,w r � 1; otherwise, ε ue,w r � 0. Equation (15) ensures that the sum of all path fows through link a is equal to q ue a .δ ue,w a,r are path-link relations.If link a is on path r, then δ ue,w a,r � 1; otherwise, δ ue,w a,r � 0. Equation (16) indicates that the variables are positive.
Tis study used the BPR function as the performance function: where α and β are fxed parameters, which are assumed to be α � 2.05 and β � 4. We used the least squares method combined with GPS and historical trafc data to determine the values of α and β by ftting the function.We used calibration values to ensure that the BPR function accurately represents the observed relationship between travel time and trafc fow in the study area.Given the limited length of this study and the fact that the parameter calibration process has been widely used in previous studies [26][27][28], we will not elaborate on it again in this paper.
Te solution efciency was signifcantly reduced because of the numerous roads in the study area while solving the BP model.Surrogate models (e.g., polynomial regression, radial basis function, neural networks, and Kriging models) have been applied to transportation problems in recent years because of their superior computational speed and good analytical properties [29,30].Some studies have indicated that the Kriging model has the highest ftting accuracy [31].Terefore, this study adopted the Kriging model to solve the BP model.A single-projection algorithm was used to solve the lower-level model.Refer to [32][33][34] for more details.Te specifc modelling and solving steps of the Kriging model are as follows [35,36].
Te Kriging model is an interpolation method, and its interpolation result is defned as the linear weighting of the response value of all sample points.
As long as the weighting coefcient ϖ is provided, the performance prediction value of any scheme in the design space can be obtained.To calculate the weighting coefcients, the unknown function was regarded as a Gaussian process.
where β 0 is an unknown constant and the mathematical expectation of Y(x) and Z(x) is a stochastic process with a mean of 0 and a variance of σ 2 .In the design space, these random variables are correlated and their covariance is where R(x, x ′ ) is the correlation function.Tis study adopts the Gaussian exponential model: where θ is the hyperparameter.Te maximum likelihood estimation was used for hyperparameter training to increase the solving accuracy of the model.Because the values of hyperparameters cannot be provided analytically, hyperparameter training is generally solved using numerical optimisation methods such as genetic, simulated annealing, and nongradient algorithms.In this study, a simulated annealing algorithm was used.Based on the above assumptions, the Kriging model aims to fnd the optimal weighting coefcient ϖ so that the following mean square error is minimised and unbiased condition is satisfed.

MSE[􏽢 y(x)]
Using the Lagrange multiplier method, the Kriging model can be obtained through derivation: Te Kriging model is an interpolation model, and the infll criterion generally adopts expected improvement (EI) maximisation to generate new sample points.Let the current optimal objective function value be y best and let the Kriging prediction result satisfy the normal distribution with mean  y(x) and standard deviation s(x).
Te expected improvement at point x is expressed as follows: where I(x) is the improvement at point x and Φ and ϕ are functions of the standard normal cumulative distribution and standard normal distribution probability density, respectively.Te new sample point can be obtained by solving the suboptimization problem that maximizes the E[I(x)].
Because the road capacity constraint is a strong constraint, this study adopts the penalty method to incorporate it into the objective function while solving the BP model, and the objective function is as follows: where τ is the penalty coefcient.Te specifc solution algorithm is given below: Step 1. Enter the maximum number of iterations N max , maximum consecutive successes C max succ , maximum consecutive failures C max fail , and the initial perturbation probability p 0 select .Generate an initial set of evaluation points I 0 using Latin hypercube sampling and set the number of iterations n � 0.
Step 2. Obtain the vector of objective function values for all evaluation points Z � [Z(Q i ), Q i ∈ I 0 ] and the current optimal solution Q best .
Step 3. Based on the evaluation point set I n , the simulated annealing method is used to obtain the hyperparameters and update the Kriging model.
Step 4. Based on the current optimal solution y best and perturbation probability p n select , the simulated annealing method is used to solve the maximum EI to obtain the candidate point y n+1 .
Step 5. Te single-projection algorithm is used to solve the lower-level mixed trafc equilibrium model and then calculate Z(y n+1 ). Step

Experiments and Analysis
Tis section presents a case study that demonstrates the efectiveness of the proposed methodology.Tis could provide essential parameter values for implementing the TRS in a specifc environment, which provides a theoretical and practical basis for the promotion of the strategy.

Study Area and Data Description.
As the most important political and cultural centre in Northwest China, Xi'an is the beginning of the ancient Silk Road and an important node of the "One Belt, One Road" policy.Terefore, improving the intellectual level of urban trafc management and relieving urban trafc congestion are urgent.Tis study took part of the urban road network in Xi'an, China, as the study area, including 19 nodes and 56 directional links, as shown in Figure 1 Te label on the link in Figure 2 is denoted by (t 0 a , C a ).Links 3-8-11-16 and 16-11-8-3 are commuter roads with large trafc fow and obvious congestion during peak hours.Terefore, we chose these links as reservation roads, and the reservation period was from 8:00 to 9:00 in the morning rush hour.During this period, only reserved vehicles could enter the reserved roads.

6
Journal of Advanced Transportation Tis study used the geomagnetic sensor data during the working days in Xi'an, China, in June 2019, and obtained the initial trafc demand (as shown in Table 2) by OD matrix inversion.
Parameter values of the BP algorithm are as follows: maximum number of iterations N max � 200, maximum consecutive successes C max succ � 3, maximum consecutive failures C max fail � 3, and initial perturbation probability p 0 select � 0.5.Parameter values of the single-projection algorithm of the lower-level model are as follows: the projection step size was 0.01, convergence accuracy was 1 × 10 − 5 , and path set update frequency was 20.

Calculation of Reservation Road ORV.
As mentioned before, from the perspective of road network resilience, we used the previous research results to determine the critical threshold of trafc congestion difusion by using the percolation theory to determine the critical speed of the road [21,22].Te critical speed of reserved road was approximately 30 km/h.Referring to the research on the road capacity distribution function in highway scenarios [19], this study frst used the PLM to estimate the road capacity distribution function (F(q)) according to equation (1).Te PLM is a nonparametric estimation method that cannot estimate the complete F(q).Terefore, three types of functions with analytical solutions (Weibull, logistic, and Gumbel distributions) were used in this study to ft the distribution function of road capacity.Finally, the best-ftting function was selected, and the optimal volume was determined by estimating the corresponding SFI based on equation (3).Te calculation results are presented in Figure 3 and Table 3.
As shown in Table 3, the logistic distribution function has the largest adjusted R-squared value and the smallest residual sum of squares compared to the other two distribution functions.Moreover, combined with Figure 3, the logistic distribution function was more consistent with the evolution law of the road capacity distribution function.Te calibration parameters c and ϑ under the logistic distribution were 2292.323 and 188.513, respectively.By substituting the parameters into Table 3, the optimal volume of the logistic distribution and the corresponding breakdown probability were calculated to be 1879 vph and 0.1, respectively, as shown at point B in Figure 3.When the fow rate was 1879 vph, SFI reached a maximum value of 1691 vph.Te link performance reaches its maximum value, as shown at point A in Figure 3. Te breakdown probability increased when the fow rate was greater than 1879 vph, which increased the probability of congestion.Although the  breakdown probability is reduced when the fow rate is less than 1879 vph, the low fow rate indicates that road resources are not completely utilised.In summary, the ORV for link 8-11 was 1879 vph.
Te same method was used to determine the best-ft function and optimal road volume.Table 4 shows the results for the diferent links.As shown in Table 4, the Weibull and logistic distribution functions provided the best ftting efect for urban roads.Te adjusted R-squares were all above 0.95, and the residual sum of squares was below 0.2.Considering the highways, most roads were suitable for the Weibull distribution.Table 4 also shows the best-ft distribution, calibrated parameters, optimal volumes, and breakdown probabilities of diferent links.Te results indicated that the optimal volume range is 0.79-0.89times the road capacity.Previous studies have demonstrated that the breakdown probability of highways is approximately 5% [19].Te breakdown probability of urban roads mostly exceeded 10%, indicating that urban roads face a greater risk of trafc congestion than highways.Tis is primarily to the complex road conditions of urban roads, such as signal lights, poor driving habits, and the infuence of pedestrians and nonmotor vehicles [37].
Tis study assumed that the trafc demand of the road network remained unchanged before and after the implementation of TRS.Te rationality of ORV is verifed by comparing the original trafc assignment (no reservation roads in the network) results based on user equilibrium and the results after implementing TRS.As mentioned above, links 3-8-11-16 and 16-11-8-3 are set as reservation links during the morning rush hour.Terefore, only reserved vehicles can drive into these links during this period.Te ORV and road capacity constraints (C a ) of each link are substituted into the BP model, and the calculation results are listed in Table 5.
As shown in Table 5, the average V/C ratio of the arterial road is 0.931 before strategy implementation, indicating severe trafc congestion.However, the average V/C ratio of the secondary road was only 0.278, indicating that road resources were not completely utilised.After the strategy implementation, the congestion of the arterial road is alleviated, the usage of secondary road resources increases, and the total travel cost is reduced.It has been proven that implementing this strategy reduces trafc congestion on the road network and alleviates the imbalance of trafc fow in spatial distribution.Further observation shows that under the ORV constraint, the travel cost is lower and the V/C ratio of the arterial road is lower, which proves that it is reasonable to determine the road reservation capacity using the SFI method.

Verifcation of TRS Efectiveness.
Tis section explores the efect of diferent factors (e.g., reservation tendency and number of reservation roads) on the road network capacity, service level, and travel cost, as shown in Figures 4(a  Table 3: Goodness of ft for diferent distribution functions.Journal of Advanced Transportation in the original trafc assignment.It can be observed that under the same conditions, the larger the number of reservation roads, the smaller the road network capacity.Setting reservation roads is equivalent to adding capacity restrictions to these roads.Although it can improve the congestion of reservation roads to a certain extent, it also increases the trafc pressure on surrounding roads and causes the congestion to spread to other roads.Given this, it is unreasonable to set up numerous reservation roads during peak hours.Terefore, selecting the most congested critical roads in the network is necessary to implement the TRS.With the same number of reservation roads, the network capacity decreases and fnally becomes stable with the increase in λ.Te network capacity is stable at a value less than the original capacity, indicating that the network trafc surges in the morning peak hour, the road network is congested, and there is no spare capacity in the network.Some travellers need to adjust their departure time.When λ reaches 0.4, the network capacity reaches a stable state.Combined with Figures 4(b) and 4(c), it can be found that when λ reaches 0.4, the V/C ratio and travel cost of the network both reach a stable state.Terefore, the TRS achieves the best beneft when the demand for reservation travel reaches at least 40%.Tis stable network capacity is the optimal capacity for travel during peak hours under the reservation policy.Te road network capacity is controlled within this range to achieve peak load shifting management of trafc fow so that the queuing of vehicles on the road during the extreme congestion period becomes the queuing of online reservations, thereby reducing road trafc congestion.Terefore, adopting the TRS for critical roads with extreme congestion is a dynamic constraint that can realise the dynamic adjustment of travel demand.

Distribution type
We further discuss the service level change of diferent grade roads in the network with diferent λ values under the six reservation roads, and the results are shown in Figure 4(b).With an increase in λ, the diferent grades of roads indicated diferent evolution trends.Te average V/C ratios of all roads, arterial roads, and secondary roads decreased gradually with an increase in λ.When the value of λ is small, the reservation travel demand is low, and the resources of the reservation roads are not efectively utilised, resulting in the remaining roads bearing part of the trafc volume of the reservation roads with a large V/C ratio of the remaining roads.With an increase in λ, the V/C ratio of the reservation roads frst increased and fnally stabilised.After λ � 0.4, the average V/C ratios of all roads and reservation roads are stable at approximately 0.726 and 0.685, respectively.Because the ORV is set on the reservation roads, the trafc fow exceeding the ORV cannot enter the reservation roads; therefore, the V/C ratio of the reservation road will remain stable and acceptable.In conclusion, combined with the results in Table 5, it can be found that the TRS can alleviate the time-space mismatch between supply and demand to a certain extent and reduce queuing on the road under limited resource conditions.
Figure 4(c) shows the travel cost change of diferent modes in the network with diferent λ values under the six reservation roads.With an increase in λ, the travel cost of the ordinary mode frst decreased and fnally stabilised, while the travel cost of the reservation mode frst increased and then stabilised.Initially, drivers who adopt the reservation mode can enjoy smooth travel.However, with an increase in reservation users, the trafc fow of the reservation road gradually increases, which leads to an increase in the V/C ratio and travel costs.After λ � 0.4, owing to the ORC  Journal of Advanced Transportation limitation, some users cannot make a reservation and choose to take a detour; therefore, the travel cost of the reservation mode is stable.In addition, with an increase in λ, the total travel cost of the road network frst decreased and fnally stabilised.It shows that implementing TRS can alleviate urban trafc congestion to a certain extent and reduce travel costs for residents.Moreover, we explored the efects of varying degrees of driver compliance on the results.Specifcally, we set six reservation roads and λ � 0.4 and evaluated the impact on road network service level and travel cost by considering cases where a certain percentage of drivers did not comply with the proposed policy.Te default ratio of 5% means that 5% of travellers who have not obtained the right-of-way violate the regulations and pass through the reserved roads.As shown in Figure 5(a), the travel cost in both reservation and ordinary modes gradually increases with the increase of the default ratio.It can be understood as defaulting by some travellers leading to the reservation road's trafc volume exceeding the ORV, exacerbating congestion and delays on the reservation roads, and afecting other travellers' itineraries.As depicted in Figure 5(b), the driver's default behavior can have an adverse efect on the V/C ratios of the entire road network.Te roads in the road network are interconnected.Terefore, when the congestion on the reservation roads increases sharply, it afects the surrounding roads.Tese roads are also afected, with reduced travel speed, vehicle queues, and more severe congestion.In summary, the driver default behavior can lead to trafc congestion and increase the V/C ratios and travel costs of all roads in the network.Terefore, we must adopt appropriate reward and punishment measures and improve the supporting monitoring system to encourage drivers to abide by the rules and avoid default behaviors.

Conclusion
Tis study aimed to determine the optimal reservation volume (ORV) of urban roads and verify the efectiveness of the travel reservation strategy (TRS).We employed the sustained fow index (SFI) to estimate the ORV from the degree of trade-of between road breakdown probability and capacity.A bilevel programming model based on ORV constraints was established to explore the infuences of the number of reservation roads and reservation tendency on road network capacity, V/C ratio, and travel cost under the implementation of TRS.Te efectiveness of this strategy was verifed by mining the road trafc characteristics using geomagnetic sensor data.Te contributions of this study are summarised as follows: (1) it is one of the few data-driven attempts to discuss the implementation of TRS on urban roads; (2) the proposed ORV can be widely used in general studies of urban road network scenarios, such as optimisation of trafc control strategies and vehicle routing during emergencies; (3) it verifes the efectiveness of TRS on urban roads, which helps alleviate the spatial-temporal mismatch between the supply and demand, reducing queuing on the road under limited resource conditions and helping promote sustainable city construction; and (4) crucially, the research elucidates that during peak hours, TRS should target key congested road segments and becomes most efective when at least 40% of vehicles are reserved, providing actionable insights for urban trafc policymakers.
Te case analysis results indicate that (1) the Weibull distribution is more consistent with the evolution tendency of the road capacity distribution function, and the ORV range is 0.79-0.89times the road capacity; (2) it is more efective to select the most congested critical roads in the network to implement TRS than a large area; (3) the TRS can achieve best steady beneft when the demand for reservation travel reaches at least 40%; (4) the TRS cannot increase the road network capacity within a specifc period; however, it can dynamically reduce travel costs and alleviate trafc congestion by improving the network resource supply and demand matching efciency; and (5) the driver default behavior can increase the V/C ratios and travel costs of all roads in the network.
Te results and conclusions have theoretical and practical signifcance for the implementation of TRS on urban roads.Teoretically, it is proven that the reservation transportation mode is benefcial for promoting the timespace balance between supply and demand to alleviate trafc congestion and realise the transition from user to system optimal transition.Practically, the proposed ORV and reservation tendency ratio can help managers and decision makers determine the critical parameters for implementing the TRS.
Tis study utilised the SFI to estimate the ORV for urban roads, but the dynamic nature of road capacity and fuctuating trafc demands may afect the accuracy of this estimation.Furthermore, while the TRS shows potential in improving the resource supply-demand match in urban networks, it does not consider shifts in residents' daily travel behavior after implementation.Tis omission can infuence the strategy's efcacy, especially given that driver default behavior has the potential to alter V/C ratios and travel across the network.Future research will aim to capture these dynamic behaviors more comprehensively.

Figure 1 :
Figure 1: Schematic diagram of theoretical maximum volume.

Figure 4 (Figure 3 :
Figure 3: Capacity distribution function and sustained fow index of the link 8-11.

Figure 4 :
Figure 4: Sensitivity analysis.(a) Road network capacity under diferent scenarios.(b) Average V/C ratios under diferent conditions.(c) Travel costs for diferent situations.

Figure 5 :
Figure 5: Te efects of varying degrees of driver compliance on the results.(a) Travel costs under diferent default ratios.(b) Average V/C ratios under diferent default ratios.

Table 1 :
Description of parameters and variables.Te trafc fow on the path r on the OD pair w Te capacity of the link a the travel cost function of link a. Equation (8) represents the fow conservation constraints scaled by the OD matrix multiplier μ. μ is a variable from the upper-level program.d so w 6.If Z(y n+1 ) < Z(y best ), then y best � y n+1 , Z(y best ) � Z(y n+1 ), continuous success count C succ � C succ + 1, and consecutive failure count C fail � 0; otherwise, C fail � C fail + 1 and C succ � 0. Step 7. Update the set of evaluation points I n+1 � I n ∪ If n ≥ N max , stop, and output y best .Otherwise, return to Step 3.

Table 5 :
V/C ratio and travel costs after implementing the TRS.