# Predicting Freeway Work Zone Capacity Distribution Based on Logistic Speed-Density Models

Authors: Chaoru Lu; Jing Dong; Anuj Sharma; Tingting Huang; Skylar Knickerbocker
Journal: Journal of Advanced Transportation
 (2018)

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## Abstract

Speed-volume-density relationship and capacity are key elements in modelling traffic operations, designing roadways, and evaluating facility performance. This paper uses a modified five-parameter logistic model to describe the speed-density relationship. The calibrated speed-density models show that the stop-and-go speed (Vb) and shape parameters (θ1 and θ2) are similar for work zones and the nonwork zone site. Accordingly, an operational capacity prediction method is proposed. To demonstrate the effectiveness of the proposed method, the predicted operational capacities are compared with the field data, Highway Capacity Manual method, the output of WorkZoneQ software, and the ensemble tree approach under different work zone scenarios. Furthermore, a lifetime distribution prediction framework for stochastic capacity of work zones is proposed. The predicted lifetime distribution can well capture the tendency of the observed work zone capacities.


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## Body

## 1. Introduction

The conflict between the aged infrastructure and the continuous growth of traffic demand makes the maintenance and repair activities on roadways become commonplace. These activities usually cause traffic delays and safety concerns [1, 2]. Traffic control and management strategies, such as imposing reduced speed limits and coordinating lane closure schedules, have been applied to alleviate the impact of work zones [3–5]. To properly design work zone management strategies, accurate prediction of work zone capacity is crucial. Numerous statistical and simulation-based methods have been proposed to estimate or predict work zone capacity [3, 6–12]. However, only a few work zone capacity estimation methods were derived from speed-flow relationships [3, 10, 11], although speed, volume, and density relationship have been widely used to estimate the capacity of freeways [13].Moreover, work zones usually induce bottlenecks and cause traffic breakdown. Traffic breakdown does not always occur at the maximum flow rate because of its stochastic nature [14, 15]. Therefore, predicting work zone capacity lifetime distribution, referred to as prebreakdown distribution in some literature, is crucial for transportation agencies to evaluate the traffic flow reliability at the work zone site. Several studies have investigated capacity distributions based on lifetime data analysis [14, 16, 17]. However, none of the existing methods focus on predicting work zone capacity lifetime distribution.In this study, speed-volume-density relationships of both work zones and nonwork zone sites are developed. The work zone capacity is predicted based on the speed-volume-density models and the relationship between free-flow speed and work zone characteristics. The work zone capacity lifetime distribution is predicted based on the capacity distribution before work zone and work zone characteristics.


## 2. Literature Review

In the literature, the definition of capacity can be categorized into operational capacity and stochastic capacity [18, 19]. Operational capacity is usually derived from the maximum flow rate [20], the queue discharge rate [6, 9, 21], and the speed-volume curve [11, 22, 23]. Queue discharge flow rate, which is measured after traffic breakdown, is generally lower than the sustaining flow rate before breakdown [24, 25]. Maximum flow rate is a single measurement and might be unreliable [24, 26]. The speed-volume curve is developed based on observations measured before and after traffic breakdown and describes the characteristics of traffic flow at the study site. Thus, the capacity derived from speed-volume curve is defined as the operational capacity in this paper. Furthermore, Kondyli et al. [24] and Highway Capacity Manual (HCM) 6th edition [27] recommended using prebreakdown flow rate as freeway capacity. However, previous research has shown that prebreakdown flow rates follow probability distributions [17, 19, 28]. In this paper, the distribution of prebreakdown flow rates is defined as stochastic capacity [18, 22].Work zone operational capacity estimation and prediction methods can be categorized into three groups: simulation based, nonparametric, and parametric methods [2]. Microscopic traffic simulation models, such as CORSIM and VISSIM, have been applied to estimate the operational capacity of work zones with different lane closure configurations [29–31]. To replicate real world traffic conditions, the microscopic simulation models need to be calibrated to local conditions, which is usually a tedious and expensive procedure. Nonparametric methods, including neural-fuzzy logic, decision tree, and ensemble tree models, have been used to predict work zone operational capacity [12, 32, 33]. These nonparametric methods usually need extensive historical traffic data to provide reliable prediction. Parametric approaches use predetermined coefficients of the predictors that are calibrated based on the data collected from the work zone site to predict work zone operational capacity. For example, Krammes and Lopez [9] and Kim et al. [8] developed multiregression models to predict short-term work zone operational capacity. Al-Kaisy and Hall [6] and Al-Kaisy et al. [34] proposed a generic multiplicative model to predict the long-term work zone operational capacity based on the traffic data collected from Ontario, Canada. They investigated the effects of grade, the day of week, and weather condition on work zone operational capacity.In addition, Highway Capacity Manual (HCM)6th edition [27] proposed a method to calculate the work zone operational capacity. First, the work zone queues discharge rate is calculated as follows:(1)QDRwz=2093-154×fLCSI-194×fBr-179×fAT+9×fLAT-59×fDNwhere QDRwz is the work zone queue discharge rate (pc/hr/ln); fLCSI is the lane closure severity index, fLCSI=totalnumberoflanes/numberofopenlanes2; fBr is the barrier type, 0 for concrete, 1 for plastic cone or drum; fAT is the area type, 0 for urban areas, 1 for rural areas; and fAT is the lateral distance from the edge of travel lane adjacent to the work zone to the barrier, barricades, or cones; and fDN is 1 for daytime, 0 for night time.Since the unit ofQDRwz is passenger car per hour per lane, the capacity adjustment factor is applied to convert it to vehicle per hour per lane.(2)CAFMIX=CAFAO-CAFT-CAFGwhere CAFMIX is the mixed-flow capacity adjustment factor; CAFAO is the capacity adjustment factor for the auto-only case, which defaults to 1.0; CAFT is the capacity adjustment factor for the percentage of trucks in mixed-flow conditions, CAFT=0.53×TruckPercentage0.72; and CAFG is the capacity adjustment factor for grade for segment j in mixed-flow conditions, which is 0 in this study.Therefore, the work zone operational capacity can be calculated as follows:(3)qc=QDRwz100-αwz×100×CAFMIXwhere αwz is the percentage drop in capacity at the work zone due to queuing conditions (%). The recommended value is 13.4% for freeway work zone. qc is the work zone operational capacity.Over the years, several researchers tried to derive work zone operational capacity from speed-volume-density relationship to provide a reliable estimate [3, 10, 11, 35, 36]. Numerous models have been developed to describe the speed-volume-density relationship, including single-regime and multiregime models. Two parameter single-regime models, such as Greenshields model [37] and Newell’s model [38], usually cannot fit traffic data under congested and uncongested conditions at the same time. Multiregime models, such as Edie model [39], modified Greenberg model [40], and the cluster analysis based model [41], use two or more curves to fit traffic data in different regimes separately. The main challenge of applying multiregime model is to determine breakpoints in a systematic way [42]. To overcome these limitations, MacNicholas [43] proposed a five-parameter logistic speed-density model, which can fit the congested and uncongested regimes using one curve. Wang et al. [42] pointed out that the five-parameter logistic speed-density model outperforms the three-parameter logistic speed-density model and the existing models, such as Greenshields model, Greenberg model, Modified Greenberg model, Van Aerde model, and Newell’s model. Thus, the five-parameter logistic speed-density model is adopted in this paper to fit the traffic data collected at work zone sites.Furthermore, work zone capacity distribution estimation and prediction methods have been introduced in recent years. For example, Weng and Yang [44] proposed a method to determine work zone capacity distribution based on probabilistic two-regime speed-flow relationships. Weng and Yan [45] developed a truncated lognormal distribution model to predict work zone capacity, where the distribution parameters are formulated as a linear function of work zone characteristics. Lu [46] proposed a framework to predict work zone capacity range based on single-regime speed-flow relationships. In addition, previous studies found that traffic flow breakdown occurs with some probability at various flow rates that are lower than the maximum flow rate, referred to as stochastic capacity [14, 15, 47]. A lifetime distribution has been used to describe the probabilistic distribution of prebreakdown capacity. Different factors influencing freeway stochastic capacity have been investigated, including weather [17] and incidents [48]. In this study, the lifetime distribution is used to describe stochastic capacity of work zones.In practice, HCS, QuickZone, QUEWZ, and WorkZoneQ are the commonly used software packages to estimate work zone operational capacity [49, 50]. In addition, the operational capacity estimation method proposed by Highway Capacity Manual 6th (HCM) [27] is also widely used by transportation agencies. In this paper, the results based on WorkZoneQ and HCM are compared with the proposed model. Moreover, the proposed model is compared with the work zone operational capacity prediction model proposed by Weng and Meng[12], which is based on the ensemble tree approach.


### 2.1. Objective and Contribution

The objective of this paper is to predict work zone operational capacity based on speed-volume-density relationship and predict the lifetime distribution of work zone capacity. The operational capacity prediction model captures the relationship between operational capacity and work zone characteristics, considering the impact of work zone on free-flow speed. The stochastic capacity is predicted based on the capacity distribution before work zone and work zone characteristics and is described using a Weibull distribution.The contribution of this paper is two-fold. First, we propose a method to predict the speed-volume-density relationship and the operational capacity under work zone conditions by incorporating work zone characteristics in the five-parameter logistic speed-density model. Second, we propose a method to predict work zone capacity lifetime distribution considering the stochastic nature of flow breakdown. The proposed methods could help traffic engineers to predict work zone capacity and its distribution and design appropriate traffic management strategies to avoid long periods of oversaturation caused by work zones.


## 2.1. Objective and Contribution

The objective of this paper is to predict work zone operational capacity based on speed-volume-density relationship and predict the lifetime distribution of work zone capacity. The operational capacity prediction model captures the relationship between operational capacity and work zone characteristics, considering the impact of work zone on free-flow speed. The stochastic capacity is predicted based on the capacity distribution before work zone and work zone characteristics and is described using a Weibull distribution.The contribution of this paper is two-fold. First, we propose a method to predict the speed-volume-density relationship and the operational capacity under work zone conditions by incorporating work zone characteristics in the five-parameter logistic speed-density model. Second, we propose a method to predict work zone capacity lifetime distribution considering the stochastic nature of flow breakdown. The proposed methods could help traffic engineers to predict work zone capacity and its distribution and design appropriate traffic management strategies to avoid long periods of oversaturation caused by work zones.


## 3. Data Description

Traffic flow rate, speed, and occupancy data were collected by Iowa Department of Transportation (DOT) using Wavetronix radar sensors placed in the work zone areas. Four work zones in Iowa during the 2015 and 2016 construction seasons are investigated in this study.(i)
Work zone data: Traffic volume, speed, and occupancy data were collected when the work zones were active in 2015 and 2016. The date and time of work zones are determined by combining traveler information data from Iowa DOT and the contractor reports and plans.(ii)
Baseline data: Traffic volume, speed, and occupancy data were collected from May to August in 2014 from the sensors located in the same or nearby locations of the work zone sites in Iowa City and Council Bluffs. Traffic data were also collected from May to August in 2015 at a freeway section in Des Moines where no work zone presented. These data are treated as the baseline for the work zones in Quad Cities and Sioux City because the Des Moines location has the same speed limit and similar geometric characteristic as the Quad Cities and Sioux City sites.The configurations and layouts of the work zones are summarized in Table1 and Figure 1, respectively. The sensors are located at the upstream of the merging area in order to measure prebreakdown flow rates [28, 51].Table 1
Summary of work zone configurations.
City Work Zone Truck Percentage Speed limit (mph) Work Type Control Type Barrier Type Area Type Work Time Original Work zone Quad Cities I80 WB 29.63% 65 55 Bridge deck overlay Lane closure Plastic Drum Urban Daylight Quad Cities I80 EB 29% 65 55 Bridge deck overlay Lane closure Plastic Drum Urban Daylight Sioux City I29 NB 13.24% 65 55 Pavement Reconstruction Lane closure and shift Plastic Drum Urban Daylight Iowa City I380 NB 15.97% 70 55 Bridge deck overlay Lane closure /shift Plastic Drum Urban Daylight Council Bluffs I80/I29 WB 17.23% 65 55 Road construction Lane closure /shift Concrete Urban DaylightFigure 1
Work zone layouts.
(a)
Quad Cities (b)
Sioux City (c)
Iowa City (d)
Council BluffsBased on the volume and speed measurements, the density is calculated using(4)k=qVwhere V is the speed (mi/h); k is the density (veh/mile/ln); and q is the volume (veh/hr/ln).To focus on the traffic impact of work zones, the data collected on rainy days were removed. In addition, erroneous measurements were removed from the raw data. In particular, the average effective vehicle length (AEVL) is used to identify anomalies [52]. AEVL is calculated using speed, volume and occupancy collected from Wavetronix sensors [53], as shown in (5). The observations that result in AEVLs out of the normal vehicle length range, namely, 10 to 75 ft (3.048 to 22.86 m) [54], were removed from the dataset. The overall data reduction rate is 24.5%.(5)AEVL=5280×V×Oqwhere AEVL is average effective vehicle lengths (feet); and O is the occupancy.


## 4. Methodology

### 4.1. Operational Capacity

According to MacNicholas’ model [43], the function of general logistic speed-density relationship is expressed as (6)V=Vb+Vf-Vb1+exp⁡k-kt/θ1θ2where Vf is the free-flow speed (mi/h); Vb is the average speed at stop-and-go condition (mi/h); kt is the turning point that the speed–density curve transitions from free-flow to congested flow (veh/mi/ln); θ1 is the scale parameter; and θ2 is the parameter controls the lopsidedness of the curve.By rearranging (6), traffic density can be written as follows:(7)k=kt+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1where ln⁡· is the natural logarithm.By substituting (7) into (4), the speed-volume function is derived as follows:(8)q=ktV+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1VThe relationship between turning point (kt) and inflection point (kIP) in the five-parameter logistic model is written in [42](9)kt=kIP+θ1ln⁡θ2where kIP is the inflection point, where the logistic speed-density curve switches from being concave to convex.Wang et al. [42] pointed out that kt has linear relationship with θ1 and θ2. In order to remove the collinearity between kt and θ1 and θ2, α is introduced. In other words, kt is written as a function of α, θ1, θ2, Vb, and Vf. When θ2 equal 1, kt=kIP=kc [42]. Therefore, we assume(10)kc=kIP+αθ1ln⁡θ2where kc is the density at operational capacity (veh/mi/ln).As a result,(11)Vc=Vb+Vf-Vb1+θ2α-1θ2where Vc is the speed at operational capacity (mi/h).Using the speed-volume relationship, the operational capacity is reached when the following two conditions are met:(12)∂q∂V=∂k∂V·V+k=0(13)∂2q∂V2=∂k∂V+∂2k∂V2·V+∂k∂V<0The derivations of the first condition and the second condition are shown in Appendix A.Based on the first condition (i.e., (12)), the inflection density is derived as follows:(14)kIP=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb-αθ1ln⁡θ2As a result,(15)kt=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2The derivation of (14) is shown in Appendix B.Therefore, the modified five-parameter logistic speed-density relationship and speed-volume relationship are shown as follows:(16)k=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1(17)q=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1VBased on (15), the turning point changes with free-flow speed, average speed at stop-and-go condition, θ1, θ2, and α. θ1 and θ2 are the shape parameters of the speed-density curves. In this study, two assumptions are made:(i)
Vb, θ1, and θ2 remain the same during the work zone as the ones before work zone.(ii)
The same type of work zones has the same value ofα.From the second condition derived from (13), the exponential function is always greater than 0. Since θ1 is positive, when exp⁡ln⁡Vf-Vb/V-Vb/θ2-1×θ2×V-2Vb+V is greater than 0, the condition stated in (13) is satisfied. As a result, we assume that(18)exp⁡ln⁡Vf-Vb/Vc-Vbθ2-1×θ2×Vc-2Vb+Vc>0When Vc≥2Vb, the second condition is satisfied.As a result, the operational capacity can be calculated as follows:(19)qc=max⁡q,Vϵ2Vb,Vf


### 4.2. Stochastic Capacity

In previous research, flow breakdown is identified when speed drops from free-flow speed and the low speed is sustained for a certain time duration (i.e., minimum breakdown duration) [51, 55]. Accordingly, Kim et al. [17] proposed a criterion to classify breakdown and stochastic capacity, as follows:If the speed in time intervali is above a threshold speed and the speed in the next time interval i+1 is below the threshold speed, and the low speed is sustained for at least 15 min, breakdown is assumed to start in time interval i+1, and the flow rate in time interval i is defined as the stochastic capacity.According to Brilon et al. [51], when a breakdown occurred, the prebreakdown flow rate is considered a true estimate of stochastic capacity, which is a censored value. When the breakdown did not occur, the flow rates are created as uncensored values. Considering both the censored and the uncensored values, a survival analysis approach is applied to estimate the lifetime distribution of the capacity using the maximum likelihood estimation (MLE). The likelihood function is defined as follows [56]:(20)L=∏i=1nfqiδi·1-Fqi1-δiwhere n is number of observations; δi is 1, if uncensored, and 0, otherwise; f· is probability density function; and F· is cumulative distribution function.Weibull distribution has been calibrated and suggested for freeway reliability analysis by Al-Deek and Emam [57] and Brilon et al. [51] and is used to fit the observations in this paper.(21)Fq=1-e-q/σswhere σ is scale parameter; and s is shape parameter.The mean of the stochastic capacity distribution is given by(22)Eq=σ·Γ1+1swhere Γ· is the gamma function.To investigate the relationship between the scale parameter and the operational capacity and the relationship between the scale parameter and the mean stochastic capacity, data from the study of [22] and the data collected in Iowa are summarized in Table 2. The mean of stochastic capacity is derived based on (22).Table 2
Operational capacity, mean and parameters of capacity distribution from an existing study and field data.
Source Location s σ Operational capacity(veh/hr/ln) E q (veh/hr/ln) Liu et al. (2011) SR-91 7.55 1950 1550 1831 17.68 2565 2051 2489  Iowa I-35/80 17.96 2238 2150 2173 I-29 13.43 2368 1980 2278 I-380 8.63 2350 2235 2221  Work Zone I80 WB 11.04 1478 1376 1412 I80 EB 15.16 1503 1267 1452 I29 NB 14.45 1475 1206 1423The relationship between scale parameter and operational capacity is plotted in Figure2. The scale parameter and operational capacity is highly correlated, with an R-square of 0.88. As a result, the linear relationship between scale parameter (σ) and operational capacity is written as(23)qc=A·σ+Bwhere A, B are coefficients, which are 0.8729 and -10.888, respectively.Figure 2
Relationship between the scale parameter and the operational capacity.The relationship between mean stochastic capacity andσ is plotted in Figure 3. The scale parameter and the mean stochastic capacity is highly correlated, with an R-square of 0.99. Accordingly, the linear relationship between scale parameter (σ) and mean of stochastic capacity distribution is shown as follows:(24)Eq=C·σ+Dwhere, C, D are coefficients, which are 0.9743 and -22.644 respectively.Figure 3
Relationship between the scale parameter and the mean capacity.According to (24) and (22), the relationship between the shape parameter (s) and the scale parameter (σ) is derived as follows:(25)s=1Γ-10.97436-22.644/σ-1


### 4.3. Work Zone Capacity Distribution Prediction

The work zone capacity and its lifetime distribution prediction framework is proposed, as follows.Step 1.
Estimate the five-parameter logistic model and the capacity distribution using traffic data collected before work zones. The function “nls” in R statistics package [58] is utilized to fit the speed-density curve based on the traffic data.Step 2.
Determine the mean work zone free-flow speed based on HCM (2016):(26)Vf=9.95+33.49×fsr+0.53×fs-5.6×fLCSI-3.94×fBr-1.71×fDN-1.45×fNrwhere fsr is the ratio of the normal speed limit to work zone speed limit; fs is speed limit of work zone (mi/h); and fNr is the number of ramps within 3 miles (4.8 km) upstream and 3 miles (4.8 km) downstream.Step 3.
Determineα based on work zone type.Step 4.
Calculate the work zone operational capacity using Equation (17) and (19). In particular, Vb, θ1, and θ2 are determined in Step 1. α is determined in Step 3. The free-flow speed is determined in Step 2.Step 5.
Calculate the scale parameter of the work zone capacity distribution based on (23) and the operational capacity determined in Step 4.Step 6.
Calculate the shape parameter of the work zone capacity distribution based on (25) and the scale parameter from Step 5.


## 4.1. Operational Capacity

According to MacNicholas’ model [43], the function of general logistic speed-density relationship is expressed as (6)V=Vb+Vf-Vb1+exp⁡k-kt/θ1θ2where Vf is the free-flow speed (mi/h); Vb is the average speed at stop-and-go condition (mi/h); kt is the turning point that the speed–density curve transitions from free-flow to congested flow (veh/mi/ln); θ1 is the scale parameter; and θ2 is the parameter controls the lopsidedness of the curve.By rearranging (6), traffic density can be written as follows:(7)k=kt+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1where ln⁡· is the natural logarithm.By substituting (7) into (4), the speed-volume function is derived as follows:(8)q=ktV+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1VThe relationship between turning point (kt) and inflection point (kIP) in the five-parameter logistic model is written in [42](9)kt=kIP+θ1ln⁡θ2where kIP is the inflection point, where the logistic speed-density curve switches from being concave to convex.Wang et al. [42] pointed out that kt has linear relationship with θ1 and θ2. In order to remove the collinearity between kt and θ1 and θ2, α is introduced. In other words, kt is written as a function of α, θ1, θ2, Vb, and Vf. When θ2 equal 1, kt=kIP=kc [42]. Therefore, we assume(10)kc=kIP+αθ1ln⁡θ2where kc is the density at operational capacity (veh/mi/ln).As a result,(11)Vc=Vb+Vf-Vb1+θ2α-1θ2where Vc is the speed at operational capacity (mi/h).Using the speed-volume relationship, the operational capacity is reached when the following two conditions are met:(12)∂q∂V=∂k∂V·V+k=0(13)∂2q∂V2=∂k∂V+∂2k∂V2·V+∂k∂V<0The derivations of the first condition and the second condition are shown in Appendix A.Based on the first condition (i.e., (12)), the inflection density is derived as follows:(14)kIP=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb-αθ1ln⁡θ2As a result,(15)kt=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2The derivation of (14) is shown in Appendix B.Therefore, the modified five-parameter logistic speed-density relationship and speed-volume relationship are shown as follows:(16)k=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1(17)q=θ1Vb1+θ2α-1θ2+Vf-Vb1+θ2α-1θ2αVf-Vb+1-αθ1ln⁡θ2+θ1ln⁡exp⁡ln⁡Vf-Vb/V-Vbθ2-1VBased on (15), the turning point changes with free-flow speed, average speed at stop-and-go condition, θ1, θ2, and α. θ1 and θ2 are the shape parameters of the speed-density curves. In this study, two assumptions are made:(i)
Vb, θ1, and θ2 remain the same during the work zone as the ones before work zone.(ii)
The same type of work zones has the same value ofα.From the second condition derived from (13), the exponential function is always greater than 0. Since θ1 is positive, when exp⁡ln⁡Vf-Vb/V-Vb/θ2-1×θ2×V-2Vb+V is greater than 0, the condition stated in (13) is satisfied. As a result, we assume that(18)exp⁡ln⁡Vf-Vb/Vc-Vbθ2-1×θ2×Vc-2Vb+Vc>0When Vc≥2Vb, the second condition is satisfied.As a result, the operational capacity can be calculated as follows:(19)qc=max⁡q,Vϵ2Vb,Vf


## 4.2. Stochastic Capacity

In previous research, flow breakdown is identified when speed drops from free-flow speed and the low speed is sustained for a certain time duration (i.e., minimum breakdown duration) [51, 55]. Accordingly, Kim et al. [17] proposed a criterion to classify breakdown and stochastic capacity, as follows:If the speed in time intervali is above a threshold speed and the speed in the next time interval i+1 is below the threshold speed, and the low speed is sustained for at least 15 min, breakdown is assumed to start in time interval i+1, and the flow rate in time interval i is defined as the stochastic capacity.According to Brilon et al. [51], when a breakdown occurred, the prebreakdown flow rate is considered a true estimate of stochastic capacity, which is a censored value. When the breakdown did not occur, the flow rates are created as uncensored values. Considering both the censored and the uncensored values, a survival analysis approach is applied to estimate the lifetime distribution of the capacity using the maximum likelihood estimation (MLE). The likelihood function is defined as follows [56]:(20)L=∏i=1nfqiδi·1-Fqi1-δiwhere n is number of observations; δi is 1, if uncensored, and 0, otherwise; f· is probability density function; and F· is cumulative distribution function.Weibull distribution has been calibrated and suggested for freeway reliability analysis by Al-Deek and Emam [57] and Brilon et al. [51] and is used to fit the observations in this paper.(21)Fq=1-e-q/σswhere σ is scale parameter; and s is shape parameter.The mean of the stochastic capacity distribution is given by(22)Eq=σ·Γ1+1swhere Γ· is the gamma function.To investigate the relationship between the scale parameter and the operational capacity and the relationship between the scale parameter and the mean stochastic capacity, data from the study of [22] and the data collected in Iowa are summarized in Table 2. The mean of stochastic capacity is derived based on (22).Table 2
Operational capacity, mean and parameters of capacity distribution from an existing study and field data.
Source Location s σ Operational capacity(veh/hr/ln) E q (veh/hr/ln) Liu et al. (2011) SR-91 7.55 1950 1550 1831 17.68 2565 2051 2489  Iowa I-35/80 17.96 2238 2150 2173 I-29 13.43 2368 1980 2278 I-380 8.63 2350 2235 2221  Work Zone I80 WB 11.04 1478 1376 1412 I80 EB 15.16 1503 1267 1452 I29 NB 14.45 1475 1206 1423The relationship between scale parameter and operational capacity is plotted in Figure2. The scale parameter and operational capacity is highly correlated, with an R-square of 0.88. As a result, the linear relationship between scale parameter (σ) and operational capacity is written as(23)qc=A·σ+Bwhere A, B are coefficients, which are 0.8729 and -10.888, respectively.Figure 2
Relationship between the scale parameter and the operational capacity.The relationship between mean stochastic capacity andσ is plotted in Figure 3. The scale parameter and the mean stochastic capacity is highly correlated, with an R-square of 0.99. Accordingly, the linear relationship between scale parameter (σ) and mean of stochastic capacity distribution is shown as follows:(24)Eq=C·σ+Dwhere, C, D are coefficients, which are 0.9743 and -22.644 respectively.Figure 3
Relationship between the scale parameter and the mean capacity.According to (24) and (22), the relationship between the shape parameter (s) and the scale parameter (σ) is derived as follows:(25)s=1Γ-10.97436-22.644/σ-1


## 4.3. Work Zone Capacity Distribution Prediction

The work zone capacity and its lifetime distribution prediction framework is proposed, as follows.Step 1.
Estimate the five-parameter logistic model and the capacity distribution using traffic data collected before work zones. The function “nls” in R statistics package [58] is utilized to fit the speed-density curve based on the traffic data.Step 2.
Determine the mean work zone free-flow speed based on HCM (2016):(26)Vf=9.95+33.49×fsr+0.53×fs-5.6×fLCSI-3.94×fBr-1.71×fDN-1.45×fNrwhere fsr is the ratio of the normal speed limit to work zone speed limit; fs is speed limit of work zone (mi/h); and fNr is the number of ramps within 3 miles (4.8 km) upstream and 3 miles (4.8 km) downstream.Step 3.
Determineα based on work zone type.Step 4.
Calculate the work zone operational capacity using Equation (17) and (19). In particular, Vb, θ1, and θ2 are determined in Step 1. α is determined in Step 3. The free-flow speed is determined in Step 2.Step 5.
Calculate the scale parameter of the work zone capacity distribution based on (23) and the operational capacity determined in Step 4.Step 6.
Calculate the shape parameter of the work zone capacity distribution based on (25) and the scale parameter from Step 5.


## 5. Results

### 5.1. Operational Capacity

The estimated traffic speed-volume-density relationships for different work zones are compared with the baseline non-work zone conditions. The calibrated parameters are summarized in Table3.Table 3
Parameters of logistic speed-density models.
Data Source V f(mi/h) V b(mi/h) k t (mi/h) θ 1 θ 2 α Non-work Zone 69.39 5.14 34.95 7.61 0.35 0.489 Work Zone I80 WB 63.74 5.32 21.02 7.53 0.37 -0.263 I80 EB 64.46 5.06 24.54 7.82 0.37 -0.274 I29 NB 51.67 5.17 20.54 7.51 0.38 -0.289Table3 shows that free-flow speeds and turning density are smaller at the work zone sites compared to the ones at the nonwork zone site. The stop-and-go speed (Vb) and shape parameters (θ1 and θ2) are similar for work zones and non-work zone sites. Moreover, the values of α at work zone sites are significantly smaller than the one at the nonwork zone site. However, the values of α at different work zones are similar. Note that all the work zones considered in this study are lane closure ones. Other types of work zones might result in different α values. In the subsequent analysis, α of work zones is set as -0.27.The speed-volume-density relationships before and after work zone started are compared in Figures4 and 5. In order to investigate the relationship between work zone capacity and the free-flow speed, Vb, θ1, and θ2 that are calibrated based on the before work zone data are assumed to remain the same during the work zone. Based on the predicted free-flow speed, the predicted operational capacity is close to the capacity estimated from the work zone data. The predicted speed-volume-density relationship follows the pattern of the field data collected at work zone sites.Figure 4
Speed-volume relationships before and during work zone.
(a)
Council Bluffs (b)
Iowa CityFigure 5
Speed-density relationships before and during work zone.
(a)
Council Bluffs (b)
Iowa CityAdditionally, the estimated and predicted operational capacities are compared with the capacity estimates by WorkZoneQ software, HCM, the maximum 15-minute flow rate, and the operational capacities predicted by the model proposed by Weng and Meng [12]. As shown in Table 4, the modified five-parameter logistic model generates similar results as WorkZoneQ and HCM. The maximum 15-minute flow rate tends to overestimate capacity, compared to logistic model and WorkZoneQ results. The work zone in Iowa City, which has a larger free-flow speed reduction, has a lower operational capacity. The work zones that have similar free-flow speed reduction demonstrate similar capacities. Moreover, the predicted operational capacities based on the proposed method match the estimated values better than the Weng and Meng [12] method.Table 4
Operational capacities from the proposed method, HCM, Weng and Meng method [12], WorkZoneQ and maximum 15-minute flow rate.
City Work Zone Maximum 15-min flow rate (veh/hr/ln) Estimated operational capacity (veh/hr/ln) Predicted operational capacity (veh/hr/ln) HCM Logistic Model WorkZoneQ Proposed method Weng and Meng[12] Quad Cities I80 WB 1572 1239 1267 1296 1252 1383  Quad Cities I80 EB 1488 1245 1206 1296 1318 1383  Sioux City I29 NB 1512 1394 1413 1313 1257 1138  Iowa City I-380 NB 1296 1199 1058 1424 1008 1383  Council Bluffs I80/I29 WB 1524 1517 1376 1313 1281 1138In Figure6, the range of the predicted operational capacity of work zone (i.e., the shaded area) is compared with the results reported in the literature. The predicted capacities tend to be lower than the work zone operational capacities in the literature. One of the reasons is that maximum 15-min flow rate was used as operational capacities in most of the previous studies.Figure 6
Comparison of the predicted capacity and the results from the existing studies.


### 5.2. Capacity Distribution

Based on the predicted operational capacity from Table4, the scale parameter of work zone capacity lifetime distribution can be calculated. The parameters of stochastic capacity distributions before and during work zone started are shown in Table 5. The scale parameters of work zone are significantly smaller than the ones before work zone.Table 5
Parameters of Fitted and Predicted Distribution Functions.
City Before Work zone Estimated Estimated Predicted s σ s σ s σ Iowa City 8.64 2350 7.82 1199 11.56 1167  Council Bluffs 13.44 2368 11.04 1478 13.03 1480The predicted and estimated work zone stochastic capacity distribution are compared, as shown in Figure7. The predicted work zone stochastic capacity distribution captured the tendency of the estimated work zone stochastic capacity distribution well.Figure 7
Predicted and estimated stochastic capacity distributions.
(a)
Iowa City (b)
Council BluffsThe errors in the proposed capacity distribution prediction approach could be attributed to several factors. First, the difference between predicted operational capacity and the ground truth operational capacity may cause errors in predicting the scale parameter of work zone. Second, the data used to calibrate the relationship between scale parameter and shape parameter is limited.


## 5.1. Operational Capacity

The estimated traffic speed-volume-density relationships for different work zones are compared with the baseline non-work zone conditions. The calibrated parameters are summarized in Table3.Table 3
Parameters of logistic speed-density models.
Data Source V f(mi/h) V b(mi/h) k t (mi/h) θ 1 θ 2 α Non-work Zone 69.39 5.14 34.95 7.61 0.35 0.489 Work Zone I80 WB 63.74 5.32 21.02 7.53 0.37 -0.263 I80 EB 64.46 5.06 24.54 7.82 0.37 -0.274 I29 NB 51.67 5.17 20.54 7.51 0.38 -0.289Table3 shows that free-flow speeds and turning density are smaller at the work zone sites compared to the ones at the nonwork zone site. The stop-and-go speed (Vb) and shape parameters (θ1 and θ2) are similar for work zones and non-work zone sites. Moreover, the values of α at work zone sites are significantly smaller than the one at the nonwork zone site. However, the values of α at different work zones are similar. Note that all the work zones considered in this study are lane closure ones. Other types of work zones might result in different α values. In the subsequent analysis, α of work zones is set as -0.27.The speed-volume-density relationships before and after work zone started are compared in Figures4 and 5. In order to investigate the relationship between work zone capacity and the free-flow speed, Vb, θ1, and θ2 that are calibrated based on the before work zone data are assumed to remain the same during the work zone. Based on the predicted free-flow speed, the predicted operational capacity is close to the capacity estimated from the work zone data. The predicted speed-volume-density relationship follows the pattern of the field data collected at work zone sites.Figure 4
Speed-volume relationships before and during work zone.
(a)
Council Bluffs (b)
Iowa CityFigure 5
Speed-density relationships before and during work zone.
(a)
Council Bluffs (b)
Iowa CityAdditionally, the estimated and predicted operational capacities are compared with the capacity estimates by WorkZoneQ software, HCM, the maximum 15-minute flow rate, and the operational capacities predicted by the model proposed by Weng and Meng [12]. As shown in Table 4, the modified five-parameter logistic model generates similar results as WorkZoneQ and HCM. The maximum 15-minute flow rate tends to overestimate capacity, compared to logistic model and WorkZoneQ results. The work zone in Iowa City, which has a larger free-flow speed reduction, has a lower operational capacity. The work zones that have similar free-flow speed reduction demonstrate similar capacities. Moreover, the predicted operational capacities based on the proposed method match the estimated values better than the Weng and Meng [12] method.Table 4
Operational capacities from the proposed method, HCM, Weng and Meng method [12], WorkZoneQ and maximum 15-minute flow rate.
City Work Zone Maximum 15-min flow rate (veh/hr/ln) Estimated operational capacity (veh/hr/ln) Predicted operational capacity (veh/hr/ln) HCM Logistic Model WorkZoneQ Proposed method Weng and Meng[12] Quad Cities I80 WB 1572 1239 1267 1296 1252 1383  Quad Cities I80 EB 1488 1245 1206 1296 1318 1383  Sioux City I29 NB 1512 1394 1413 1313 1257 1138  Iowa City I-380 NB 1296 1199 1058 1424 1008 1383  Council Bluffs I80/I29 WB 1524 1517 1376 1313 1281 1138In Figure6, the range of the predicted operational capacity of work zone (i.e., the shaded area) is compared with the results reported in the literature. The predicted capacities tend to be lower than the work zone operational capacities in the literature. One of the reasons is that maximum 15-min flow rate was used as operational capacities in most of the previous studies.Figure 6
Comparison of the predicted capacity and the results from the existing studies.


## 5.2. Capacity Distribution

Based on the predicted operational capacity from Table4, the scale parameter of work zone capacity lifetime distribution can be calculated. The parameters of stochastic capacity distributions before and during work zone started are shown in Table 5. The scale parameters of work zone are significantly smaller than the ones before work zone.Table 5
Parameters of Fitted and Predicted Distribution Functions.
City Before Work zone Estimated Estimated Predicted s σ s σ s σ Iowa City 8.64 2350 7.82 1199 11.56 1167  Council Bluffs 13.44 2368 11.04 1478 13.03 1480The predicted and estimated work zone stochastic capacity distribution are compared, as shown in Figure7. The predicted work zone stochastic capacity distribution captured the tendency of the estimated work zone stochastic capacity distribution well.Figure 7
Predicted and estimated stochastic capacity distributions.
(a)
Iowa City (b)
Council BluffsThe errors in the proposed capacity distribution prediction approach could be attributed to several factors. First, the difference between predicted operational capacity and the ground truth operational capacity may cause errors in predicting the scale parameter of work zone. Second, the data used to calibrate the relationship between scale parameter and shape parameter is limited.


## 6. Conclusions

The effects of work zones on traffic speed-volume-density curves and the roadway capacity are investigated using traffic data collected on freeways in Iowa, USA. A modified five-parameter logistic model is developed to describe the speed-density relationship. The calibrated speed-density models show that the free-flow speed and turning density are smaller when work zone is active compared to non-work zone conditions. Moreover, based on the logistic speed-density model, an operational capacity prediction method is proposed considering the relationship between free-flow speed and work zone characteristics. The performance of the proposed work zone capacity prediction method is evaluated using field data. The logistic model-based method can predict the speed-density relationship that is close to the estimated one from the field data. The predicted work zone operational capacities are similar to the results from WorkZoneQ and HCM and are generally smaller than the maximum 15-min flow rate. Moreover, the predicted operational capacities based on the proposed method is closer to the estimated values than the one proposed by Weng and Meng [12].Furthermore, a work zone distribution prediction framework is proposed. Based on existing studies and the field data, a linear relationship between the scale parameter(σ) and the operational capacity is established. The predicted capacity distribution can well capture the tendency of the distribution estimated based on the field data.There are some caveats in the present paper. First, only work zones with lane closure are considered in the study. In the future, other types of work zones need to be investigated. Second, only free-flow speed andα are considered as dependent variables that influence the speed-density curve. Future studies should examine the impact of work zone intensity, traffic control type, and road configuration on the other parameters of the logistic model. Third, the linear relationship between scale parameter (σ) and operational capacity is based on the limited data from existing studies and the field data. In the future, comprehensive datasets, including different impacting factors needs to be considered.



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*Source: 9614501-2018-11-01.xml*