Analysis of Work Zone Gaps and Rear-End Collision Probability

Analysis of Work Zone Gaps and Rear-End Collision Probability

DAZHI SUN 1,*
RAHIM F. BENEKOHAL 2

ABSTRACT

This paper studies platooning and headway/gap characteristics of traffic flow in highway short-term and long-term work zones under various car-following patterns. The relationship between traffic volume and the percentage of vehicles in platoons is developed, along with some statistical models for platoon size and headway/gap size distribution. An in-depth analysis of data reveals that vehicles in work zones with higher speed limits maintain shorter car-following time gaps than those in work zones with lower speed limits, even though more time is needed to stop a faster vehicle. This unusual combination of higher speeds and shorter car-following time gaps in work zones may contribute to the high proportion of rear-end collisions among all work zone-related accidents. This paper also presents a new method for evaluating rear-end collision potential, including the probability and the number of vehicles involved in rear-end collisions, by analyzing platoon and gap characteristics for locations without crash records during a construction period.

KEYWORDS: Car-following patterns, rear-end collisions, platoons, work zones.

INTRODUCTION

The number of fatalities in motor vehicle crashes in work zones has risen from 693 in 1997 to 1,181 in 2002 in the United States. Rear-end crashes are one of the most common kinds of work zone crashes and account for more than 30% of all crashes in work zones. By investigating gap characteristics of platooning vehicles in work zones, researchers may be better able to evaluate the risk of rear-end collisions for vehicles in platoons and understand driver behavior in this situation.

In the past, numerous investigations have looked at the headway characteristics of highway traffic, but limited studies exist on gap characteristics, particularly in work zones. Wasielewski (1979) reported on headway characteristics of highway traffic and concluded that the headway distribution was independent of the traffic volume. Luttinen (1992) studied the independence of consecutive headways using geometric bunch size distribution for two-lane highways in Finland. May (1990), Griffiths and Hunt (1991), Mei and Bullen (1993), and Akcelik and Chung (1994) applied different models to study the distribution of time headways in highway and urban traffic flow conditions.

Most existing headway studies investigated normal traffic flow conditions rather than work zone conditions. Work zone traffic flow has different characteristics due to lower speed limits, work activities, lane closures and channelization plans, and other geometric and traffic factors. The presence of queues and platoons of vehicles is more prevalent in work zones than on regular sections of highway. Therefore, it is necessary to explore drivers' car-following behaviors while they are traveling through construction areas.

Benekohal and Sadeghhosseini (1991) and Sadeghhosseini and Benekohal (1995) investigated the platooning and time headway characteristics of highway work zones. They examined the effects of traffic volume on distribution of time headways and on the percentage of platooning vehicles. Although headway characteristics have been used widely in these analyses, gap characteristics provide a better measure of car-following behaviors and safety-related issues. This paper focuses on quantifying the variations of time headways or gaps for different car-following patterns and work zone types and the relationship between car-following characteristics and the accident risks/safety performances for work zones.

To address these issues, we analyzed field data from 11 work zone sites. In the case study, we proposed and implemented a new gap-analysis-based safety performance evaluation methodology for work zones. Work zone safety performance is difficult to evaluate due to the lack of reliable work zone crash data. This new method provides an alternative approach to evaluating accident risk by analyzing crash predisposition under nonaccident situations.

DATA COLLECTION AND REDUCTION

Field data were collected at 11 work zones sites on Interstate highways in Illinois. Three of the sites investigated were short-term work zones and eight were long-term. In this study, a short-term work zone is defined as a construction or maintenance site that lasted less than a few days and the closed lane was delineated using cones, barrels, or barricades (but not barriers). A long-term work zone is defined as a construction or maintenance site that lasted more than a few days and the closed lane was delineated using concrete barriers. The short-term work zones studied had a posted speed limit of 45 mph, while all the long-term work zones had a posted speed limit of 55 mph.

All 11 sites had two lanes in each direction; one lane was closed due to construction and the other was open. A video camera was used to capture the times at which a vehicle passed over two specific markers placed at a fixed distance. The distance between the markers was about 250 feet, but varied for different sites. Data were collected over a time period of two to four hours for each site, depending on the traffic conditions. Initially, the videotapes were time coded. The time coding of the videotapes allowed us to read the travel time more accurately. Time headway, time gap, space, speed, and volume data were obtained from the tapes. The headway for each vehicle was computed based on the time measured at marker 2 (the marker closest to the camera) when the front bumper of a vehicle passed over the line of sight between the camera and the marker. The time headway for the following vehicle was the time difference between the passing of the front bumper of the leading and following vehicles over the line of sight. The gap is the time difference between the passing of the rear bumper of the leading vehicle and the front bumper of the following vehicle over the line of sight. The time measurements are accurate to within 1/30 seconds.

Vehicles were classified into platooning or nonplatooning based on their speed and spacing. A platoon is a group of vehicles traveling close to one other with short headways. The literature gives four definitions of platoons based on either time or space headway, a combination of time headway and speed, or a combination of space headway and speed. Different thresholds in time headway, ranging from 2.5 to 6 seconds, have been used in the past to identify platooning vehicles. Keller (1976) and Benekohal and Sadeghhosseini (1991), for example, used five seconds as the threshold of time headway to separate platooning vehicles from the traffic flow. Sumner and Baguley (1978) used a gap of two seconds and speed differences of less than 10% as the platooning threshold. Horban (1983) suggested four seconds for the time headway threshold in a level-of-service study. Our analysis focuses only on vehicles in platoons and employs data on over 15,000 vehicles to investigate platoon and gap characteristics in work zones.

PLATOONING AND GAP CHARACTERISTICS IN WORK ZONES

Analyses of platooning characteristics, including the percentage of vehicles in platoons and platoon size distributions, are discussed below. Then we analyze gap characteristics for platooning vehicles to determine the effect of different car-following patterns and work zone types on gap size, determine the effect of platoon size on gap size, and establish a gap size distribution.

Platoon Analysis

Percentage of Platooning vs. Volume

Figure 1 shows how the percentage of vehicles in a platoon varies as the traffic volume changes. Two different platooning criteria were applied to examine the effect of the threshold, using four-second and three-second headways.

We constructed 96 sets of data from the initial observation with each corresponding to a 15-minute period of observation. The average traffic volume and percentage of vehicles in the platoon were computed for each set (plotted in figure 1). Figure 1A shows that the percentage of vehicles in the platoon varied from 55% to 75% under low volume conditions (less than 600 vehicles per hour (vph)). The percentage increased to 95% when traffic volume reached about 1,200 vph.

Figure 1B uses a three-second headway as the platooning criteria, which presents a lower percentage of platooning than the four-second headway seen in figure 1A, yet there is still about 43% to 70% platooning at low volume conditions. Only 80% of vehicles were identified as platooning, with a volume of 1,400 vph using the three-second headway criterion. As the volume rises, the percentage increase in platooning slows, indicating that the three-second criterion does not accurately reflect the reality of platooning. Therefore, using a four-second headway as the platooning criterion is more appropriate and provides a greater margin of safety than the three-second headway. The methodology discussed in this paper is independent of the headway threshold; only the numerical values change. As a result, we define a platoon as a group of vehicles separated by a time headway of no longer than four seconds. The remaining discussion of platooning vehicles is based on this definition.

We examined the relationship between traffic volume and percentage of vehicles platooning and found that a logarithmic function fit the data better than other forms. As volume increases, the percentage of platooning vehicles increases and ultimately all vehicles will be considered as part of a platoon. The logarithmic function is expressed as

y = − 1.377 + 0.327 ln(x)    (1)

where

x is the hourly flow rate (vph), 400 ≤ x ≤ 1400, and

y is the percentage platooning (number of vehicles in a platoon/volume).

Platoon Size Distribution

The type of vehicle leading a platoon and the number of vehicles in each platoon were determined. Then platoons were classified into two groups: truck-leading platoons and nontruck-leading platoons. Truck-leading platoons have a large truck at the front of the platoon. Platoons with the same number of vehicles were further grouped into platoon size groups. The relative frequencies of the platoon size groups are shown in figure 2A and figure 2B for short-term and long-term work zones, respectively. These figures show that 70% to 80% of platoons had only two or three vehicles.

Difference models were evaluated to see which of the observed frequencies fit better. The goodness of fit was determined in terms of the root mean square (RMS) error. As a result, a shifted negative exponential function best fit the model in terms of having the least RMS error. For short-term work zones, equations (2) and (3) describe the relationship between platoon size (x) and the percentage of vehicles belonging to that platoon size p(x). Equation (2) represents truck-leading platoons, and equation (3) represents nontruck-leading platoons.

lowercase p (lowercase x) = 1 over 1.6899 exp (negative (lowercase x minus 1.5406) divided by 1.6899) (2)

lowercase p (lowercase x) = 1 over 1.1993 exp (negative (lowercase x minus 1.5075) divided by 1.1993) (3)

Similar relationships were found for the long-term work zones as expressed by equation (4) for truck-leading platoons and equation (5) for nontruck leading platoons.

lowercase p (lowercase x) = 1 over 1.6244 exp (negative (lowercase x minus 1.4853) divided by 1.6244) (4)

lowercase p (lowercase x) = 1 over 1.3 exp (negative (lowercase x - 1.4916) divided by 1.3) (5)

Table 1 shows the platoon size frequency, average headway, and average gap for short-term and long-term work zones. The table shows that more small platoons are led by cars. For example, the relative frequency of nontruck-leading two-vehicle platoons is 0.55 for short-term work zones and 0.52 for long-term work zones, while the relative frequency of truck-leading two-vehicle platoons is only 0.45 for both short-term and long-term work zones.

To determine if the platoon size distributions of the four cases shown in equations (2) through (5) differ significantly, the two most commonly-used two-independent-samples teststhe Mann-Whitney U test and the Kolmogorov-Smirnov z test in SPSSwere applied for the following combinations:

  1. nontruck-leading platoon vs. truck-leading platoon in short-term work zones,
  2. nontruck-leading platoon vs. truck-leading platoon in long-term work zones,
  3. truck-leading platoon in short-term work zones vs. in long-term work zones, and
  4. nontruck-leading platoon-in short-term work zones vs. in long-term work zones.

The results of these tests show that the significance is less than 0.05 for combinations 1 and 2; therefore, the hypothesis H 0: the two distributions are identical is rejected. For combinations 3 and 4, we cannot reject the null hypothesis because the p-value is considerably above 0.05. This indicates that the type of lead vehicle has a significant impact on the platoon size distribution, while the type of work zone does not.

Gap Analysis

To examine car-following safety in work zones, we analyzed the time gap instead of the time headway, because the time gap represents the actual time available for the following car to avoid a rear-end collision. We also studied the effects of the combination of the leader and follower on gap size as well as the effects of the leader of a platoon on platoon size. We determined the gap size distributions and used them to predict the probability of rear-end collisions.

Effect of Car-Following Patterns

Will the gap size be affected by the combination of leader and follower? For instance, a car following a truck may tend to keep a larger time gap than a car following a car. Also, the probability of a rear-end collision may depend on the brake features of the following vehicle and the time gap available to it. To answer our question, we studied the relationship between car-following patterns and time gaps. We investigated the average gaps under different car-following patterns. The four possible car-following patterns analyzed are: car-car, car-truck, truck-car, truck-truck (leader-follower).

Table 2A and 2B detail the mean gap, the mean headway, and the frequency of four car-following patterns in short-term and long-term work zones. These tables show that the average gap is the shortest when a car follows another car. The next shortest gap is when a car follows a truck. The gap is longer when a truck follows a car or a truck. When a truck follows a car or a truck, the gap sizes are not as different as when a car follows a car or a truck. This would appear to indicate that car drivers are more sensitive to what type of vehicles they are following than truck drivers. The table also shows that the gaps in short-term work zones are longer than the gaps in long-term work zones for the same combination of leader and follower. Thus, it is important to know which one of these differences is statistically significant.

This study presents only the aggregate analysis of the mean headway/gap size for different car-following patterns. The data appear to show that each vehicle's car-following behavior is determined primarily by the vehicle directly in front of it, particularly in highway work zones with only one lane open. Of course, other vehicles may also impact driving behavior. The interdependence of different car-following patterns is a complicated problem that would benefit from a more extensive dataset and disaggregate analysis on numerous combinations. Our current dataset does not support this analysis, which we plan to address in a future study.

A "two-sample means z-test" was conducted to evaluate the difference between the mean time gap under different car-following patterns. We compared the gaps in short-term and long-term work zones for the same car-following pattern using a 95% confidence level to test the hypotheses. The results of the tests of 16 different hypotheses are presented in table 3. The z-test shows no significant difference in the time gap between truck-truck and car-truck following patterns in either short-term or long-term work zones. This further supports our findings in table 2. All the other null hypotheses were rejected indicating that, with a 95% confidence level, there is a significant difference in the time gap.

Safety Paradox

The analysis in the previous section shows significantly smaller time gaps for all car-following patterns in long-term work zones with a speed limit of 55 mph compared with gaps in short-term work zones with a speed limit of 45 mph. Although the measured average speeds in the two types of work zones that post the same speed limit vary slightly, the average speeds of nonplatoon and platooning vehicles in long-term work zones tend to be significantly higher than those in short-term work zone. For example, the measured average speed for a nonplatoon vehicle was 42.39 mph, with platooning vehicles averaging 39.80 mph in short-term work zones. In long-term work zones, the measured average speed was 53.29 mph for nonplatoon vehicles, and 50.78 mph for platooning vehicles. Table 2 shows that for a car-car pattern, the average time gap for vehicles with an average speed of 39.0 mph was 1.610 seconds, while the gap for vehicles with an average speed of 50.78 mph was 1.384 seconds. That is, the time gap decreased by 14% at an average speed increase of 22%.

The above numbers indicate a safety paradox: even though people know they need a greater safety buffer when they are driving at a higher speed, our data show that the actual time gap they maintained in a higher speed work zone was significantly shorter than that maintained in a lower speed work zone. The problem may be that people do not recognize this reduction in the safety buffer in terms of the time gap. Rather, people may judge their safety buffer in terms of the space gap. For example, for the car-car following pattern, the average space gap in a long-term work zone (103 feet) was still greater than that for a short-term work zone (94 feet), but the real available time gap for the vehicle traveling in the long-term work zone was reduced by 22%.

The Effect of Platoon Size

The number of vehicles involved in a rear-end collision depends on the size of the platoon. The discussion here is limited to platoons of seven vehicles or less, because we had only a few observations for larger platoons. The variation in the time gap with respect to platoon size is illustrated in figure 3.

Figure 3 shows that in short-term work zones, gap size generally declines for truck-leading platoons as the platoon size increases, except for the slight upturn as the platoon size increases from six to seven. The declining trend also exists for truck-leading platoons in long-term work zones, although the trend is not as clear as in short-term work zones. For car-leading platoons, the average gap size does not seem to depend on platoon size. For platoons consisting of two or three vehicles, we found the average gap size of truck-leading platoons was greater than that of car-leading platoons in short-term and long-term work zones. For platoons with four or more vehicles, the gap sizes of car-leading platoons were, in general, greater than those of truck-leading platoons in short-term work zones, whereas no significant difference was observed for long-term work zones.

The above findings make it clear that as the platoon size of truck-leading platoons increases, drivers tend to follow more closely and thus become more vulnerable to a rear-end crash.

Gap Size Distribution

Gap size can better measure the probability of a rear-end crash than headway. The observed mean gap size was 1.73 in short-term work zones and 1.49 in long-term work zones. The median gap size was 1.68 and 1.42 for short-term and long-term work zones, respectively. In order to find the gap size distribution, we grouped the observed gaps into intervals of 0.25 seconds, starting from 0 seconds and ending at 4 seconds. Figure 4 is a relative frequency histogram of gap sizes for short-term and long-term work zones.

To find equations for the gap size distributions, the 10 widely applied mathematical models were assessed using BestFit with respect to RMS errors. Weibull offers the best fitted function followed by the BetaGeneral function. Gamma, InvGauss, and log-normal ranked as the third, fourth, and fifth best-fitted model, respectively. The appendix presents a brief introduction to these five models to show the density function as well as the model parameters.

The probability distribution function (PDF) of Weibull is

lowercase f (lowercase x) = lowercase sigma divided by lowercase alpha ((lowercase x minus lowercase mu) divided by lowercase alpha) superscript {lowercase sigma minus 1} exp (negative ((lowercase x minus lowercase mu) divided by lowercase alpha) superscript {lowercase gamma})

      xμ; γ, α > 0          (6)

where

γ is the shape parameter; γ = 2.396 for short-term work zone and γ = 2.051 for long-term work zone;

μ is the location parameter; μ = 0.215 for short-term work zone and μ = 0.286 for long-term work zone; and

α is the scale parameter; α = 1.716 for short-term work zone and α = 1.373 for long-term work zone.

The cumulative distribution function (CDF) of the three-parameter Weibull is

uppercase f (lowercase x) = 1 minus exp (negative ((lowercase x minus lowercase mu) divided by lowercase alpha) superscript {lowercase gamma})  for lowercase x greater than or equal to lowercase mu (7)

For short-term work zones, the PDF and CDF of the resulting Weibull models are

lowercase f (lowercase x) = 2.396 divided by 1.716 ((lowercase x minus 0.215) divided by 1.716) superscript {1.396} dot exp (negative ((lowercase x minus 0.215) divided by 1.716) superscript {2.396})  for x greater than or equal to 0.215 (8)

uppercase f (lowercase x) = 1 minus exp (negative (lowercase x minus 0.215) divided by 1.716) superscript {2.396})  for lowercase x greater than or equal to 0.215 (9)

For long-term work zones, the PDF and CDF of fitted Weibull models are

lowercase f (lowercase x) = 2.051 divided by 1.373 ((lowercase x minus 0.286) divided by 1.373) superscript {1.051} dot exp (negative ((lowercase x minus 0.286) divided by 1.373) superscript {2.051})  for lowercase x greater than or equal to 0.289 (10)

uppercase f (lowercase x) = 1 minus exp (negative ((lowercase x minus 0.286) divided by 1.373) superscript {2.051})  for lowercase x greater than or equal to 0.286 (11)

ESTIMATION OF SAFETY PERFORMANCE USING GAP AND PLATOONING ANALYSIS

Nearly one in three work zone crashes are rear-end crashes. Rear-end crashes in a work zone can occur when a vehicle suddenly decelerates due to an unexpected situation. The next section looks at the probability of one or more collisions as a vehicle suddenly decelerates.

Probability of at Least One Rear-End Collision

The risk of a rear-end collision is relative to the time gap, platoon size, and position of the problem vehicle in a platoon. In this section, we develop a model to compute the probability of rear-end collisions when a platooning vehicle suddenly decelerates.

The probability, (p) that a platooning vehicle has a less than critical gap can be obtained via the gap relative frequency histogram (figure 4) or the fitted Weibull CDF (equations (9) and (11)) introduced above. For example, if the critical gap takes a value of 1.0 seconds, we obtain a probability of 0.23 according to equation (11).

Here, we define a rear-end collision as the collision of a following vehicle with the leading vehicle due to an unsafe gap, when the leading vehicle suddenly decelerates. There are at most (i 1) rear-end collisions as the first vehicle in a platoon of size i makes a sudden deceleration.

Figure 5 shows our calculation of the probability of at least one rear-end collision as the jth vehicle of a platoon of i vehicles suddenly decelerates or stops. For a platoon of j + 1 vehicles, the probability of having a rear-end collision is p; and the probability of no rear-end collision is 1 p. For a platoon of j + 2 vehicles, the probability of one rear-end collision is 2 p (1 − p); the probability of two rear-end collisions is p 2; and the probability of no rear-end collisions is 1 − p 2 − 2p (1 − p). For a platoon of j + 3 vehicles, the probability of one rear-end collision is C31p (1 − p)2 = 3p (1 − p)2; the probability of two rear-end collisions is C32p 2 (1 − p)2 = 3p 2 (1 − p); the probability of three rear-end collisions is C33p 3; and the probability of no rear-end collisions is 1 − 3p (1 − p)2 − 3p 2 (1 − p) − p 3.

As such, for a platoon of i (where ij) vehicles, the probability of a rear-end collision is

uppercase c superscript {lowercase i minus lowercase j} subscript {1} times lowercase p (1 minus lowercase p) superscript {lowercase i minus lowercase j minus 1} plus uppercase c superscript {lowercase i minus lowercase j} subscript {2} times lowercase p superscript {2} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus 2} plus ... plus uppercase c superscript {lowercase i minus lowercase j} subscript {lowercase i minus lowercase j} times lowercase p superscript {lowercase i minus lowercase j} (1 minus lowercase p) superscript {0} 

Thus, we are able to generalize the equation for calculating the rear-end collision probability when the problem vehicle is the jth vehicle in a platoon of size i as

lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) = {summation from lowercase m = 1 to (lowercase i minus lowercase j) of (lowercase i minus lowercase j) factorial divided by ((lowercase i minus lowercase j minus lowercase m) factorial lowercase m factorial) lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m} if all lowercase i element of uppercase n, 0 otherwise} (12)

The probability of no rear-end collision is

lowercase p (lowercase y bar superscript {lowercase j} subscript {lowercase i}) = {1 minus summation from lowercase m = 1 to (lowercase i minus lowercase j) of (lowercase i minus lowercase j) factorial over ((lowercase i minus lowercase j minus lowercase m) factorial lowercase m factorial) lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m} if all lowercase i element of uppercase n, 1 otherwise} (13)

where

lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) is the probability that at least one rear-end collision occurs as the jth vehicle of a platoon of size i makes a sudden deceleration.

lowercase p (lowercase y bar superscript {lowercase j} subscript {lowercase i}) is the probability that no rear-end collision occurs as the jth vehicle of a platoon of size i makes a sudden deceleration.

p is the probability that a vehicle has a gap less than the critical gap value (probability of having a rear-end collision);

j is the position of the problem vehicle in the platoon;

i is the platoon size;

m is the number of rear-end collisions that take place; and

N is the vector of all possible platoon sizes [2,3,4...].

When a nonplatooning vehicle keeps a headway of at least four seconds, the probability of rear-end collisions due to the sudden deceleration of a nonplatooning vehicle is considered to be zero (p (y1) = 0).

Probability that the jth Vehicle in a Platoon is a Problem Vehicle

The number of vehicles involved in a rear-end collision is relative to the platoon size and the position of the problem vehicle in the platoon. For example, if the leading car in a longer platoon is involved in an accident, the probability of a multivehicle rear-end collision is higher than that when any other vehicle in the platoon is involved.

Equation (14) gives the probability that a problem vehicle is in a platoon of size i:

lowercase p (lowercase x subscript {lowercase i}) = lowercase i lowercase psi (lowercase i) divided by uppercase v (14)

where

p (x i) is the probability that the problem vehicle belongs to a platoon of size i;

ψ (i) is the number of platoons of size i, which is obtained from figures 1 and 2 for a given volume; and

V is the traffic volume.

The problem vehicle has an equal chance of being at any position within a given platoon. Thus, equation (15) was constructed to represent the probability that the problem vehicle is the jth vehicle in a platoon of size i:

lowercase p (lowercase x superscript {lowercase j} subscript {lowercase i} = lowercase p (lowercase x subscript {lowercase i}) divided by lowercase i = (lowercase i dot lowercase psi (lowercase i)) divided by (lowercase i dot uppercase v) = lowercase psi (lowercase i) divided by uppercase v for all lowercase j less than or equal to lowercase i (15)

Finally, equation (16) was developed to calculate the probability of having rear-end collision(s) when any vehicle in a traffic flow makes a sudden deceleration:

uppercase p (rear-end collisions) = summation from lowercase i = 1 to uppercase n of summation from lowercase j = 1 to lowercase i of lowercase p (lowercase x superscript {lowercase j}) subscript {lowercase i} lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) (16)

As the probability of a rear-end collision caused by the sudden deceleration of a nonplatooning vehicle is considered to be zero (p (y1) = 0), equation (16) can be modified:

uppercase p (rear-end collisions) = summation from lowercase i = 2 to uppercase n of summation from lowercase j = 1 to lowercase i of lowercase p (lowercase x superscript {lowercase j}) subscript {lowercase i} lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) (17)

where

p(rear-end collisions) is the probability of one or more rear-end collisions as a problem vehicle suddenly decelerates;

lowercase p (lowercase x superscript {lowercase j} subscript {lowercase i}) is the probability that the jth vehicle in a platton of size i is a problem vehicle, which is given by equation (15);

lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) is the probability of rear-end collisions when the jth vehicle in a platton of size i suddenly decelerates, which is given by equation (12).

Number of Vehicles in Rear-End Collisions

In order to predict the number of vehicles involved in rear-end collisions caused by the sudden deceleration of a problem vehicle in a work zone, it is necessary to know the mean number of vehicles (κi) involved in rear-end collisions for each platoon size i. We also need to know the probability (pi) that the problem vehicle belongs to a platoon of size i. The mean number of vehicles in rear-end collisions can be computed by summation with lowercase i element of uppercase n of lowercase p subscript {lowercase i} lowercase kappa subscript {lowercase i}.

Finding κi

For a particular platoon (iN), the number of vehicles involved in rear-end collisions depends on the position of the problem vehicle and platoon size. All vehicles in the platoon have an equal chance of being the problem vehicle, but each has a different number of following vehicles.

The number of vehicles involved in rear-end collisions also depends on the type of collision. If m rear-end collisions are continuous, there will be m + 1 vehicles involved. On the other hand, if these rear-end collisions are discrete, there will be at most 2m vehicles involved. To make a comprehensive prediction, we used the average value (m + 1 + 2m)/2 as the number of vehicles involved in m rear-end collisions.

We defined κ ji as the mean size of the rear-end collision if the problem vehicle is the jth vehicle in a platoon with i vehicles. Assuming that the problem vehicle is the first vehicle in a platoon, the following examples demonstrate how to find κ1i.

A platoon of two vehicles involves only one possible rear-end collision, thus κ 12 = 2.

For a platoon of three vehicles, the probability of a rear-end collision is 2 p(1 − p); a probability of having two rear-end collisions is p2; thus

lowercase kappa superscript {1} subscript {3} = (2 times 2 times lowercase p (1 minus lowercase p) plus 3 times lowercase p superscript {2}) divided by (2 times lowercase p (1 minus lowercase p) plus 3 times lowercase p superscript {2}) 

Likewise, for a platoon of four vehicles, the probability of having a rear-end collision is

C31p(1 − p)2 = 3p(1 − p)2;

the probability of having two rear-end collisions is

C32p2 (1 − p)2 = 3 p2 (1 − p);

the probability of having three rear-end collisions is

C33 = p3;

thus

lowercase kappa superscript {1} subscript {4} = (2 uppercase c superscript {3} subscript {1} lowercase p (1 minus lowercase p) superscript {2} plus 3.5 uppercase c superscript {3} subscript {2} lowercase p superscript {2} (1 minus lowercase p) plus 4 uppercase c superscript {3} subscript {3} lowercase p superscript {3}) divided by (uppercase c superscript {3} subscript {1} lowercase p (1 minus lowercase p) superscript {2} plus uppercase c superscript {3} subscript {2} lowercase p superscript {2} (1 minus lowercase p) plus uppercase c superscript {3} subscript {3} lowercase p superscript {3}).

We estimate the equation for κ1i as follows:

lowercase kappa superscript {1} subscript {lowercase i} = (summation from lowercase m = 1 to (lowercase i minus 1) of ((3 lowercase m plus 1) divided by 2) uppercase c superscript {lowercase i minus 1} subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus 1 minus lowercase m}) divided by (summation from lowercase m = 1 to (lowercase i minus 1) uppercase c superscript {lowercase i minus 1) subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus 1 minus lowercase m}) (18)

Similarly, the general formula for the mean number of vehicles in the rear-end collision when the jth vehicle is the problem vehicle in a platoon of size i is

lowercase kappa superscript {lowercase j} subscript {lowercase i} = (summation from lowercase m = 1 to (lowercase i minus lowercase j) of ((3 lowercase m plus 1) divided by 2) uppercase c superscript {lowercase i minus lowercase j} subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m}) divided by (summation from lowercase m = 1 to (lowercase i minus lowercase j) uppercase c superscript {lowercase i minus lowercase j) subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m}) (19)

Now, we can compute κ i from the following equation:

lowercase kappa subscript {lowercase i} = 1 divided by (lowercase i minus 1) summation from lowercase j = 1 to (lowercase i minus 1) of lowercase kappa superscript {lowercase j} subscript {lowercase i} (20)

Therefore, the mean number of vehicles involved in a crash caused by sudden deceleration can be obtained from

lowercase kappa = summation with lowercase i element of uppercase n of lowercase p subscript {lowercase i} lowercase kappa subscript {lowercase i} (21)

The pi can be calculated easily using the percentage of platooning and the platoon size distribution.

CASE STUDY

Our case study attempts to predict the probability of rear-end collisions and the mean number of vehicles involved at a long-term work zone. We developed equations to make the prediction using two input variables. The input variables are: 1) work zone type (long-term or short-term), and 2) hourly volume.

Assume that there is a sudden deceleration in a work zone traffic flow, the proposed methodology presented here can be used to answer the following questions:

  1. What is the probability of a rear-end crash?
  2. How many vehicles might be involved in this crash?

The prediction uses equations (1), (5), (11), (17), and (21) to answer the above questions. Predictions are for a long-term work zone with volumes of 400 to 1,600 vehicles per hour at increments of 200.

Solution to Question 1

Assume the maximum platoon size is 15 vehicles.

uppercase p (rear-end collissions) = summation from lowercase i = 2 to 15 of summation from lowercase j = 1 to (lowercase i minus 1) of lowercase p (lowercase x superscript {lowercase j} subscript {lowercase i}) lowercase p (lowercase y superscript {lowercase j} subscript {lowercase i}) = summation from lowercase i = 2 to 15 of summation from lowercase j = 1 to (lowercase i minus 1) of summation from lowercase m = 1 to (lowercase i minus lowercase j) of (lowercase psi (lowercase i) divided by uppercase v) (lowercase i minus lowercase j) factorial divided by ((lowercase i minus lowercase j lowercase m) factorial lowercase m factorial) lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m}

In a long-term work zone, with the assumption of a critical gap of 1.0 seconds, equation (11) gives us a p of 0.23. This is the conditional probability of having a rear-end collision given a sudden stop or deceleration of a platoon vehicle due to an incident, error maneuver, or some other unexpected reason. This probability may seem high; however, this rear-end collision probability is defined differently from the frequency of rear-end collisions in accident statistics. To get a real overall probability or frequency of real rear-end collisions on a given highway, this probability must be multiplied by the sum probability of all other types of accidents involving only a single vehicle at this location.

Using equations (1) and (7), and the average platoon size μ as 3.2, we can compute ψ(i) from

lowercase psi (lowercase i) = uppercase v divided by lowercase mu dot [negative 1.377 plus 0.327 ln (uppercase v)] dot {1 divided by 1.4079 exp [negative (lowercase x minus 1.4856) divided by 1.4079]}

Now we can compute the conditional probability of rear-end collisions if one vehicle suddenly stops or decelerates. Figure 6A shows that the mean probability of a rear-end collision in a long-term work zone is 18.74% if a vehicle suddenly decelerates or stops. The results also show that the risk of rear-end collisions increases as the volume increases.

Solution to Question 2

We also calculated the mean number of vehicles involved in rear-end collision(s) for different platoon sizes:

lowercase kappa subscript {lowercase i} = 1 divided by (lowercase i minus 1) summation from lowercase j = 1 to (lowercase i minus 1) of ((summation from lowercase m = 1 to (lowercase i minus lowercase j) of (3 lowercase m plus 1) divided by 2 uppercase c superscript {lowercase i minus lowercase j} subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m}) divided by (summation from lowercase m = 1 to (lowercase i minus lowercase j) of uppercase c superscript {lowercase i minus lowercase j} subscript {lowercase m} lowercase p superscript {lowercase m} (1 minus lowercase p) superscript {lowercase i minus lowercase j minus lowercase m}))

Figure 6B shows that the mean number of vehicles, κi, will increase from 2.0 to 3.6 when the platoon size grows from 2 up to 15. The figure also shows that the mean number of vehicles involved for overall traffic will increase from 2.0 to 2.1 as the maximum platoon size of a traffic flow grows from 2 to 15. Obviously, the change in κ is not significant while the maximum platoon size changes significantly, even though the possible mean number of involved vehicles, κi, for a platoon size of 15 is about 1.8 times that for a platoon size of 2. This implies that using only the mean value may be misleading when we want to understand safety performance in work zones, because the change in the maximum platoon size will make a significant difference in the consequence of the worst case.

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

This paper presents an investigation of the platooning and gap characteristics in Interstate highway work zones, as well as of the gap sizes under different car-following patterns and work zone types. The study is based on data covering more than 15,000 observations. Models of the platoon size and gap size distribution for long-term and short-term work zones were developed. An in-depth analysis of the data reveals a safety paradox, which may indicate that drivers do not understand the safety implications of time and space gaps relative to speed limit increases at work zones. All the findings with respect to car-following characteristics provide practitioners a better understanding of drivers' behaviors in work zone areas.

We propose a new methodology to predict the probability of rear-end collisions in a work zone and the mean number of vehicles involved. Only two simple inputs are required to predict rear-end collisions using gap and platooning models. Because it is sometimes impossible to evaluate work zone safety performance using real crash data, this new methodology provides an alternative approach to assessing the safety performance in Interstate highway work zones. We present a case study to demonstrate the implementation of the new prediction methodology.

Some areas for future research include integrating the effect of heavy vehicle and work activity intensity on safety as an interesting extension to our methodology. It will also be important to conduct some disaggregate analysis to address the interdependence of different car-following patterns. A study of this nature may need to consider the impact of various groups of drivers, such as age group, gender, driving habits, etc., which may require more extensive data collection.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous referees for their helpful comments during the development stage of this paper.

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ADDRESSES FOR CORRESPONDENCE

Corresponding author: D. Sun, Texas Transportation Institute, Texas A&M University System, 1100 NW Loop 410, Suite 400, San Antonio, TX 78213. E-mail: d-sun@tamu.edu

R.F. Benekohal, Newmark Civil Engineering Laboratory, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 6180. E-mail: rbenekoh@uiuc.edu