Analysis of Time Headways on Urban Roads: A Case Study from Riyadh

Transcription

1 1 Analysis of Time Headways on Urban Roads: A Case Study from Riyadh By Ali S. Al-Ghamdi Associate Professor, King Saud University, College of Engineering, P.O. Box 800, Riyadh 11421, Saudi Arabia asghamdi@ksu.edu.sa Phone # (966-1) Fax # (966-1) Abstract: The headway between vehicles in a traffic stream is of fundamental importance in traffic engineering applications. Previous research in this subject has focused on modeling theoretical distributions for low and medium traffic flow conditions. Yet little research has studied congested traffic conditions, that is, the high traffic flow state. In the same context, there appears to be a lack of clear-cut boundaries for the three flow states (low, medium, and high). This study attempts to determine such boundaries on the basis of traffic conditions observed at the study sites. Although observed headways at arterial sites follow a gamma distribution, distributions that fit freeway headways differ according to the traffic flow state. The Erlang distribution provided a good fit to the observed headways at sites with high traffic flows. Background Headway is defined as the time that elapses between consecutive vehicles. The distribution of these headways has been studied, with the primary interest being application to capacity estimation, safety analysis, and generation of vehicles in microscopic simulations.

2 2 Several models have been proposed for the distribution of headways, for example, Buckley`s semi-poisson model (1968), Cowan`s M3 distribution model (1975), the log-normal distribution (Mei and Bullen 1993), and the double displaced negative exponential distribution (Sullivan and Troutbeck 1994). Al-Ghamdi (1999b) found that the gamma and shifted exponential distributions describe headways for low and medium traffic flow conditions. He suggested that headway data not be combined from different sites to find the proper distribution. Instead, the data should be collected from one site to cancel out the variability related to different sites. The study found that one would use the minimum allowable headway from the observed data for the parameter of the shifted exponential distribution. Although there have been many studies of distributions at low and medium traffic flow rates, very few studies have been devoted to headway distribution during high flow, in which all vehicles are in the car-following state. This study aims at analyzing headway data for low, medium, and high flow states. Since a clear-cut definition for these three flow states does not exist, the study devotes a portion of the analysis to defining the boundaries of these states, in other words, establishing limits in terms of traffic flow rate for each flow level. Data Characteristics Headway data (time from front bumper of one vehicle to front bumper of the following vehicle in units of 1/100 s) relating to single lanes of traffic were collected from 20 urban sites (13 freeways and 7 arterials) throughout Riyadh, Saudi Arabia. The TDC-8 device (User`s Manual 1997) was used together with a program written in Basic for data summary. All the selected road sections were reasonably level and straight, with no steep hills, intersections, traffic

3 3 signals, or median openings. The speed limits ranged from 50 to 90 km/h for the arterial sites and from 100 to 120 km/h for the freeway sites. All the samples were collected during daytime and normal weather conditions. Samples with 5 percent or more of heavy vehicles were excluded from further analysis. The data that passed the preliminary screening stage of this study consisted of 10,122 observations (6,904 freeway observations and 3,218 arterial observations). Since the study aims at analyzing headways over a full range of volumes, the data were initially collected during periods ranging from 15 to 35 min, depending on the traffic demand at the site. The flow rates in vehicles per hour (vph) were then calculated. Tables 1 and 2 summarize the study data on freeways and arterials, respectively. Methodology and Analysis To decide which of a significant number of candidate distributions to fit to the headway data collected, the following criteria were applied: The data collected at each site should be statistically sufficient for distribution fitting. The fitting process should take place at each flow level on the basis of 100- or 200-vph increments so that any change in data behavior with respect to the fitted distributions could be detected. For practicality, the fitted distribution should be as simple as possible. Taking these criteria into consideration, the chi-square testing technique was used to obtain the best fit among the distributions attempted. Although there are other goodness-of-fit tests, including the Kolmogorov-Smirnov, Anderson-Darling statistic test, and other nonparametric tests, the author believes that each of the study samples was large enough to rely

4 4 on the chi-square goodness of fit. Many theoretical distributions were attempted ranging from simple to complex ones. A list of candidate distributions was initially developed from previous research (e.g., Gerlough and Huber 1975; Luttinen 1992). Sometimes, but not too frequently, several distributions were found to reasonably fit the observed data; however, the one that gave the minimum chi-square values was selected. Grouping of distributions with respect to flow levels was attempted in this study. This grouping was helpful to roughly define the boundaries for the three traffic flow states low, medium, and high for traffic conditions in Riyadh. The interpretation of the pattern of the parameters for the fitted distributions obtained in the study was of great value in defining such boundaries, as will be seen in the following analyses. Freeway Data After data were collected at each site, they were subjected to analysis in order to determine the proper distribution that reasonably described the data. The data set for each site ranged from 122 observations to 967 observations. Each data set was subjected to chi-square analysis. An example of this test procedure is presented in Table 3 and Figure 1. Table 4 presents a summary of distributions fitted to observed headways for all freeway sections. These distributions cover the flow range from less than 400 vph to greater than 2,200 vph. A review of the fitted distributions in Table 4 suggests that three distributions can classify the range of flow involved. The first the negative exponential distribution is for flow rates less than 400 vph. The next two are the shifted exponential and gamma distributions and include flow rates of 400 to 1,200 vph. The last distribution is the Erlang, for flows above 1,200 vph. It should be noted that a large flow volume (above 2,200 vph) is not surprising in Riyadh. This

5 5 large volume is due to aggressive driving behavior. That is, motorists tend to follow other vehicles closely in an aggressive manner, which in turn increases the traffic flow rate (Al-Nafa 1988; Al-Ghamdi 1998; Al-Ghamdi 1999b). According to the type of grouping just described, the three flow states, as described in the literature, may be defined as follows: Low flows: Medium flows: High flows: less than 400 vph 400 to 1,200 vph above 1,200 vph From the results reached in this study, Table 5 presents the suggested distribution family that properly fits the headways observed at each of the foregoing three flow levels. The term ``family`` is used to emphasize the different parameter values for the fitted distribution. For example, the shifted exponential distribution is suggested to fit medium traffic flows, but the values of parameters α and β would vary at different flow levels. It should be mentioned that Al- Ghamdi (1999b) found that the shifted exponential distribution provided a decent fit to the observed freeway headways. Also, Buckley (1968) and Gerlough and Huber (1975), on the basis of data from the Hollywood Freeway, noted that although the shifted exponential distribution fits data for low flows, it is not suitable for high flows. It can be noted from Table 4 that when traffic increases for the medium flow state, the parameter values of the shifted exponential distribution decrease, as shown in Figure 2. However, the k-parameter for the Erlang distribution seems constant. The value of k, a parameter that

6 6 determines the shape of the distribution, may be estimated from the mean and variance of the observed data: 2 µ kˆ = 2 σ where µ = mean of the observed headways, and σ = standard deviation of the observed headways. The value of k should be rounded off to the nearest integer when it is used with the Erlang distribution. It should be said that when k = 1, the Erlang distribution reduces to an exponential distribution, which reasonably fit the headways for flow below 1,200 vph (most of which is said to in a be random state). Gerlough and Huber (1975) state that ``the values of k may be a rough indication of the degree of randomness. When k = 1, the data appear to be random; as k increases, the degree of non-randomness appears to increase.`` The data in the current study appear partially consistent with this context. From Table 4, it appears that the values of k (4 and 5) are all far from 1, and thus far from random as Gerlough and Huber (1975) state. However, k seems stable, not decreasing as Gerlough and Huber (1975)suggested, for all traffic flow levels (from 1,200 to greater than 2,200), which indicates that the value of k does not necessarily decrease with the increase of flow within high flow conditions. This study also found that the Erlang distribution with small k-values (less than 4) did not provide a reasonable fit for headways of flow below 1,200 vph, especially for the medium flow state (400 to 1,200 vph).

7 7 Arterial Data As mentioned previously (Table 2), arterial headway data covered the volume range from around 600 to 1,200 vph. It was difficult to collect usable data for high traffic flows for this study. As in the freeway analysis, the chi-square testing technique was used on the observed data for arterials. One example is shown in Table 6 and Figure 3. Unlike the case with freeways, one distribution, namely, the gamma distribution, provided a good fit for all the samples from arterial sites, as shown in Table 7. Yet the two parameter values for this distribution vary from one flow level to another. In another study, Al-Ghamdi (1999b) found that the same distribution fits arterial data. It would be of interest to examine the pattern of the two gamma parameters in Table 7 versus flow rate, as shown in Figures 4 and 5. From Figure 4, the scale parameter (α) looks quite stable up to a flow of 1,100 vph. It then jumps significantly, which might indicate that 1,200 vph is the boundary for high flows on arterials. On the other hand, the pattern of the shape parameter, shown in Figure 5, decreases overall; nevertheless, it can be said that the parameter values are roughly stable up to the flow level of 1,000 vph. The values then drop, which again supports the belief that the boundary for the high flow state on arterials is 1,200 vph. Mean and Standard Deviation Association It seemed relevant to investigate whether any simple relationships exist between the variables in Table 1, for example, the standard deviation of observed headways (s) versus the mean headway for freeways and arterials, together with the least squares best-fit line. The regression equations are as follows:

8 8 For freeway sites: s = (mean headway) (r 2 = 90%) For arterial sites: s = (mean headway) (r 2 = 90.1%) These relationships have important consequences since, in effect, they produce a method of estimating the standard deviations of headways directly from the mean observed flow, which is the reciprocal of the mean headway (headway = 1/flow). Griffiths et al. (1999) found a similar linear trend between standard deviation and mean as those shown in Figures 6 and 7. Coefficient of Variation The sample coefficient of variation (CV) is the proportion of sample standard deviation and sample mean. In distribution functions CV is the proportion of standard deviation to expectation. The negative exponential distribution has a CV equal to 1. Figure 8 depicts CV values for a freeway site and an arterial site. All CV points of the freeway site stay above those of the arterial site indicating higher variability of headways at the freeway site. Luttinen (1992) found that polynomial curves fit the same data for high-speed and low-speed roads. He observed that under heavy traffic, the proportion of freely moving vehicles is small. The variance of headways is accordingly small [CV <1 at high flow levels (1,000 to 1,500 vph)]. The CV data in the current study do not have the same interpretation as do Luttinen`s data. It is clear from Figure 8 that all CV values are less than 1. However, similar to Luttinen`s findings, this study suggests, as shown in Figure 8, that low-speed roads (arterials, 50 to 90 km/h) have lower values than do high-speed roads (freeways, 100 to 120 km/h).

9 9 From international research, the CV values fall in the range of 0.5 to less than 1.5 over a range of flow rates from less than 500 to greater than 2,000 vph. The data sets of Breiman et al. (1977), Buckley (1968), and May (1965) are for a freeway lane. The data of Dunne et al. (1968) come from a two-lane rural road, and the data set of Luttinen (1992) is from low-speed and highspeed roads (50 to 100 km/h). The CV is greater than 1 in all samples from Dunne et al., less than 1 in all samples from May, near 1 in the samples from Buckley, and near and greater than 1 in the samples from Luttinen. Over the same range of flow rates (500 to above 2,200 vph), this study shows that the CV is less than 1 in all samples (the range is from 0.32 to 0.82). Therefore, the CV from this study is generally shorter than corresponding values from international research, indicating that a motorist in this developing country leaves a shorter headway from the car ahead than corresponding drivers in the developed world in which the past research was conducted. This finding may reflect the difference in traffic conditions, particularly driving behavior, in Saudi Arabia (a developing country) and those in developed countries. Such differences may be attributable to the fact that driving behavior in Saudi Arabia tends to be more aggressive (Al-Saif et al. 1990). In studying driving behavior at signalized intersections, Al-Ghamdi (1999a) found that the mean of discharge headways is shorter in Riyadh than that in other countries, and accordingly, the saturation flow rate levels are higher. In addition, the occurrence of traffic accidents due to cars following each other too closely is a typical problem in this country. Thus, it is not surprising to find a substantial difference in CV values between this study and other international studies. Figures 9 and 10 show second-degree polynomial curves that have been fitted to arterial and freeway data, respectively. Although the polynomial curve for freeway data opens upward (is

10 10 concave), the curve opens downward (is convex) for arterial data. This result might indicate that motorists on arterials drive closer together than those on freeways. This result again is not consistent with the similar analysis by Luttinen (1992), who found convex curves for both low and high speed; again, this may be attributed to differences in driving behavior. Freeways Versus Arterials It is obvious from the previous analysis that headways on freeways are not similar to headways on arterials, perhaps because of differences in speed limit as well as geometric conditions. Therefore, in analyses of time headways on urban lanes, the type of road, freeway or arterial, should be clearly specified. Conclusion A large sample of data relating to headways of vehicles in single lanes of traffic was collected at 20 urban sites in Riyadh. Despite this sample, the modeling of headways for high traffic flows is still vague. There appears to be little research on this type of situation. The majority of published material relating to headway distributions has focused on low traffic conditions in which independence between successive vehicles is satisfied. An attempt was made in this study to fit distributions for all three flow states low, medium, and high particularly on freeways. Although negative exponential, shifted exponential, and gamma distributions were found to reasonably fit the low and medium states of flow on freeways, the Erlang distribution seemed to properly fit the high traffic flow state. On the other hand, the gamma distribution gave a decent fit for a large range of flows on arterials (around 600 vph to 1,200 vph).

11 11 The study also established boundaries for the three states of flow based on the distribution families that fit the three groups of flow on freeways. The low flow state consists of traffic flows less than 400 vph. The range for the medium flow state is 400 to 1,200 vph. The congested state starts at 1,200 vph. Shape and scale parameters were used to help define these boundaries. The data analysis in this study showed that driving behavior, in terms of time headway, in Riyadh is different from that in the developed world. That is, motorists in Riyadh follow one another more closely compared with drivers in developed countries. It should be said that the flow boundaries, as well as the results from the study, are valid for the collected data in Riyadh and should not necessarily reflect traffic conditions elsewhere. Acknowledgment The author would like to express his appreciation to Khalid Ayed for his efforts in collecting the needed data for this study. References Al-Ghamdi, A.S. Statistical Comparison between Severe Accidents and PDO Accidents in Riyadh. In Proc., Safety on Roads: An International Conference. Bahrain, 1998, pp Al-Ghamdi, A.S. 1999a. Entering Headway for Through Movements at Urban Signalized Intersections. In Transportation Research Record 1678, TRB, National Research Council, Washington, D.C., pp

12 12 Al-Ghamdi, A.S. 1999b. Modeling Vehicle Headways for Low Traffic Lows on Urban Freeways and Arterial Roadways. Proc., 5th International Conference on Urban Transport and the Environment for the 21st Century, Rhodes, Greece, Sept. Al-Nafa, A., and K. Al-Saif. Analysis of Psychological and Social Characteristics Related to Behavior of Driving in Saudi Arabia. King Abdulaziz City of Science and Technology, Riyadh, Saudi Arabia, Al-Saif et al. Al. Investigation of the causes of Traffic Accidents in Makkah and Eastern Regions and their Countermeasures. King Abdulaziz City of Science and Technology, Riyadh, Saudi Arabia, Breiman, R. L., Goodwin, D., and Bailey, B The Statistical Process of Freeway Traffic. Transportation Research, Vol. 11, pp Buckley, D.J A Semi-Poisson Model of Traffic Flow. Transportation Science, Vol. 2, No. 2, pp Cowan, R.j. Useful Headway Models. Transportation Research, Vol. 9, 1975, pp Dunne, M.C., Rothery, R.W., and Potts, R.B A Discrete Markov Model of Vehicular Traffic. Transportation Science, Vol. 2. Gerlough, D.L., and Huber, M.J Special Report 165: Traffic Flow Theory: A Monograph. TRB, National Research Council, Washington, D.C. Griffiths, J. D., and Hunt, J. G. Vehicle Headways in Urban Areas. Journal of Traffic Engineering + Control, pp , October 1991.

13 13 Luttinen, R.T Statistical Properties of Vehicle Time Headways. In Transportation Research Record 1365, TRB, Washington, D.C. May, A. D Gap Availability Studies. In Highway Research Record 72, HRB, National Research Council, Washington, D.C., pp Mei, M., and Bullen, G.R. Lognormal Distribution for High Traffic Flows. In Transportation Research Record 1398, TRB, National Research Council, Washington, D.C., 1999, pp Sullivan, D.P., and Troutbeck, R.J. The Use of Cowan s M3 Headway Distribution for Modeling Urban Traffic Flow. Traffic Engineering and Control, Vol. 35(7), 1994, ppl User`s Manual Jamar Technologies, Inc., Horsham, PA.

14 14 Table 1. Time Headway Data: Freeway Sections Site Lane Direction Flow Rate Mean Std. Dev`n CV (vph) (s) (s) Makkah Rd Centr Westboun Northern Ring Centr Westboun Northern Ring Centr Eastbound Eastern Ring Media Southboun Eastern Ring Media Southboun Eastern Ring Media Northboun King Fahad Rd Centr Northboun King Fahad Rd Centr Southboun King Fahad Rd Centr Northboun King Fahad Rd Centr Southboun Khurais Rd Centr Westboun Khurais Rd Centr Eastbound Khurais Road Centr Eastbound

15 15 Table 2. Time Headway Data: Arterial Sections Site Lane Type Direction Flow Rate (vph) Mean (s) Std. Dev`n (s) Jareer St Media Eastbound Batha St Media Southboun Batha St Media Northboun King Abdulaziz Media Northboun Prince Abdullah Media Eastbound Olaya St Media Northboun King Abdulaziz Media Southboun CV

16 16 Table 3. Chi-Square Test for Data from Khurais Road (Westbound). Headway Observed Data Theoretical Data for Chi-Square Exponential Distribution Test Statistics (t) sec Frequency Cumulative No. Cum. >= t Cum. % Prob. Freq. >= t Value Total * *significant at 5% level.

17 17 Table 4. Summary of Statistical Distributions Fitted To Freeway Headways. Flow Rate Fitted Statistical Coefficient(s) Distribution veh/h <400 Negative Exponential Shifted Exponential* and α = 2.9 Gamma α = 6.37, β = Shifted Exponential* α = Shifted Exponential α = Shifted Exponential α = Shifted Exponential α = Erlang k = Erlang k = Erlang k = Erlang k = Erlang k = Erlang k = 4 >2200 Erlang k = 4 * The shift of the exponential curve with respect to the origin is estimated to be 0.63 sec, which is the minimum actual (not necessarily allowable) headway.

18 18 Table 5. Distribution Family for Corresponding Flow Level. Flow Description Flow Range (vph) Distribution Family Low Medium High Less than to 1,200 Above 1,200 Negative Exponential Shifted Exponential and Gamma Erlang

19 19 Table 6. Chi-Square Test for Data from Olaya Street (Northbound). Headway Observed Data Theoretical Data for Chi-Square Gamma Distribution Test Statistics (t) sec Frequency Cumulative No. Cum. >= t Cum. % Prob. Freq. >= t Value Total 611 4* *significant at 5% level.

20 20 Table 7. Summary of Statistical Distributions Fitted To Freeway Headways. Flow Rate Fitted Statistical Coefficient(s) Distribution veh/h <600 Gamma Gamma α = 6.43, β = Gamma α = 5.407, β = Gamma α = 5.513, β = Gamma α = 5.54, β = Gamma α = 4.954, β = Gamma α = 9.241, β = Gamma α = 7.295, β = 0.406

21 21 Cumulative Headways Vehicular Headway, t (sec) Northern Ring Road, 1 km West of Othman. Rd Urban Freeway Traffic Volume = 773 veh/h Shifted Exponential Distribution Observed Theoretical Figure 1. Observed and corresponding fit for data for one freeway site.

22 22 Shift Exponential Parameter Traffic Flow (vph) Figure 2. Decrease trend for shifted exponential distribution parameter.

23 King Abdulaziz Rd, fronting Land Forces HQ. bldg. (SB) Arterial Road. Traffic Volume = 1216 veh/h Gamma Distribution Cumulative Headways Observed Theoretical Vehicular Headway, t (sec) Figure 3. Observed and corresponding fit for data from one arterial site.

24 24 Scale Paramete Traffic Flow Rate (vph) Figure 4. Pattern of scale parameter (α).

25 25 Shape Paramete Traffic Flow Rate (vph) Figure 5. Pattern of shape parameter (β).

26 26 Standard deviation (sec) Mean (sec) Figure 6. Standard deviation of arterial headways versus mean with best fit line.

27 27 Standard deviation (sec) Mean (sec) Figure 7. Standard deviation of freeway headways versus mean with best fit line.

28 28 Coeff Arterial Freeway Flow rate (vph) Figure 8. Coefficient of variation of freeway headway versus flow rate.

29 29 Coeff. of variation Flow rate (vph) Figure 9. Polynomial curve for coefficient of variation for arterial headways.

30 30 Coeff. of variation Flow rate (vph) Figure 10. Polynomial curve for coefficient of variation for freeway headways.