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55 Chapter Five CRITICAL GAP AND FOLLOW-UP TIME This chapter documents the analysis of factors that may affect the critical gap and follow-up time measurement results. The relationship between the critical gap and follow-up time is then investigated. Finally, the recommended values of critical gap and follow-up time are given based on the analysis results and eng~neenug judgment. ANALYSIS OF FACTORS AFFECTING CRITICAL GAP AND FOLLOW-UP TIME Critical gap and follow-up time measurements for each site were based on its unique intersection geometry and traffic flow conditions. It was important to identify those factors that could affect the cntical -A and follow-up time values, so Hat appropriate values could be recommended for general use. The mean values of critical gap and follow-up time were calculated according to categories such as movement type, intersection geometry, major street spool, and geographic sector; however, this approach only provided a general sense of what the values of the critical gap and foDow-up bme under various U.S. conditions were. Other investigation methods were used to identify some of the factors which may cause larger variations In the cntical gap and follow-up time. Regression analysis was used to iden~ these general factors. More specific analyses were then conducted to investigate the effects of delay, major street movement type, and vehicle type on the critical gap and follow-up time. Regression Analysis A database was established for conducting the regression analysis. To increase the sample size, the entire intersection data set was divided Into several sub-sections if the number of observations was more in 200 for the cntical gap measurement. For example, if more than 200 minor street vehicles were observed In a 2-hour penod, the cntical gap estimation was conducted for both the entire penod end for each 15-minute period. Some measured site attributes that could potentially affect cntical gap were then added into the database. These attributes included geographic sector, number of legs, multi- lane or single lane on major street, number of lanes a minor movement crosses, major street volumes, major street right turn percentage, percentage of traffic on the conflicting lane for multi-lane sites, minor movement with shared lane or exclusive lane condition, Once oftwo-way left-turn lane on the major street, distance to upstream signals, grade of minor approach, and average vehicle delay. The database used for the regression analysis was provided as Appendix m of Working Paper 16 (NCHRP 3-46, 1995~. Step-ur~se linear regression was then conducted. Through the step-wise regression process, those factors that did not show a significant effect on the critical gap were eliminated Regression models were then developed to estimate the cntical gap given certain traffic flow and intersection geometry conditions. Initially, the regression analysis factors were selected based purely on statistical significance. Based on these preliminary results, He factors were judged and included only if they were applicable to "real-worId" conditions. The independent variables considered are described below. Client is the minor movement for which the critical gap was measured. MajET - major sheet left tom, MinorLT - minor street left turn, MinorRT - minor street left turn, and MinTH - minor street through. to is the mean critical gap measured for Mat site estimated using the maximum likelihood method. S. D. is the standard deviation of He critical gap for that site. Obs is the member of observations of the critical gap measurement, which is also the number of subject vehicles observed. The geographic sector information was described
56 using dummy variables. . . . Over as combo when there are no exclusive left turn lanes on the major street. MajVolL Is the tragic volume from the left side on the major street. For the major street left turn movement, this volume represents the total conflicting volume. MajVolR is the traffic volume Mom the right side on the major street. %RT-Left is the percentage of right turn movement on the major street from the leR. %RT-Right is the percentage of right turn movement on the major street from the right. %ConfLnVol is the percentage of traffic volume on the conflicting lane at a multi- lane site (calculated only for the through movement). If traffic on the non- conflichug lane also affects the minor street driver, the higher the percentage on the non-conflicting lane, the larger the critical gap. MajSp~ is the posted speed limit on the major street, mph. Excl is a dummy vanable that has be value ~ if the minor approach has an exclusive lane and O if the minor approach has a shared lane. 1~11 N is a dummy variable that has be value ~ if there is a two-way left turn lane on the major street. Otherwise TVVETLN has the value of 0. The existence of a TWLTLN makes it difficult to accurately define the gap event, because some of be drivers simply merge into the TWLTEN and seek another gap among the traffic from the right side. MajDir is a dummy variable that has the value of ~ for one way and 2 for two way on the major street. UpSigL is the distance to the nearest Upstream signalized intersection on be left side, miles. UpSigR is the distance to the nearest upstream signalized intersection on be right side, miles. Gradle% is the approach grade at be location of the stop line. TurnAngle is another indicator of be Sector NE is a dumm r variable which has the value ~ if the site is In be northeast sector and 0 otherwise. Sector CE is a dummy variable which has the value ~ if the site is In the central sector and 0 otherwise. Sector NWis a dummy variable which has the value ~ if the site is in the Northwest sector and 0 otherwise. Sector SE is a dummy variable which has the value ~ if the site is In the Southeast sector and 0 otherwise. If a site is In the SW sector, all other variables (Sector NE, Sector CE, Sector NW, Sector SE) are zero. Independent variables include: NoLeg is the number of legs of the Intersection. Mmaj is a dummy variable that has the value of ~ if the site is Multi-lane. It has the value of zero if it is a single-lane site. A multi-lane site usually has more than two through lanes on the major street for both directions. For one way on the major street with two lanes, Me site is defined as multi- lane site for minor sheet left and nght turn measurements, but as a single-lane site for minor sheet through movement. A multi- lane site usually requires that traffic of a particular movement travel on more than one lane, and only the vehicles on the conflicting lane are considered when measuring the critical gap. InCrs is the number of lanes a minor movement needs to cross. This is perhaps a better indicator of the difficulty of a movement maneuver compared to using just a multi-lane or single lane. For example, be existence of exclusive left turn lanes on the major street requires Me minor streetleR turn movement to cross atleast a two-lane width and the minor street through movement to cross at least a four-lane width, which may prove to be more difficult . . . . . . .
57 difficulty of a turning movement maneuvre. This variable has the value of ~ for a small angle and O for a perpendicular angle. It is considered easier to make a turn at sites with a small angle approach. Sites CET305 and CET314 have small turn angles. The nght turn movement at these intersections are similar to a freeway on- ramp and the critical gap is smaller. Delay is the average delay for those minor stream vehicles dming the period of critical gap measurement, sec/veh. Other factors thalmay affect the critical gap include the city size and intersection location (rural, urban). Because data regarding Me city size and location are limited, deer have not been listed; however, it should be noted that most of the intersections In the SE sector are located In aural areas. . The following set of equations was initially obtained based on the regression analysis: Major Street Left Turn: tc = 4~3 ~0.31*MMaj `~0~' Minor StreetRight Turn: tc = 6.27+0.82*A~aj-0.0345*°/R~e.~- 109 0.38*3Leg+0.123*G=de-2.43*Tun2~gle Minor Street Left Turn: tc = 6.79 -0.72 *3Leg-0.0216 *°/oRTLeJi+ 0.22*G~de+0.39 *LnCrs- 1.03 *1WLTLN Minor Street Through Movement: it = 4.91+0.0611 *°/oRTLe.,~+0.104*(MaySpd-30) `1 1-v The results of variance analysis of the above regression equations are given in Table 23. Through We regression analysis, Me following conclusions were reached: . The following factors that have significant effects on die critical gap were identified: Anal - whether the intersection has multi- lane or single lane approaches on the major street SORT - percentage right turn movement on the major street NoLeg - whether Me intersection has 3 legs or 4 legs Grade - the grade of the minor steam approach Tu~nAngle - whether the minor stream movement has a small turn angle The regression analysis included some sites Ninth unusual geometric charactenstics, such as a one-way major street or a two- way left-turn lane. These may have caused some bias in the regression equations; therefore, judgement needs to be used in recommending final cntical gap values. Some factors were studied at a detailed level to determine Weir effect on critical gap according to each gap characteristic, including the accepted and rejected gaps. The following factors were reviewed: (110) . the delay effect; the effect of heavy vehicles; the effect of major street traffic from Me left side and Dom the right side of Me minor street approach.
58 Table 23. Summary of Variance Analysis of Different Regression Equations , -; ... .. . ; . ;;;; -; ; ; .; .;; ................ .. ; ;; .;;;;; . -;;; - -;; - ~ . ~f,~ ............ ............... ~ . . ~= . ~ MaJor Left Turn Regression 1 1.99 1.99 3.37 Residual 8S 45.37 0.53 Total 86 47.36 if 1 0.042 1[ Minor Right Turn Regression S 140.73 28.1S Residual 127 97.69 0.77 Total 132 238.42 R2 0.4S9 Minor Left Turn Recession 1 5 78.96 1 15.79 16.78 1[ Residual 12 1 113.88 0.94 otal 1 126 1 192.84 R2 0.409 . ~ Minor Through Regression 1 2 1 25.30 1 12.65 13.49 Residual 23 21.S8 0.94 Total 2S 46.88 R2 1 0.409 The Effect of Delay on Critical Gap Investigation of any of single factor should ensure that other factors Hat may affect Be critical gap have held constant. For this reason, it is better to analyze groups of sites with similar intersection characteristics. Two "groups" of sites were selected for analyzing the delay effect on the critical gap. These sites were actually videotaped from two intersections during different periods with a high number of observations. The entire period was divided into 15 minute sub-sections and the critical gap was estimated for each sub-section. Figure 27 illustrates the critical gap measured for each period and the associate average vehicle delay during that period A regression line for aD of the data points is shown In Figure 27. It can be clearly observed that, vv~th~e increase of delay, the critical gap decreases. This implies that an iterative process would be required if this delay effect is to be taken into account since delay is not known until an Initial capacity estimate has been obtained. Alternatively, 36.ss the cntical gaps could be related to the major street flow so that the iteration could be avoided. However, since the cntical gap is also related to over factors, the relationship between major street flow rate and cntical gap is always difficult to establish. Figure 27 reveals a concern regarding estimates of critical gap and follow-up time obtained Tom obsenabons curing understated conditions. Such estimates are valid for estimating capacity and then delay for given traffic conditions. However, they may underestimate capacity and overestimate delay at the same site when a site has higher traffic volumes (near capacity) when important decisions such as signal warrants or mitigation through lane improvements may be necessary. Therefore, it is recommended that site critical gap and follow-up time be obtained (rather than using national
59 8.0 87.0 6.5 6.0 85.5 :~ ~ 5.0 o 20 40 60 80 Average Delay, Seth o Group1 o Group2 | 100 120 Figure 27. Analysis of He Delay Effect on Critical Gap averages) when it appears that the critical approach is near capacity, say v/c > 0.90. Another solution would be simulation models. The Effect of Vehicle Type on Critical Gap and Follow-up Tune Because of the limited number of observations, it was impossible to estimate the critical gap for heavy vehicles at each site. To conduct this analysis, the gap events related to heavy vehicles were extracted from each site and then aggregated based on intersection geometry and movement type. The maximum likelihood method was then applied to calibrate the cntical gap. Table 24 lists the results. As indicated, critical gaps for heavy vehicles are significantly higher than those for passenger cars; large variations in the values also exist for heavy vehicles. Table 24. Critical Gap Measured for Heavy Vehicles by Geometry and Movement Type .. ... . . . .. . . . art::::::: :-:: .~ ~ | 3-leg Single | LT r 7.2 | 2.9 T 166 | 6 l RT 6 2.3 65 5.2 Leg Single LT 7.6 2.1 58 7.1 TH 6.3 2.9 24 6.4 RT 6.7 3 37 5.9 3-leg I LT I 1 2.1 1 9 1 7.2 | Multi-lane RT 9.4 3 .8 104 6.9 Leg LT 9 4.5 13 7.4 Multi-lane TH 9 .5 6 .3 17 7.6 RT 8.2 2.7 25 6.8 Mulli-lane 1 LT | 9 | 3.3 1 22 1 7.3 l TH 9.5 6.3 17 7.6 RT 8.8 ~3.3 90 6.8 ~ Singl~lane LT 7.4 2.5 224 6 .4 TH 6.3 2.9 24 6.4 RT 6.4 2.7 102 5.5 I MajLT 5.5 1.7 202 4.1 The foHow-up time for heavy vehicles was also calculated Whenever a heavy vehicle is involved in a foDow-up time event, i.e., either Me first vehicle or the second or both In a follo~up time event are heavy vehicles, the foRow-up time was classified as a heavy vehicle related follow-up time. Table 25 lists the results measured for heavy vehicles, for passenger cars, and for all vehicles grouped together. The measurements are grouped based on intersection geometry and vehicle movement type. Figure 2g illustrates the follow-up time measurement results averaged for passenger cars
60 and heavy vehicles across all sites. It can be observed ~at ~e foDow-up times for heavy vehicles Table 25. Follow Up Time Measuremen~ Based on Vehicle Type and Geomeby MajLT are about ~ second higher than those for passenger cars. Mcan S.D Obe. ~......... ~::::: ::: <( :::3~::: :: :::: : :: :: ::::~::: :: ::: ::::: :~ :::: :: ::::::: : : : :--: :-:-:::::::::::: :: :-:-:-:-:-: ~:::- : :::::::::: :::-::-::::: :-:::::::-::::: :::::::::::: .::::: - - ·:-: .. - . . ~ . . . ~ . ~: . .- .. - . . .......... 22 3.1 2.3 l.9 ~ 0.7 0.9 0.8 05 * 388 27 41S 2S * 33 4.0 3.4 3.4 43 12 12 12 2.4 1.1 7S@v 26 784 71 S.0 * 3.? 4.8 * * * 1.4 0.0 * * * 60 1.0 3.4 4.0 3.4 33 4.4 12 12 12 15 1.1 90g 27 83S 331 17 1.9 05 2S 35 2.4 76 3.7 1.4 61 3.4 15 ~Ao 22 0.8 72 2.8 03 3.0 22 0.8 7S 2S 0.8 66 4.0 1.8 181 4.1 1.4 2S9 4.0 1S SS1 3.2 0.1 _3.Q 2S 0.? 6'*~ 2.2 0.8 SS1 3.1 0.8 33 MinRT Meen S.D Obe. 3.1 1A S49 3.1 1.4 13 3.1 1.4 S62 S.1 22 11 4.0 1.8 192 3.3 1.4 1S59 4.2 1.6 SS MinTH Mean S.D Obe. * * _ _ * * * * 4.7 0.8 S 4.1 1.4 264 4.0 1.4 319 4.7 0.7 6 MinLT Mean S.D Obe. 3S 13 2760 4.4 1.6 82 3.6 13 2842 2.2 2.1 18 4.1 1.6 S69 3.S 13 44SO 4.4 1.6 144 Notes: PC is passenger care; li~y i8 heavy vehicles. Categories in top row are geometric types. - 3 o 5 _ 1 H^T ~T - ~ ~T ~-o~ 1 Figure 28. Weighted Average Values of Follow-Up Times Measured for Passenger Cars and Trucks The Effect of Direction of the Major Street Flow In the max~m~ ~e~ihood procedure, ~e maximum rejected gap of each vehicle dete~m~nes the lower limit of the critical gap value. To investigate ~e effect of different major street movements on the minor street driver, the maximum rejected gaps were extracted and classified by different ending gap movement type. The data were based on the minor street left tum movement at 3-leg intersections. Table 26 shows ~e results. It can be obsened ~at the major street left turn always yields ~e highest value. This may be explained by the "slow down effect" of the left turn movement. However, it is clear that the through movement from the right side always y~elds a higher value than that from the leR side. This means that vehicles from ~e right side usually put more pressme on~e minor street driver, because the minor slleet driver needs to accelerate to the desired speed if he or she decides to enter the intersection.
61 Table 26. Maximum Rejected Gaps by Different End Gap Movements For Minor Street LeD Tum ~-~1 it, ~ 1~1~ ~ 1~.~:-:$~: 1.7 1S7 * * * * * * 4.0 1.7 1.S 192 * * * * * * 4.0 1.S 1.6 1S3 * * * * * * 4.0 1.7 2.1 27 * * * * * * 4.8 2.1 1.9 109 4.0 2.3 123 4.8 2.7 18 4.6 2.1 2.0 108 4.0 2.7 117 S.9 2.3 49 3.9 2.0 2.1 111 3.7 2.0 183 S.4 2.1 18 4.4 1.7 2.1 44 3.9 2.2 1 4.3 1.3 4 4.1 1.8 1.9 901 3.9 2.3 424 S.1 2.1 89 4.2 1.8 , _ ..... -.- - :::::::::::::::::::- ::::::- ::-. 1 .................. 1 ................. SWT008 SWT017 SWTOlg CET306 NWT402 NWT40S SW1010 NET209 Average 3.9 3.7 3.6 4.3 3.4 3.7 3.7 3.3 3.7 240 196 303 176 27 73 9S 74 1184 A RELATIONSHIP BETWEEN CRITICAL GAP AND FOLLOW-UP TIME In the process of studying the critical gap and foDow~up time measurements, a consistent ratio was found between the critical gaps and the follow-up times. Table 27 lists He ratio of the follow-up time and the cntical gap measurements for each movement. Figure 29 shows the relationship between cntical gaps and the follow-up times measured for all He sites. Figure 30 shows the Table 27. Ratio of Critical Gaps and Follow-Up Times by Movement distribution of the critical gap/follow-up ratio. It can be observed that the ratios of He follow-up time and the critical gap range between 0.4 and 0.9. The majoribrofallobsewabons are around 0.6, which is the value used in the old version of the HCM. Further investigation of this relationship may be warranted As mentioned earlier, the follow-up time can be mms~1 directly in the field. Thus it may be feasible for the practitioner to measure the follow- up time and compute He critical gap from this measured value. .Mo ~tM ~ ~d ~ 06~Ob~o" MajLT 0.55 0.08 MinRT 0.62 0.11 MinTH 0.58 0.07 MinLT . 0.58 0.12 All 0.59 0.11 . 14 25 32 79
62 10 9- 8. on 7 ~ 6- F 5 ~ 4- _ 3 o LO 2 O o Figure 29. Relationship Between Critical Gap and Follow- Up Time 0.5 0.4 0.3 0.2 0.1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 ~ 1 1 1 1 1 1 ~51T1 11 1 1 1 1 1_111~: 1 1 . 1 1 1 1 11111 1 1 1 1 1 1 1 11115_1 1 1 . I I 1 1 111111111 1 1 . I 1~ 11111111 1 1 : I I I l An I 0 0.10.20.30.40.50.60.70.80.9 1 Or heal Gap/Follow-up Time Ratio 1 Figure 30. Distribution of Critical Gap/Pollow-Up Time Radio RECOMMENDED VALUES Both the regression analysis and the more detailed (microscopic levep analysis were conducted to identify those factors that may affect He critical gaps and follow-up times. Overall, Be regression technique provides more insights regarding these relationships. Generally, the following factors have been found to have a significant effect on the cntical gap and follow-up time: . . major street volume or minor stream vehicle delay intersection geometry, including the number of lanes on Be major street and the number of legs of the intersection right turn volume proportion on Be major street minor stream approach grade movement turn angle With the increase of major stream volume or minor stream vehicle delay, He critical gap and follow-up time tend to decrease. However, the critical gap value cannot be reduced to the minimum threshold determined by the follow-up tune value or the minimum accepted gap value. With the increase in the number of lanes on the major street or the number of legs at the intersection, He critical gap tends to increase because of the increased difficulty of the movement maneuver. With the increase of the right turn volume on the major street, the critical gap tends to decrease because the right turn movement on the major street causes less conflict to the minor street vehicles compared to the through movement. With He increase of the approach grade, the critical gap tends to increase. With a small turn angle, the movement maneuver is easier compared to a perpendicular angle or a large angle, and the critical gap tends to decrease. Based on the regression analysis, the microscopic analysis, and practical engineering judgement, the recommended critical gap and follow-up time values are listed in Tables 28 and 29. A factor addressing the proportion of right turns is not included in this set offactors. This factor is handled through the use of weighUngLactors when calculating the conflicting steam for a minor movement.
63 Table 28. Recommended Critical Gap Values .... ; : : ,:.,:.,,,,., ,, ,,,,,,,,., , , ~, . , ,,, , ............................................................................................ ., .................................................... ................................................ .. ...................................................................... ............................................ ................................................. ....................... ...................... ...................... ............. ........... ,. ............ ..................... ...................... 2~''""""'' ""'' 1''' '1 "'"I ~.t CridchlGap p~asse~ger Cars) Adjustment Facto" _ Heavy Vehicle* Grade% Three-Leg 4.1 ..... ,., ,,., ,,,,,,., ,, , ,,, , ..................................................................... ,,., """""""' . ........................................... ........... .................................. ......................... .. . . . .. .: . . . ..... .... . .. 2 ~: . ~my'' , , . , ~....... , 6.2 65 7.1 4.1 6.9 6.S 1.0 1.0 1.0 2.0 2.0 2.0 0.1 0.2 0.2 0.1 0.2 ft q ~v./ 7.5 1.0 - 2.0 0.2 -0.7 Notes: *Combined critical gap is computed based on the proportion of passenger cars and heavy vehicles Table 29. Recommended Follow-Up Time Values ,.. .. ...................................... Mowm ~ J~T ~ . ., Follow-up Thne 2.2 3.3 4.0 (Passenger Caps) Adjushnent Factors Heavy Vehicle. 0.9/1.0 0.9/1.0 0.9/1.0 ............ ;;;; ; .;;; ; ; . 3.S 0.9/1.0 Note: *O.9-second adjustment for single-lane sites and 1.0-second adjustment for multi-lane sites. Combined follow-up time is computed based on the proportion of passenger cars and heavy vehicles
64
