Delay Time Model at Unsignalized Intersections (original) (raw)

Delay Time Model at Unsignalized Intersections

Ciro Caliendo 1{ }^{1}

Abstract

A simulation process is presented for providing a point estimate of average traffic delay on minor roads at urban unsignalized intersections during peak hours. A performing process of microsimulation was carried out, and a good level of conformity was obtained between the traffic simulated in the peak hours and the corresponding amount of traffic measured in the field. Traffic performance measures expressed in terms of average delay suffered by vehicles on minor roads were used in the statistical analysis. A negative binomial regression model, jointly applied to conflict traffic volume (VHPc)\left(\mathrm{VHP}_{c}\right) and traffic volume entering intersection from minor roads (VHPe)\left(\mathrm{VHP}_{e}\right), was used to model the average delay. A dummy variable DD was also introduced to test the adequacy of the regression model through the range of hourly conflict traffic volumes. Regression parameters were estimated by the maximum likelihood method and the significance of the covariate was evaluated by using the likelihood ratio test (LRT). The results show that the average delay on minor roads is positively associated with VHPc,VHPe\mathrm{VHP}_{c}, \mathrm{VHP}_{e}, and DD, and these variables are all statistically significant. The regression model was found to fit well the average delays computed in simulation. Additionally, a stop-time model and a maximum queue length model were also developed. DOI: 10.1061/(ASCE)TE.1943-5436 .0000696. © 2014 American Society of Civil Engineers.

Author keywords: Simulation; Unsignalized intersections; Delay time; Stop time; Maximum queue; Negative binomial model.

Introduction

Delay is generally defined as the excess time consumed in a transportation facility compared to that of a reference value. More specifically, it is the difference between the time it would take to traverse a road section under ideal conditions and the actual travel time. Delay is often considered as one of the most important measures of the effectiveness of unsignalized intersections as perceived by road users.

Unsignalized intersections operate on the basis of the priority of traffic movements. The left-turning movement (in contrast with the straight on or right-turn movements) from the minor street has, for example, the lowest priority according to the corresponding traffic laws in many countries. The performance of an unsignalized intersection is strongly influenced by the delay caused by low-priority movements on minor roads.

Different models exist in the literature for estimating delay. One of the first delay models appears to be that of Kimber et al. (1977). These authors proposed relationships between delay and traffic intensity (the ratio of demand flow to capacity) for the minor roads of unsignalized intersections. Troutbeck (1986) developed a delay equation model at unsignalized intersections as a function of the following parameters: the average delay when the minor stream flow is low (also known as Adams’ delay), the degree saturation of the minor stream (entry flow/entry capacity), and a form factor that quantifies the effect of queueing in the minor stream. Khattak and Jovanis (1990) compared two main approaches to capacity and delay estimation that are probabilistic and deterministic. Heidemann (1991) provided a formula at priority junctions for calculating the delay in the flow of traffic that has to give way (minor road)

[1]as a function of the degree saturation. Kyte et al. (1991) divided the total delay for two-way stop-controlled (TWSC) intersections into two parts, namely queue delay and service delay. Queue delay is defined as the time between the vehicle’s arrival at the end of the queue and the time when the vehicle arrives at the stop line. Service delay is the time between the arrival at the stop line and the departure from the stop line. Queue delay was found to be mainly affected by the approach traffic volume, whereas service delay was mainly dependent on the conflicting traffic volume. Horowitz (1993) updated an existing queuing model of delay at all-way stop-controlled (AWSC) intersections to reflect empirical evidence of driver behavior. Madanat et al. (1994) developed a probabilistic delay model, reflecting the gap-acceptance behavior of drivers, at stop-controlled intersections, that is applicable to right-turning traffic at a T-intersection. Heidemann and Wegmann (1997) described a general queuing theory model for traffic flow at unsignalized intersections. Akcelik et al. (1998) compared three existing delay models for unsignalized intersections, showing the differences in results and proposing modified forms for these models based on the simulation tool. Al-Omari and Benehokal (1999) developed two different models for estimating queue delay and service delay, respectively, at TWSC intersections; in particular, service delay is estimated as a function of conflicting traffic volumes, while the service delay average and variance of service delay are used as inputs for estimating queue delay. Kaysi and Alam (2000) investigated the influence of driver behavior on delay (impatience, aggressiveness, experience) at unsignalized intersections. Tian et al. (2001), on the basis of simulation, developed a generalized form of the delay model that appeared to give better results than the simple exponential form delay-total traffic volume at AWSC intersections. Luttinen (2004) developed relationships between delay and traffic flow on minor roads, while Chodur (2005) investigated capacity models and parameters for unsignalized urban intersections. Chandra et al. (2009) presented a service delay model based on the microscopic analysis of delay data under mixed traffic conditions. The proportion of heavy vehicles in the conflicting traffic was found to influence service delay significantly. The Transportation Research Board (TRB 2010) in the more recent version of the Highway Manual Capacity also presents a procedure for estimating delay

1. 1{ }^{1} Associate Professor, Dept. of Civil Engineering, Univ. of Salerno, Via Giovanni Paulo II, 132, Fisciano, Salerno 84084, Italy. E-mail: ccaliendo@ unisa.it
   Note. This manuscript was submitted on October 25, 2013; approved on March 25, 2014; published online on May 15, 2014. Discussion period open until October 15, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Transportation Engineering, © ASCE, ISSN 0733-947X/04014042(13)$25.00. ↩︎

at unsignalized intersections. Delay is only estimated for each minor-street movement, and for major-street left turns. According to their study, delay includes delay due to deceleration to a stop at the end of the queue, move-up time within the queue, stopped delay at the front of the queue, and delay due to acceleration. In other words, delay is defined as the total time elapsing from the time a vehicle stops at the end of the queue to the time the vehicle departs from the stop line. The average delay for any minor movement is estimated as a function of the capacity for movement and the degree of saturation, where capacity is expressed as a function of conflicting traffic flow.

Most of the aforementioned studies were conducted in countries where traffic and geometric characteristics, as well as driving behavior and traffic rules, differ from those in Italy. Under these circumstances, the results of the cited literature on delay might not be directly transferable to a geographic area different from that in which they have been obtained. Therefore, there is evidence that local conditions should be better investigated to provide additional information to our knowledge.

When the real system concerning intersections is also complex, measuring delay in the field becomes much more complicated. On the other hand, field measurements are often costly and time consuming; thus, the microsimulation approach may prove to be a convenient tool. Simulation also permits taking into account within a short time the way in which delay is sensitive to the variability of local traffic conditions. In the transportation area, the microsimulation approach is often used for estimating not only delay at unsignalized intersections, but also for measuring performance of different types of intersections [for further information on this subject see also the results found by Fang and Elefteriadou (2005), Stevanovic and Martin (2008), Chevallier and Leclercq (2009), Chen et al. (2010), Caliendo and Guida (2012), Caliendo and De Guglielmo (2013), and Yousif et al. (2013)]. However, the use of this tool for more complex geometric configurations containing one or more adjacent single intersection appears to have been investigated to a lesser degree. Therefore, the present paper also makes, in contrast with microsimulation studies based on a single intersection, a contribution to the state of the art by evaluating the average delay on minor roads that belong to intersection scenarios covering a much wider geometric area.

Regression models have been widely used in the literature to predict road crash occurrence. Regarding past studies, Lord and Mannering (2010) and Savolainen et al. (2011) provide a listing of the methods applied together with their strengths and weaknesses. For the purposes of this paper, the negative binomial (NB) regression model is of special interest. In this respect, Caliendo et al. (2007,2013)(2007,2013) and Caliendo and Guida (2014) have provided a demonstration of the application of the NB model as a methodological approach in analyzing accident frequency occurring on Italian motorways and road tunnels, respectively. However, the possibility of using these regression models also for estimating the delay at road intersections, even if this appears to have an interesting appeal, has hitherto been less investigated. From the perspective of statistical modeling, delay count data consist of nonnegative integer values, which are random and discrete in nature. In the light of these considerations, the use of the NB may be appropriate and provide new insights into the way that independent variables influence the delay time (dependent variable) at road intersections, contributing to the state of our knowledge of this subject.

As a result of the considerations discussed, there are at least three main reasons for justifying this paper. The first, given the shortage in Italy of predictive delay models for unsignalized intersections, relationships that were not known before should be developed. In fact, there is a prior reason for believing that the results of
the aforementioned foreign studies concerning delay models might not be directly transferable to Italy. The second reason, given that microsimulation can help a road designer and/or a traffic operator to overcome the difficulties due to the measuring delay in the field, is to encourage the use of this tool. Finally, in order to have a better understanding of the more statistically significant variables that tend to increase or decrease the delay time at unsignalized intersections, regression models such as the NB might be applied.

This paper aims at investigating urban unsignalized intersections with the intent of identifying a model for predicting the average delay on minor roads. The tool used for this analysis is based on the microsimulation approach. For this purpose, three intersection scenarios of the city of Salerno (Italy), containing one or more adjacent single intersections, which were identified as isolated in simulation (i.e., not affected by upstream or downstream intersections), were investigated. Traffic performance measures expressed in terms of average delay suffered by vehicles on minor roads, computed in the simulation in six 1-h time periods corresponding to the peak traffic volumes, are used in the statistical analysis. An NB regression model is used, jointly applied to conflict traffic volume (VHP1)\left(\mathrm{VHP}_{1}\right) and traffic volume entering an intersection from minor roads (VHP2)\left(\mathrm{VHP}_{2}\right). A dummy variable (D)(D) is also introduced to test the adequacy of the regression model through the range of hourly conflict traffic volume. Regression parameters are estimated by the maximum likelihood method and the significance of the covariate is evaluated by using the likelihood ratio test (LRT). Additionally, a stop-time model is also developed. A queuing model for showing the effects of the aforementioned independent variables on the maximum queue length is also proposed.

It is to be stressed that the present study was carried out for the following main objectives: (1) to develop a microsimulation approach for assessing the performance measures at unsignalized intersection scenarios when the real system is much complex for the measurements in the field; (2) to individualize an appropriate statistical model for the analysis of simulation outcomes; and (3) to propose simple mathematical equations for estimating delay time, stop time, and maximum queue length on the minor roads of unsignalized intersections.

In light of the above considerations, the present paper is organized as follows. Following the in-depth overview of past research studies, the subsequent section presents a short description of the data set. The third section shows the calibration process developed for implementing the microsimulation software used (AIMSUN). The fourth section involves the description of the statistical modeling applied for the analysis of the results computed in the microsimulation. The results of the statistical analysis are then presented and discussed; consequently, mathematical equations for estimating delay time, stop time, and maximum queue length are proposed. Finally, the conclusions and directions for future research are reported.

Data Description

Geometric Configurations

The geometry of the three intersection scenarios investigated (identified as being considered isolated), priority of manoeuvres, and speed limits, were obtained by observation during site visits in the city of Salerno. The city of Salerno has more than 130,000 residents, and it is the 29th more populous city of Italy. It was investigated as a reference and for representing a wide sample of Italian cities having similar characteristics.

Scenario I comprised four single intersections, two intersections being of the four-arm type with the remaining two of the T-type

(Fig. 1). Four single intersections were represented in this scenario, three of which were unsignalized intersections (Intersections 1, 2, and 3) while one was a signalized intersection (Intersection 4). Scenario I also contains a signalized intersection because during the simulation it was found that the performances of the remaining three unsignalized intersections were affected by the proximity of this intersection. However, for the present paper only the data concerning unsignalized intersections were subsequently considered in the statistical analysis. All single intersections had a major and a minor road. Both major and minor roads were characterized as having unidirectional traffic only (one-way streets). Major or minor roads consist of two or three lanes (excluding the auxiliary lane for buses, that was not, however, in conflict with motor vehicles flow).

Scenario II includes two single intersections only. Both these intersections are of the T-type and are unsignalized as shown in Fig. 2 (Intersections 5 and 6). Also, in this case each single intersection had a major and a minor road. A major road is always with bidirectional traffic, while the minor road has unidirectional traffic for Intersection 5 and was bidirectional for Intersection 6. With reference to Intersection 5 (minor road is a one-way street), a lane is used for right turns from a minor to a major road only, while the other lane is used for left turns from a minor to a major road only. At Intersection 6, a lane is used for the left and right turns from major to minor roads only, while the other lane is used for left and right turns from a minor to a major road only. The number of lanes on the major road ranges from two to three, while on minor roads there are two lanes.

Scenario III is formed by three unsignalized intersections of the T-type as reported in Fig. 3 (Intersections 7, 8, and 9). In this case each single intersection has a major and a minor road, and the traffic is bidirectional. The minor road of Intersection 7 is a two-lane median-divided road. The minor road of Intersection 8 is a twolane median-undivided road, while the minor road of Intersection 9 is a four-lane median-undivided road. Traffic islands are also present at Intersections 8 and 9. The number of lanes on the major roads ranges from two to four.

A average lane width of 3.75 m and a negligible longitudinal grade were found for all the aforementioned nine intersections. The imposed speed limit was 50 km/h50 \mathrm{~km} / \mathrm{h} for all cases investigated. The aforementioned short description shows that the real system of intersection scenarios investigated is complex, so that the measuring delay in the field becomes very complicated. Therefore, in this respect, the microsimulation approach appeared to be a desirable tool.

Measured Traffic

Capacity and other traffic analyses commonly focus on the peakhour traffic volume because it represents the most critical period for operations at intersections. However, the peak-hour volume is not a constant value; it varies by hour of the day as trip purposes and the number of persons desiring to travel varies. Regarding the cases investigated, a detailed analysis of traffic showed that the highest hourly traffic volumes occurred at three 2-h time periods during the day. In particular, the most critical peak-traffic volumes were found to occur between 7:00 and 9:00 a.m., between 12:00 and 2:00 p.m., and finally between 6:00 and 8:00 p.m. In other words, the choice of the aforementioned study time is justified for capturing the effects of the most critical traffic conditions during the day.

Measurements of traffic in the field were made for each single intersection contained in the three scenarios mentioned. Traffic flows were computed by using video cameras placed at each intersection. The video camera was placed on a tripod at a suitable
vantage point near each intersection to record an obstructed view of all approaches and turning movements (right turn, left turn, and straight on). Data were recorded for 15 min during all three aforementioned two-hour time periods. The videotape was played in the laboratory several times to record the entering and turning traffic volume count and to divide the traffic volume into three vehicle categories, namely cars, heavy vehicles, and two-wheelers. However, only the traffic expressed in terms of hourly volume (sum of traffic measured at four 15 -min intervals) at each single intersection in the 6 h mentioned above was considered in the subsequent analysis.

In order to shorten the length of the current paper, traffic data measured in the field are presented only as hourly traffic volumes entering (or volumes exiting) a single intersection from major or minor roads. In addition, for making a visual comparison between the traffic measured and that computed at the detectors during the simulation, some graphic representations concerning entering (or exiting) traffic volumes were specially created (instead of presenting tables of numbers of measured traffic). Figs. 1-3 show this for Scenarios I, II, and III, respectively. For example, in Fig. 1, detectors D1 and D3 reported the hourly traffic volumes both measured and computed in simulation entering Intersection 1 (sum of cars, heavy vehicles, and two-wheelers) while those exiting were recorded by detectors D2 and D4. Detector A1 in contrast reported bus traffic volume entering only, with detectors A2 and A3 recording those exiting.

A maximum entry capacity of 800 vehicles per hour (veh/h) was considered for each lane both on the major and minor roads. The ratio between the measured entry flow and entry capacity was found to be contained within the 0.2−1.50.2-1.5 range on major roads and 0.005−1.50.005-1.5 on minor roads, respectively.

Microsimulation

The AIMSUN software (version 6.1.2) was used, which was commercially available at the time of developing this research. The main steps in the process of implementing the AIMSUN software were (1) identifying the geometric background of the site studied; (2) singling out the geometric characteristics of the intersections; (3) defining the driver’s behavior; (4) assigning traffic and turning movements; and (5) calibrating the microsimulation model.

A graphic interface, which is contained in AIMSUN, was used for introducing the geometric background of the site studied and displaying it on the computer. This graphic interface is compatible with the AutoCAD 2000 software (Autodesk, United States), which contained aerial surveys of the area investigated. Intersections were modeled on the basis of geometric characteristics (e.g., number of lanes, lane width, auxiliary lanes for buses, grades, etc.), priority rules (e.g., stop or give way on minor roads) and speed limits. However, it should be noted that for some intersections, which in an exploratory analysis had been simulated as being isolated, it was found that the traffic computed in the simulation was not a reasonable approximation of the traffic measured in the field. This was attributable to the short distance between the adjacent intersections, which influenced the traffic flow entering the downstream intersection. For this reason, wider geometric configurations, containing more adjacent single intersections, were defined. In particular, three intersection scenarios, named from I to III, were identified as being isolated in the simulation (i.e., not affected by other upstream or downstream intersections).

Driver behavior was simulated by employing car-following and lane-changing models. In the implementation of the car-following model in AIMSUN, also additional constraints on the braking

Scenario I
(a)

Fig. 1. Comparison of measured and simulated peak-hour volume, geometric layout of simulation, and location of detectors for Scenario I

Fig. 1. (Continued.)
capabilities of vehicles, which are imposed on the classic safe-tostop-distance, were taken into account. The lane-changing process is modeled as a form of decision making that emulates a driver’s behavior when he is considering changing lanes. However, when drivers are waiting for an acceptable gap, the following assumptions are made: no passing or lane changing is allowed at intersections; all drivers maintain at least a minimum headway; and drivers have knowledge about the priority of movements.

The traffic was modeled on the basis of vehicle type (car, heavy vehicle, two-wheelers), vehicle characteristics (e.g., length, width, maximum desired speed, maximum acceleration and deceleration), traffic volume of all turning manoeuvres (right turn, left turn, straight ahead), and traffic composition.

The calibration process in the simulation involved not only the traffic, but also the comparison of the simulation output with the input value of each parameters used, and any subsequent adjustments. In this respect, it is to be said that if some parameters generated by the simulation were less than their minimum allowable (e.g., the generated headway was less than 1.0 s ) or more than their maximum allowable (e.g., the generated maximum acceleration was more than 3.5 m/s23.5 \mathrm{~m} / \mathrm{s}^{2} for cars), these parameters were set to the minimum or maximum, respectively, and a new simulation run was made again. Therefore, the calibration process was interrupted when no further adjustments to parameters were necessary and the differences between the traffic measured in the field and
that simulated were contained within preselected statistical tolerance levels. In order to focus the reader’s attention more closely on the main purpose of this paper and to shorten the paper’s length, not everything the author did to achieve the objective of calibration is reported in this paper, but only the findings concerning the comparison between the traffic measured and that computed in simulation. In this respect, after many simulation runs and adjustments, the GEH statistic was used

GEH=2(M−C)2M+C\mathrm{GEH}=\sqrt{\frac{2(M-C)^{2}}{M+C}}

where M=M= simulated hourly traffic volume and C=C= hourly traffic volume measured in the field in the peak hours. The use of the GEH formula (GEH is the acronym of Geoffrey E. Havers) is an acceptance criterion for travel demand forecasting models recognized by the UK Highway Agency (1996). For all nine intersections contained in the studied three intersection scenarios a recommended GEH value of less than 5 was found, which proved to represent a good level of conformity between simulated peak hourly volumes and those measured in the field. In Figs. 1-3, the results of simulation runs together with measured traffic volumes are presented for each intersection and detector.

Visual comparison confirms the close agreement between simulated and actual traffic volumes. It is to be stressed that the calibration process of simulation involved not only the attempt to replicate the traffic measured in the field, but also the adjustments when some parameter values generated by the simulation were not admissible. The microsimulation models investigated were subsequently resolved, and the results were shown in the postprocessing phase. Simulation data were obtained every 15 min for each hour of the aforementioned six 1-h time periods corresponding to the peak traffic volumes. Typically, predictions of traffic, delay time, stop time, travel time, mean and maximum queue, and mean speed were computed. Animated 2D output of the simulation runs was also obtained.

Only the average values in peak hours concerning the performance measures on minor roads are here reported. In particular, the average values of delay time, stop time, and queue maximum length are presented, as shown in Table 1. These average values corresponded to the hourly traffic volumes entering intersections from minor roads (VHPe)\left(\mathrm{VHP}_{e}\right) and corresponding conflict traffic volumes (VHPc)\left(\mathrm{VHP}_{c}\right) at intersections in the subsequent statistical analysis (the values of VHPe\mathrm{VHP}_{e} and VHPc\mathrm{VHP}_{c} are reported in Table 1).

Statistical Modeling

One way to describe the delay counts, YiY_{i}, occurring at the peak hour time on the minor roads of intersections investigated, is to assume that YiY_{i} is a random variable with Poisson probability law. Let λi\lambda_{i} be, for example, the expected delay; it is well known that a Poisson random variable YiY_{i} has E(Y)=λiE(Y)=\lambda_{i} and Var⁡(Yi)=\operatorname{Var}\left(Y_{i}\right)= λi\lambda_{i}. However, delay counts might appear to be overdispersed with respect to the theoretical variability consistent with the Poisson model (the mean of the dependent variable is equal to variance). Hence, the NB model

f(yi)=Γ(yi+φ)yi!Γ(φ)[λiλi+φ]yi[φλi+φ]φf\left(y_{i}\right)=\frac{\Gamma\left(y_{i}+\varphi\right)}{y_{i}!\Gamma(\varphi)}\left[\frac{\lambda_{i}}{\lambda_{i}+\varphi}\right]^{y_{i}}\left[\frac{\varphi}{\lambda_{i}+\varphi}\right]^{\varphi}

which has E(Yi)=λiE\left(Y_{i}\right)=\lambda_{i} and Var⁡(Yi)=λi(1+λi/φ)\operatorname{Var}\left(Y_{i}\right)=\lambda_{i}\left(1+\lambda_{i} / \varphi\right), might be more appropriate, thus allowing for the variance of delay counts to be greater than the mean, provided that 1/φ>01 / \varphi>0.

A major objective of statistical analysis of delay counts is to estimate the expected delay on a given minor road at intersections as a function of explanatory variables such as hourly traffic volume entering the intersection from a minor roads (VHPe)\left(\mathrm{VHP}_{e}\right) and conflict

Fig. 2. Comparison of measured and simulated peak-hour volume, geometric layout of simulation, and location of detectors for Scenario II

Fig. 2. (Continued.)
traffic volume (VHPT)\left(\mathrm{VHP}_{T}\right), which implies defining a regression model where the explanatory variables act as covariates.

Let x\mathbf{x} be a vector of kk covariates and β\boldsymbol{\beta} a vector of kk (unknown) coefficients. For both the Poisson and the NB model, a regression model of the expected delay is defined by λi=g(xi;β)\lambda_{i}=g\left(\mathbf{x}_{i} ; \boldsymbol{\beta}\right), where g(⋅)g(\cdot) denotes a certain function. In this paper, the widely used regression model

λi=exp⁡(xiTβ)\lambda_{i}=\exp \left(\boldsymbol{x}_{i}^{T} \boldsymbol{\beta}\right)

was adopted [see Caliendo et al. (2007,2013)(2007,2013) for a greater in-depth discussion].

In order to obtain estimates, β^\hat{\boldsymbol{\beta}} and φ^\hat{\varphi} say, of the unknown parameters β\boldsymbol{\beta} and φ\varphi, the log-likelihood function under the governing model is maximized. In light of the invariance property of maximum likelihood (ML) estimation, λ^i=λi(xi;β^)\hat{\lambda}_{i}=\lambda_{i}\left(\mathbf{x}_{i} ; \hat{\boldsymbol{\beta}}\right) is the ML estimate of the expected delay time at the peak hour time. In this respect, the procedure developed by Greene (2007) was used. Note that the loglikelihood is generally expressed as a negative integer. In this case, the higher values of log-likelihood are preferred.

There are many measures that can be used for estimating how well a model fits the data. This study uses some likelihood statistics. The LRT is a common test used to assess two competing
models. It provides evidence in support of one model (usually a full model) over another model that is restricted by having a reduced number of parameters. Sometimes the restricted model is considered to include only the constant term (intercept-model only) so that in this case the LRT is

χ2=−2[LL(β)−LL(0)]\chi^{2}=-2[L L(\beta)-L L(0)]

where LL(β)L L(\beta) is the log-likelihood of the full model and LL(0)L L(0) is the likelihood of the constant term only. The χ2\chi^{2} statistic is χ2\chi^{2} distributed with degrees of freedom equal to the difference in the numbers of parameters in the restricted and unrestricted model.

In addition, it should be noted that the LRT was also used for assessing the significance of the covariates entering the regression model (i.e., for examining whether the adding of a variable to the model was statistically significant). In this case, the LRT in the current paper is asymptotically distributed as a χ2\chi^{2} distribution with 1 degree of freedom.

The fit of competing models was also evaluated by the ρstatistic 2\rho_{\text {statistic }}^{2} given by

ρstatistic 2=1−LL(β)/LL(0)\rho_{\text {statistic }}^{2}=1-L L(\beta) / L L(0)

where LL(β)L L(\beta) is the log-likelihood of the full model and LL(0)L L(0) is the likelihood of the intercept-model only. The ρstatistic 2\rho_{\text {statistic }}^{2} is between zero and one, and when it is closer to one the greater variance of the estimated model is accounted for [see Washington et al. (2011) and Hilbe (2007) for more information on the ρstatistic 2\rho_{\text {statistic }}^{2} ].

The Akaike information criterion (AIC) proposed by Akaike (1973) is another measure used for assessing models. The AIC is computed as

AIC=2k−2LL(β)\mathrm{AIC}=2 k-2 L L(\beta)

where kk is the number of parameters in the model (this also includes the dispersion parameter) and LL(β)L L(\beta) is the log-likelihood of the full model.

The model uses also the Bayesian information criterion (BIC) proposed by Schwarz (1978) defined as

BIC=kln⁡(N)−2LL(β)\mathrm{BIC}=k \ln (N)-2 L L(\beta)

where NN is the number of observations, and kk and LL(β)L L(\beta) are as described in Eq. (5). Note that smaller values of AIC and BIC prove to be better.

In order to quantify the impact of an independent variable on the expected value of a dependent variable, the marginal effect (ME)(M E)

was also used. The ME(MExijλi=∂λi∂xij)M E\left(M E_{x_{i j}}^{\lambda_{i}}=\frac{\partial \lambda_{i}}{\partial x_{i j}}\right) reflects the effect of a oneunit change in xix_{i} on the variable of interest λi\lambda_{i}.

Estimation Results of the Negative Binomial Model

The following variables were considered as potential covariates in the regression model: hourly traffic volume entering the intersection from minor roads (VHPc)\left(\mathrm{VHP}_{c}\right); conflict traffic volume (VHPc)\left(\mathrm{VHP}_{c}\right) at the intersection for the traffic stream of minor roads; and a dummy variable (D)(D), which takes a value of 1 for VHPc≤500veh/h\mathrm{VHP}_{c} \leq 500 \mathrm{veh} / \mathrm{h}, otherwise 0 .

Table 2 presents the estimation results of the NB model with and without the dummy variable DD, based on the average values of delay time in peak hours, as well as the corresponding average marginal effects. Table 2 shows that the estimated coefficients have the expected signs for both models. The delay time is positively associated with hourly traffic volume entering intersection from minor roads (VHPc)\left(\mathrm{VHP}_{c}\right) and conflict traffic volume (VHPc)\left(\mathrm{VHP}_{c}\right). However, the NB model with the dummy variable DD proves better for showing a higher log-likelihood, and improved ρstatistic 2\rho_{\text {statistic }}^{2}, as well as smaller values of AIC and

Fig. 3. Comparison of measured and simulated peak-hour volume, geometric layout of simulation, and location of detectors for Scenario III

Fig. 3. (Continued.)

BIC when compared to the NB model without the variable DD. The estimated overdispersion parameter (α=1/φ)(\alpha=1 / \varphi) is also smaller for the NB regression with the variable DD. Regarding the NB model with the variable DD, which is proposed in the current paper to be better, a coefficient greater than 1(eαD)1\left(e^{\alpha D}\right) is associated with a small conflict traffic volume (VHPc≤\left(\mathrm{VHP}_{c} \leq\right. 500veh/h500 \mathrm{veh} / \mathrm{h} ); while it is 0 for greater conflict traffic volumes (VHPc>500veh/h)\left(\mathrm{VHP}_{c}>500 \mathrm{veh} / \mathrm{h}\right). For this regression model, all variables are statistically significant at the 10%10 \% level (LRT >2.71>2.71 ). Moreover, the regression coefficient is higher for the variable VHPc\mathrm{VHP}_{c} than for VHPe\mathrm{VHP}_{e}. This means that the delay time on minor roads increases more greatly with conflict traffic volume than with approach traffic volume. One possible explanation might be that with increasing conflict traffic volumes, the size of the available gaps that the vehicles of minor roads require in order to pass across the intersection becomes smaller, and as a consequence the delay increases. On the other hand, when more vehicles arrive at the intersection from minor streams, assuming that the conflict traffic volume is kept constant, longer queues might be expected so that the delay on minor roads also increases. The former cause might be more prevalent when compared to the latter. Also, the average marginal effects
have the expected signs. However, the average marginal effect for VHPc\mathrm{VHP}_{c} is found to be higher than that due to VHPe\mathrm{VHP}_{e}, proving that the impact of a change in VHPc\mathrm{VHP}_{c} on the expected delay is greater.

Table 3 presents the estimation results of the binomial negative models with and without the dummy variable DD based on the average values of stop time at peak hours. Table 3 also shows that the stop time is positively associated with VHPc\mathrm{VHP}_{c} and VHPe\mathrm{VHP}_{e} for both the models. However, the NB with the dummy variable appears to be superior for showing an improved log-likelihood, higher ρstatistic 2\rho_{\text {statistic }}^{2}, and smaller values of AIC and BIC when compared to the NB model without the variable DD. For the regression model with the dummy variable DD here proposed, all variables are also noted to be statistically significant at the 10%10 \% level. Moreover, the regression coefficient is higher for the variable VHPc\mathrm{VHP}_{c} than VHPe\mathrm{VHP}_{e}, showing that also the stop time increases more significantly with conflict traffic volume than approach traffic volume. Also, the average marginal effect of VHPc\mathrm{VHP}_{c} is higher than that of VHPe\mathrm{VHP}_{e}.

Table 4 presents the estimation results of the NB model based on the average values of maximum queue length at peak hours. Since in this case the dummy variable DD was not

Table 1. Microsimulation Results Concerning Traffic Performance Measures for Minor Roads

Note: VHP = peak-hour volume.
found to be statistically significant at the 10%10 \% level, only the model containing VHPc\mathrm{VHP}_{c} and VHPe\mathrm{VHP}_{e} is reported here. Table 4 shows that the maximum queue length is positively associated with VHPc\mathrm{VHP}_{c} and VHPe\mathrm{VHP}_{e}. However, the regression coefficient is higher for the variable VHPc\mathrm{VHP}_{c} than for VHPe\mathrm{VHP}_{e}, proving, in contrast with the aforementioned results concerning delay time and stop time, that the maximum length on the minor roads increases more greatly with the approach traffic volume than
with the conflict traffic volume. Also, the marginal effect of VHPc\mathrm{VHP}_{c} is higher than that of VHPc\mathrm{VHP}_{c}.

Model Equations

The equations of prediction models for estimating the average values in peak hours of delay time (s), stop time (s), and maximum queue length (m) are as follows:

Table 2. Estimation Results of Negative Binomial Model without and with the Dummy Variable DD (NB and NBD, Respectively) Based on the Average Values of Delay Time in Peak Hours

Note: LRT == likelihood ratio test; NB == negative binomial; NBD == negative binomial with the dummy variable DD.

Table 3. Estimation Results of Negative Binomial Model without and with the Dummy Variable D(NBD(N B and NBDN B D, Respectively) Based on the Average Values of Stop Time in Peak Hours

Note: LRT == likelihood ratio test; NB == negative binomial; NBD == negative binomial with the dummy variable DD.

Table 4. Estimation Results of Negative Binomial Model Based on the Average Values of Maximum Queue Length in Peak Hours

Note: LRT == likelihood ratio test.

Delay time =exp⁡{−1.29456+[0.45169(VHPe×10−3)=\exp \left\{-1.29456+\left[0.45169\left(\mathrm{VHP}_{e} \times 10^{-3}\right)\right.\right.

+1.78196(VHPc×10−3)+1.84307D]}\left.\left.\quad+1.78196\left(\mathrm{VHP}_{c} \times 10^{-3}\right)+1.84307 D\right]\right\}

Stop time =exp⁡{−3.67458+[0.71952(VHPe×10−3)=\exp \left\{-3.67458+\left[0.71952\left(\mathrm{VHP}_{e} \times 10^{-3}\right)\right.\right.

+2.79849(VHPc×10−3)+3.46372D]}\left.\left.\quad+2.79849\left(\mathrm{VHP}_{c} \times 10^{-3}\right)+3.46372 D\right]\right\}

Maximum queue ParseError: KaTeX parse error: Expected 'EOF', got '\right' at position 81: …}\right)\right.\̲r̲i̲g̲h̲t̲.

+0.56925(VHPe×10−3)]}\left.\left.\quad+0.56925\left(\mathrm{VHP}_{e} \times 10^{-3}\right)\right]\right\}

where VHPe(veh/h)\mathrm{VHP}_{e}(\mathrm{veh} / \mathrm{h}) is the traffic volume entering an unsignalized intersection from minor roads, VHPc(veh/h)\mathrm{VHP}_{c}(\mathrm{veh} / \mathrm{h}) is the conflict traffic volume at the intersection for the traffic stream of minor roads, and DD is a dummy variable, which takes a value of 1 for VHPc≤500veh/h\mathrm{VHP}_{c} \leq 500 \mathrm{veh} / \mathrm{h}, but otherwise 0 .

As an example, using as a possible reference for a comparison the arithmetic mean μ\mu [where μ=∑i=1Nxi/N\mu=\sum_{i=1}^{N} x_{i} / N is the mean of the sample of measurements x1,…,xnx_{1}, \ldots, x_{n} (predicted or computed in simulation) and NN is the size of the sample], one estimates by means of the aforementioned predictive models μdelay time =7.66 s\mu_{\text {delay time }}=7.66 \mathrm{~s} against a mean of 7.49 s computed by using the simulation results; μstop time =7.33 s\mu_{\text {stop time }}=7.33 \mathrm{~s} against a mean of 6.34 s of the simulation data; and finally, μmaximum queue =12.53 m\mu_{\text {maximum queue }}=12.53 \mathrm{~m} against a mean of 11.70 m based on the simulation outcomes. This comparison shows an acceptable level of conformity between the predicted and simulated performance measures.

Recommendations for Future Investigations

Although this paper represents an interesting advancement in the assessment of performance measures (more especially delay time, stopping time, maximum queue) on minor roads of unsignalized intersections, there still remain some points that are worth investigating. Firstly, field measures should also be made for a more consolidated verification of results obtained in the simulation, even if the issue is not whether the simulated performance measures can be replicated in a field study, but rather whether it is possible to assess the performance measures at intersections when the local situations are very complex and researchers require alternative or complementary approaches to the measurements in sites or to the results of the literature. Another point of discussion is the direction to follow for developing predictive models of performance measures at intersections. In this respect, the simulation tool associated with a statistical analysis based on the NB model is shown to be appropriate in the current paper. In the light of these findings, the costs and time necessary for future tests in the field are now more justifiable.

The regression models developed in this paper appear to be useful for many practical applications such as offering assistance to road designers and/or traffic operators in making preventive decisions based on (1) rapidly quantifying the performance modifications at unsignalized intersections due to a variation of traffic demand over time; (2) understanding in a short time the countermeasures for modifying an existing intersection in such a way that appropriate conditions of mobility are ensured; and (3) comparing different design alternatives.

With regard to transferability, the results obtained might be applied to other Italian cities having the same intersection scenarios. However, one should be wary of stating that the results might be used also for cities different from Italian ones because they have been obtained by being based on driving behavior, traffic rules, and geometric and traffic characteristics that are typical of the place studied. In this respect, it should be noted that the use of
microsimulation allows for the local conditions of a given site to be taken into account. Therefore, the general transferability of this paper lies above all in the method followed, which is based on the simulation tool and the use of the simulation outcomes in the NB model for predicting performance measures at intersections.

Summary and Conclusions

This research was motivated specifically by the need to develop a microsimulation approach for assessing the performance measures at unsignalized intersections when the real system appears to be much complex for the measurements in the field. A further point of interest was to single out predictive models of performance measures at intersections. To achieve these goals, data were collected at three intersections of the city of Salerno (Italy), defined by wide geometric configurations containing one or more adjacent single intersections, which were identified as isolated in the simulation (i.e., not affected by upstream or downstream intersections). For these intersection scenarios, a performing process of the microsimulation tool was carried out.

Traffic performance measures expressed in terms of average delay undergone by vehicles on minor roads, which had been computed in the simulation in six 1-h time periods corresponding to the peak traffic volumes, were used in the statistical analysis. A NB regression model, jointly applied to conflict traffic volume (VHPc)\left(\mathrm{VHP}_{c}\right) and traffic volume entering intersections from minor roads (VHPe)\left(\mathrm{VHP}_{e}\right), was used to model the average values of delay time at peak hours. Regression parameters were estimated by the maximum likelihood method and the significance of the covariate was evaluated by using the LRT. The candidate set of variables was VHPc\mathrm{VHP}_{c}, VHPe\mathrm{VHP}_{e}, and a dummy variable (D)(D) that was introduced to test the adequacy of the regression model through the range of hourly conflict traffic volume.

The results show that the average delay on minor roads is positively associated with VHPc,VHPe\mathrm{VHP}_{c}, \mathrm{VHP}_{e}, and DD, and that these variables are all statistically significant. Likewise, the average stop time was found to be positively associated with VHPc,VHPe\mathrm{VHP}_{c}, \mathrm{VHP}_{e}, and DD, and these variables are all statistically significant also in this model.

Additionally, a queuing model was also developed and showed that the maximum queue length is positively linked to VHPc\mathrm{VHP}_{c} and VHPe\mathrm{VHP}_{e}. In this case, the variable DD was not statistically significant. It was found that the delay time and stop time on minor roads increase more with conflict traffic volume (VHPc)\left(\mathrm{VHP}_{c}\right) than approach traffic volume (VHPe)\left(\mathrm{VHP}_{e}\right), while the contrary was obtained for the maximum queue length.

The models developed in this paper appear to produce an acceptable level of conformity between the predicted and simulated performance measures. Therefore, they can be useful for many applications such as for quick estimations of performance modifications due to a variation of traffic demand over time, as well as for the prediction of performances when adjustments of intersections or different alternatives are designed.

Although this paper represents an interesting advancement in the assessment of performance measures at unsignalized intersections, measurements in the field should also be made for a more consolidated verification of results obtained. However, the method followed in the current paper, which is based on the simulation tool associated with the use of the NB model, appears to present considerable potential when the real system is very complex to investigate. Therefore, research needs to be addressed also towards these combined actions for making further developments possible.

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