Capacity Analysis of Interchange Ramp Terminals: Final Report (1997)

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APPENDIX C CAPACITY CHARACTERISTICS FOR INTERCHANGES AND CLOSELY-SPACED INTERSECTIONS This appendix describes the development, calibration, and application of models that collectively can be used to predict the capacity of traffic movements at signalized interchange ramp terminals and other closely-spacedintersections. Specifically,ffiese models predict three important capacity characteristics: saturation flow rate, start-up lost time, and end lost time. The mode! for each characteristic is developed using theoretic constructs that incorporate Me factors that have an influence on the characters tic's magnitude or duration. These models were calibrated with data collected at twelve interchanges (the field studies are described in Appendix B). It should be noted that the traffic characteristics described in this appendix reflect passenger car performance as all heavy vehicles were excluded from the database. This appendix includes five main sections. The first four sections describe the development and calibration of models of saturation flow rate, start-up lost time, clearance lost tone, and large utilization. The last section describes the proposed form of these models and their application to capacity analysis. C.1 SATURATION FLOW RATE This section describes the development and calibration of a saturation flow rate mode} applicable to the signalized movements at interchanges and closely-spaced intersections. Separate models are developed for the left-turn and through movements at these junctions because of their unique operational character. In each case, the saturation flow rate mode} is denved from a mode] of the discharge headway process for queued vehicles. The saturation flow rate is defined as the minimum discharge headway reached and sustained by the discharging queue. The models are sensitive to factors that effect the discharge process at interchanges end closely-spacedintersections such as distance to the downstream queue and signal timing. Three topics are discussedin the remainder ofthis section. First, the basic issues related to the measurement of the minimum discharge headway and start-up lost time are descnbed. Then, the minimum discharge headway and resulting saturation flow rate models are developed for through movements. Finally, the minimum discharge headway and saturation flow rate models are developed for the left-turn movements. The development of start-up lost time models for each of these movements is described in the next section. C.. Minimum Discharge Headway and Start-up Lost Time Methods of Computation. The discharge of a traffic queue, upon presentationofthe green indication, is charactenzedby the reaction time of the queued Divers to the indication followed by C-l

their steady acceleration to a desired discharge speed. As a result, the first few vehicles have relatively long headways; however, Me headways of subsequent vehicles aradualIv decrease as Rev 1 ~1 1 · 1 it- ~1 T To, · , ~,1 ~ - ~- ~ =~ ~ J - J approach ule Desired Discharge speed. u~umate~y, the a~scnarg~ng stream converges to a relatively constant headway In He range of I.8 to 2.0 seconds per vehicle. This constant value represents the minimum discharge headway H of He queue; its Inverse represents the saturation flow rate s. Typical discharge headways observed for each of He first ten queue positions are shown in Figure Cal. As this figure suggests, the minimum discharge headway is electively reached by the sixth queue position; however, it should be noted Hat He headways for each subsequent queue position continue to decrease slightly. The first four or five queue positions. having headwords larder _ _ J. ~ ~ ~ _, ~_7 . ~ ~ · ~ · · ~ ~ r ~- ~ ~ · o ~ than me m~n''num, each Incur art Increment or lost-t~me due to the react~on-tune process and the subsequent acceleration to the desired discharge speed (i.e., the speed at saturation flow). The sum of these lost-time Increments for each vehicle represents the total start-up lost time Is of He queue. 3 5 Discharge Headway, sec Measured Headway 3.0 2.5 2.0 1.5 11 ~ Is~ 1106~---- 12: ' Avg. Minimum Discharge Headway (Ha) 1 1 1 1 ! 5 6 7 8 9 0 1 2 3 4 Queue Position Figure C-1. Discharge headway by queue position. 10 In recognition of Be relatively constant headway achieved by He higher queue positions, He 1994 Highway Capacity Manual (TICK) (1) recommends that the minimum discharge headway be estimated as the total headway of He combined higher queue positions divided by He number of vehicles observed in these queue positions. In this context, the total headway is computed as He difference between the discharge time of He last vehicle to discharge TO d that ofthe last vehicle to Incur some start-up lost time Tat p. Thus, the minimum discharge headway can be computed as: C-2

T - T _ d(J) d0-l) H J J-(J-1) 1 J J-(~-1) Ash, (C-1) where: Hj = minimum discharge headway based on specification of the jth queue position as the first to achieve the minimum discharge headway, sec/veh; Td(i) = discharge time of the ~ queued vehicle (i = j-l to by, see, h, = headway of the vehicle in the id queue position' see; j = ``specified'' first queue position to discharge at the minimum discharge headway; and J= last queue position to discharge. The HCM (1J indicates Hat the saturation flow rate s can be computed from He minnnum discharge headway using He following equation: 3,600 s. = J HJ (C-2) where: Sj = saturation flow rate for the subject lane based on specification of the jth queue position as the first to achieve the minimum discharge headway vphgpl. By definition, He start-up lost time can be estimated from the discharge time of He mth vehicle and the average minimum discharge headway as follows: ~ o.' = Too.' - j H. = where: Is I = start-up lost time based on Hj, sec. ~ hi - ~ j H ~ (C-3) For most practical applications, the HCM (1) recotrunends that the fifth and higher queue positions can be used to estimate the minimum discharge headway (i.e., j = 51. In general, these averages are sufficiently accurate as to pelt the estimation of the capacity of an intersection traffic movement; however, they are likely to be higher than the true minimum discharge headway H. As C-3

was discussed at He start of this section, H represents the average headway converged upon by the highest queue positions (say, those above position 10~. In practice, it is difficult to quantify H because of the difficulty of obtaining an adequate number of headway observations at the higher queue positions. As a result of these sampling difficulties, Equation C-l is generally used to compute H; (as an estimate of H) for the purpose of .. . . . ,. ., ~ ,.. . . ... .. . . . . .. ... capacity analysis. As the results ot this research share this purpose, the models developed in this appendix are based on H5 rawer Man H. lithe implications ofthis approach are discussed in the next section, however, subsequent to that discussion, all references to the m~nnnum discharge headway, saturation flow rate, or start-up lost the variables do not include the "5" subscript Equations C-l, C-2, and C-3 were used to compute the average discharge headway, saturation flow rate, and start-up lost time for the Trough movements included in the field studies. These averages are reported ~ Table C-~. Examination of He data In this table indicates Hat Here is some vanation in perfonnarlce among the various interchange configurations, although these differences are not statistically signtficar~t. Moreover, the limited number of sites studied for each configuration type (! to 3 sites each) requires caution In any extrapolation of the observed Rends to generalities about He discharge characteristics of various interchange configuration types. Table C-~. Through movement discharge characteristics by junction types | Min. Discharge Headway, N SaL Flow | Start-up Lost Time, to Junction Configuration Rate, s Type Obs. Mean Std. Dev. (pcphgpl) Obs. Mean (sectveh) (sec/veh) (see) nterchange |CDI I 3,027 1 1.93 1 0.53 1,865 1 687 2.48 IPUCIOA]B 1 667 1 1.~8 1 0.53 1,915 1 226 2.66 . |ParCIO B 1 1,291 1 1.90 1 0.51 1,895 1 261 2.30 I SPUI I 1,945 I 1.86 1 0.53 1,935 r 443 2.52 IllnDI3 1 1,257 I I 67 r 047 2,156 1 147 2.88 1 ImDI w/f 1 1,105 1 1.88 1 0.53 1,915 1 293 3.45 I TOtal / AVera 7e 1 9~92 1 1.87 1 0.52 1,925 1 2,057 2.65 l IterSeCtiOn |AGI 1 5,971 T 1.88 1 0.52 1,915 1 1,474 2.46 . Std. Dev. (see) 1.07 0.96 1.09 o.ss 0.81 .03 .08 .07 Notes: 1 AGI - at-grade intersection; TUD! - tight urban diamond interchange (no frontage road or U-tu~n lanes); SPIJI - single-point urban interchange; CDI - compressed diamond interchange; TUDI w/f - tight urban diamond interchange (with frontage road and U-turn lanes); Parclo B - partial cloverleaf with off-ramps beyond overpass; Parclo AB - partial cloverleaf with both off-ramps on the same side of the overpass. 2 - Averages reported do not include cycles Hat incurred spillback from a downstream intersection. Flow rates are in passenger car units (i.e., pcphgpl) as no heavy vehicles were included in He database. 3 - This interchange type represents one site. The Trough headway is low for this site because of a -3.6 percent . .. . .. . . . . . . . ~ downgrade on the approach studied; all other sites had approach grades of less than ~Z.5 percent. C-4

Potential Bias in Headway and Lost Time Estimation. A closer examination of the data was undertaken to explore the causes of the variability in the tabulated headways shown in Table C-~. This examination focussed on the possibility that the method used to calculate the minimum discharge headway introduces some bias. Specifically, the specification of queue position five as He first to achieve the minimum discharge headway is likely to introduce some bias into the resulting average (as obtained from Equation C-~) if the headway s recorded for position five at a given site are not, on average, equal to the average value of the higher queue positrons. Further exam~nahon of the data at the study sites indicated that this bias does exist and that the amount of bias is dependent on two factors. The first factor relates to the inclusion of headway s from lower queue positions, that are not discharging at the minimum headway, in the estimate of the average minimum discharge headway. Specifically, the average headway for a Tower queue position can be larger than the minimum discharge headway and, if included in its estimate, wall bias the estimate to values larger than the true minimum discharge headway. This effect can be seen in Figure C-l where the line representing average minimum discharge headway is "pulled" upward slightly by headway observations in queue positions five and six. The second factor affecting the amount of bias due to the use of headway s not at the m~nun~n value relates to the frequency of headway observation at each queue position. In general, the number of observations is highest for the lowest queue position and decreases with each increasing queue position. This van ation in observations can amplify the effect noted above (i.e., differences in average headway between queue positrons) by giving greater "weight" in the overall average to the more frequently observed lower queue positrons. The following example is posed to illustrate the effect of the bias due to the use of lower numbered queue positions in the average minimum discharge headway computation. Consider a traffic lane at each of two different intersections. Each large has a demand volume of 200 vphpT and art identical queue discharge character that yields the headway s shown in Figure C-2. These headways represent the "true" headway s that are known in advance for the purpose of this example. Inspection of this figure Indicates that the hue minimum discharge headway H of each lane is I.~S sec/veh (1,915 vphgpl). A field study is conducted to estimate the true minimum discharge headway H for each intersechon through the use of Equation C-! (i.e., Ha. Headway s are measured for Tree hours at each ~ntersechon. The cycle length is 120 seconds at one intersection and 60 seconds at the other. These cycle lengths are found to yield average queues of 6.7 and 3.3 vpcpI, respectively, and the frequencies of observation by queue position shown in Figure C-2. The average ~ninimllm discharge headway is computed for each lane and shown in Table C-2. As this table indicates, the two estimates are different solely because of the bias effects described previously. Moreover, neither estimate yields the true minimum discharge headway of I.~S sec/veh (although, the lane with the longer average queue length yields a closer estimate). C-S

Discharge Headway, see 2.5 . Frequency of Observation 2.3 2.1 1.9 1.7 1.5 Untrue Discharge Headways 06.67 vpcpl \ - ' ~ APE \~ - Frequency of Observation an _ , B : . _ -it-: . it, ---I I--~--n-- 7 8 9 n I 10 11 .. ~ ~ 2 3 4 5 6 150 120 90 60 30 Queue Position 13 o Figure C-2. Data used to demonstrate potential bias in the estimate of minimum discharge headway. Bonneson (29 has investigated the amount of bias observed in average minimum discharge headway estimates for several leh-turn and through traffic movements at several sites From this investigation, he fourld that minimum discharge headway estimates obtained from the "average of the fifth Trough last position" H5 were typically 0.13 and 0.06 sec/veh higher for Me left arid through movements, respectively, than unbiased estimates of Hobtained from other methods. This increase translates into the saturation flow rate being ur~derestunated by about ~ ~ O and 50 vphgp} for the respective movements. From a practical standpoint, these differences may be considered small. However, from the standpoint of statistically quantifying the effect of venous factors (e.g.' lane widths on minimum discharge headway or saturation flow rate, these differences can be very important. There are several implications that stem from He use of a biased estimate of the true magnum discharge headway. First, the estimate always exceeds the true minimum discharge headway, however, as mentioned in the previous paragraph, this effect may be small for practical purposes. Second, He estimate of the start-up lost time will be biased toward a value that is smaller Han the hue value because it is computed using the biased estimate of magnum discharge headway (as per Equation C-3). Third, and most important, the bias will likely cloud any statistical analysis of cause and eKect by introducing added variability in He data set. As a result, the Due effect of a treatment or factor (e.g., lane width, grade, interchange configuration, etc.) may be obscured by data from sites having different degrees of bias. In fact, it is this type of bias that causes most of the differences between the estimates of minimum discharge headway for the interchange configurations  shown in Table C-. C-6

Table C-2. Effect of headway observation frequency on the estimate of minimum discharge headway Number of Vehicles Observed: 600 .. . ... Cycle Length, see: ~120 ~60 Average Vehicles / Cycle, vpcpl: ~6.7 ~3.3 . Queue Position True Headway Observations hT; * n Observations (hTj ) sec/veh (nO,i) see (n) 3.63 180 2 2.43 83 148 3 2.18 75 116 2.05 68 85 5 1.97 61 120.36 53 1.93 54 104.36 21 7 1.91 46 87.88 0 ~ 8 1.90 39 74.03 0 1.89 32 60.52 _. 10 1.89 24 45.30 0 11 1.89 17 32.06 12 1.88 10 18.85 0 13 1.88 3.77 14 1.88- 0 - 0 ~0 15 H= 1.88 0 0 0 Sum: 285 547.13 74 . Avg. Headway (Hs = E(hT i*no i)/En): 1.92 see (1,875 vphgpl) 1.96 see (1,835 vphg~l) l hTj*n see 104.58 40.59 o o o ° 1 ° 1 O O . O 145.17 C.~.2 Saturation Flow Rate Mode! for Through Movements Factors Affecting Discharge Headway. In addition to the biases descnbed in the preceding section, the differences in headway and lost-time among He interchange configurations shown Table C-} can be partly explained by differences among He mdividua1 study sites. A review of the literature on the topic of through movement headways suggests that several site-specific factors exist that can have an elect on the discharge process. For example, Bonneson (by, in a previous study of headways at intersections arid single-point urban interchanges for NCHRP Project 340 (3), found that the number of vehicles served per cycle had an effect on the minimum discharge headway. Specifically, he found that the headways observed for each queue position were lower when there were more vehicles queued behind that position. He called this elect that of "traffic pressure." In this context, traf Eic pressure is believed to result from the presence of aggressive drivers (e.g., corrunuters) that are anxious to minimize their Gavel time In otherwise high-volume conditions. As these drivers are typically traveling during the morning and evening peak traffic periods, they are C-7

typically found to be concentrated in the large queues associated with these periods. It should be . . _ . , _ , . . ~7 ~. . ~_ _ ~ ~ ~ ~ _ _ ~ ~ ~ ~ _ ~ _ _ _ ~ A ~ ~ ~ ~ ~ ~ ~ ~ ~ - ~ · · 1 ^^ . ~ , ^~ novel Teal Blokes' 1vlesser, and clover (fly round a similar effect or trattlc queues on headways; they termed this effect "headway compression." Bonneson 62J recommended the following equation for predicting the minimum discharge headway of a single-point urb art interchange through movement as a function of traffic pressure: Hrh 1.57 0.0086 vat' (C-4) s where: Huh = through movement minimum discharge headway, sec/veh; us = speed at saturation flow, m/s; and I' = demand flow rate per lane (i.e., traffic pressure), vpcpl. The speed in Equation C-4 represents the maximum speed drivers tend to reach as they discharge from a traffic queue. In theory, it represents the speed associated with a traffic stream flowing at its saturation flow rate. This speed was found to vary between 12 and 15 m/s in the sites studied by Bonneson 62) (it was denoted by the variable "imp," in Bonneson's work). One reason offered for this variation was the proximity of some sites to adjacent intersections. Specifically, Bonneson noted that lower speeds were associated web those sites where the distance to the downstream ~ntersechon (and its associated queue) was relatively short. This suggests that discharge headway s may be lower because of lower discharge speeds that result from the impending downstream stop faced by the discharging Divers. The HCM (1J describes many additional factors that can affect discharge headway. These factors include: lane width, vehicle classification, local bus frequency, parking activity, approach grade, and area type. To avoid confounding the effect of these factors with those specifically being considered in this study (e.g., distance to back of queue), several steps were taken to avoid or remove We aforementioned factors from the data collected for this project. Specifically, the study sites all had lane widths of about 3.6 meters, approach grades of less than +2.5 percent, no local busses, and no parking activity. In addition, all heavy vehicles (i.e., vehicles wad more than two axles) and all queued vehicles that followed heavy vehicles were removed from the data base. Analysis Approach. The analysis of the through movement headway s focussed on an examination of the headway s for passenger cars in the fifth and higher queue positrons attweIve study sites. Several precautions were taken to eliminate Me elects of bias and to account for factors that might confound the analysis. Specifically, analysis of variance (ANOVA) techniques were used to control for differences in sample size (i.e., unbalanced data) and to account for extraneous differences among otherwise similar sites (e.g., bias by queue position). The ANOVA was implemented with the Statistical Analysis System's (SAS) (5) general linear model (GEM) because of its ability to provide stahshcs corrected for unbalanced data. The distnbubon of the residual errors was checked graphically to verify: (~) that they did not deviate significantly from a normal C-8

. distnbuhon, and (2) that their variance was essentially constant over the range of the independent variables. All significance tests were conducted at a 95 percent confidence level (i.e., ~ = 0.05~. The effects of bias in the examination of factors affecting headway were eliminated by ~ ~ - · . · (~' ~ ~ · 1~ , '' · . ~ · ~ Tow A · ~ ~ ~ ~ (t . · '' ·, ·, ~ - includlng queue position as a -~blocklng t actor In the AN()VA model anti by Nesting it within each potentially influential factor (e.g., traffic pressure or phase duration) being considered. By blocking and nesting on queue position, all of the ANOVA comparisons and parameter estimates are made on a queue-position-by-queue-positron basis, thereby eliminating any bias by differing sample sizes or queue positions. In modeling terms, blocking gives each queue position its own intercept whereas nesting allows the predicted headway to vary independently with each influential factor. If the factor is continuous, this latter effect is equivalent to allowing each factor to have a unique slope for each queue position. Using these techniques, the effects of potentially influential factors were examined for each queue position on an individual basis with the effect of bias eliminated. If the effect of a factor was found to be the same for each queue positron (i.e., it had the same sIope), then the nesting technique was eliminated and a common slope fitted to all positions. Similarly, if any queue positions were found to have the same coefficient (i.e., the same intercept), then they were combined, or pooled, to determine a single representative intercept. These types of combinations are desirable because they yield better estimates of the model parameter coefficients. Once the influential factors were identified from the ANOVA, regression techniques were used to fit the data (via these factors) to the proposed model. Linear or nonlinear regression techniques were used to quantify the calibration parameters, depending on the model formulation. The linear regression was implemented with the SAS 65) regression model (REG3; the nonlinear regression was implemented with the nonlinear model - Ad. Effect of Traffic Pressure. The ANOVA analysis indicated that two factors had a significant influence on Me headway s at each queue position. These factors include: traffic pressure and distance to the back of the downstream queue at the start of the upstream/subject phase. The first factor is described in this section, the second factor is described in the next section. The average headway recorded for each queue position at each site was found to decrease slightly as the number of vehicles served per cycle increased. This trend is consistent wad the findings of Bon neson 629 and Stokes ef al 649. The effect of traffic pressure on discharge headway is shown in Figure C-3. ~. ~' . . Figure C-3 shows the effect of traffic pressure as it exists in one queue position at all sites and at all sites and queue positions combined. The analysis indicated that there were no statistically significant differences In We elect of traffic pressure among sites and queue positions. Comparison of Figures C-3a and C-3 b support this finding. C-9

Minimum Discharge Headway, sev/veh 2.2 2.0 (Each data point represents the average of 35 obs) A A A B A A A A A A A A A A A A A A A A A A A A 1.8 + A A A A 1.6 ~ _+___________+___________~___________~___________+_ 5 10 15 20 25 Through Demand Flow Rate per Lane (Traffic Pressure), vpcpl a) Tra~c pressure e~ectfor queue posifion seven af aRstu~ sifes. Minimum Discharge Headway, sec/veh 2.2 ~ 2.0 + (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 100 obs) A AA AB A A A A AA A B A BA A AA A A ABAA BA A A AAB AA AA AA A A I A BBA B ; I AA A A A A 1.8 ~A A A  I AAAA A A A 1.6 + AB A A A _~___________~___~ ____+_ __________+_______ ____+_ 0 10 20 30 40 Through Demand Flow Rate per Lane (Traff~c Pressure), vpcpl b) Tra~ic press~ure effect for aR queue posifions af ad sfa'~ sites. Figllre C-3. Effecf of tra~qc press~re on fArough movemenf minimz~m ctlischarge headway. C-IO

The data shown In Figure C-3 represent the average of several observations. The number of individual observations was so large that, when they were plotted, they tended to become a black mass of ink which obscured the examination of trend-w~se effects. To overcome this problem, the data were first sorted by tane volume, segregated into contiguous groups of 35 (or 100) observations, and then used to compute average headways and lane volumes for each group. · . ~ . . ~ ~ ~ ~ . ~ · . . . Effect of Distance to the Back of Downstream Queue. As mentioned previously, the ANOVA indicated that Me distance to Me back of downstream queue had a significant effect on the queue discharge headways. This distance is measured from the subject movement stop line to the "effective" back of queue at the start of the subject (or upstream) phase. The effective back of queue represents the location of He back of queue if all vehicles on the downstream street segment (moving Or stopped) at the start of the subject phase were joined into a stopped queue. If there are no moving vehicles at the start of the phase, then the effective and actual distance to queue are the same. If there are no vehicles on the downstream segment at the start of the phase, then the effective distance to queue would equal the distance to the through movement stop tine at the downstream intersection. Figure C-4 shows the relationship between the discharge headway of the fifth and higher . . . . .. .. . . . . . .. . . queue positions and the corresponding distance to the back of downstream queue. As the headways for these queue positrons are considered to be effectively at their minimum value, the data shown represents individual estimates of the minimum discharge headway H. The trend shown in this figure indicates that the minimum discharge headway decreases with increasing distance to queue. Minimum Discharge Headway, sec/veh 2.5 + 1.5 + 1.0 (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 50 obs) A A AA B BAA A A A AA B BAA A BA A A AN A A BCDDBBACA B BAA A A AD CAB CEEDBACAC GAB B ABA A B BA AB AACA ABB BB A A A BCBAA BA A CB A A  0 75 150 225 Distance to the Downstream Queue, m 300 375 Figure C-4. Effect of disfance-fo-queue on through movement minimum discharge headway. C-11

As there are more than 7,700 headways in He data base, plots of individual data points were problematic. Therefore, the data were sorted by distance-to-queue, segregated into contiguous groups of 50 obseIvabons' and then average headways were computed for each distance group. The data shown in Figure C-4 represent these average headway s. The effect of distance-to-queue was found to vary depending on whether the downsteam traffic queue spilled back into the upstream intersection. Spillback conditions are characterized by the backward propagation of a downstream queue into the upstream intersection such that subject (or upstream) intersection movement is effectively blocked from discharging during some or all of the signal phase. In general, the headways of vehicles able to discharge prior to spillback were larger than the headway s of similar vehicles measured dunug phases in which spillback did not occur. Model of Minimum Discharge Headway. Several alternative model forms were considered in an attempt to find the best fit to the data. A simple linear model was considered, however, this form did not provide logical boundary values. For example, it is logical that the headway would tend to very large values at short distances-to-queue and would converge to a nominal minimum discharge headway for large distances-to-queue. Based on this rationalization, the following empirical model form was developed and fit to the data: H = b be (I - Is) + b2 Ash ~(C-S) where: Huh = through movement minimum discharge headway, sec/veh; D = effective distance to the back of downstream queue (or stop line if no queue) at the start of the subj ect (or upstream) phase, m; Is = indicator variable (~.0 if spillback occurs during phase, 0.0 otherwise); vat' = demand flow rate per lane (i.e., traffic pressure), vpcpl; and bo, be, be, be = calibration coefficients. Mode! Calibration. The field data were used to calibrate the minimum discharge headway model, as shown in Equation C-5. The details of the model calibration analysis and statistics describing the model's predictive performance are provided in Table C-3. As the statistics In Table C-3 indicate, the calibrated model explains only about four percent of the variability in the headway data. The remaining variability is primarily due to the random (or unexplainable) variability inherent in headway data. Some of the variability is also due to differences among the traffic lanes and sites studied. Nevertheless, the statistics in Table C-3 indicate that there is a statistically significant relationship between minimum discharge headway, traffic pressure, and distance to the back of queue. The root mean square error and number of observations can be used to estimate the minimum standard deviation (or precision) of the predicted average minimum discharge headway as +0.006 sec/veh. C-12

Table C-3. Calibrated through movement minimum discharge headway mode! Model Statistics Value R2: ~ 0.04 | Root Mean Square Error 0.56 sec/veh Observations: | 7,704 Range of Model Variables Variable | Variable Name | Units ~Minimum H', Through movement min. discharge headway sec/veh 0.61 v,' Demand flow rate per lane (traffic pressure) vpcpl ~ D Distance to back of downstream queue meters 35 Calibrated Coefficien Values Coeff. | Coefficient Definition | Value | Std. Dev. Min. discharge headway for ideal D and v,. 1.94 0.028 Effect of distance to queue (no spillback) S. 13 ~ .45 _ b2 Effect of distance to queue (win spillback) 21.8 1.56 b3 | Effect of A ~ffic pressure T 0.00453 | 0.0005 | Maximum 6.8 37 315 t-statistic 69.3 5.6 14.0 9.1 The data used for model calibration were screened to identify the effects of queue spilIback. This screening was accomplished by using Me videotape record of traffic events obtained dming We field studies. Dunng the screening process, each headway observation included In the database was visually verified as to whether it was affected by spilIback. This approach resulted in there being two categories of observations in the database. One category Included those headways Mat occurred during phases not having spiliback. The other category included Hose headways that occurred dunng phases with spilIback but prior to the first occurrence of spilIback. Headways occurring during phases with spilIback but subsequent to the first occurrence of spilIback were excluded from the database. As the coefficient values In Table C-3 indicate, the magnitude of the effect of distar~ce-to- queue is dependent on whether queue spilIback occurred dunng the phase. Phases without spilIback had a smaller regression coefficient indicating less sensitivity to distance. In general, the coefficients predict a larger minimum headway for those queues discharging prior to the occurrence of spilIback than for those that discharge without spilIback ever occulting. In addition to its intended purpose, the distance-to-queue variable D addresses the effect of intersection spacing (including differences among interchange types). In general, it suggests that discharge efficiency wait likely be lower for the inbound movements at closely-spaced intersections (e.g., interchanges with relatively short distances between ramp terminals) than it would be at overwise similar intersections but with larger spacings. The quality of fit of the calibrated mode} to the headway data is shown in Figure C-5. The plotted data points represent averages of 50 observations each. As discussed previously, this C-13 1

averag ng was performed to overcome the presentation problems incurred when plotting hundreds of data points. It should be noted that the Individual headway observations were used for the mode! calibration; averaged data were used for plotting purposes only. Predicted Minimum Discharge Headway, sec/veh I (Legend: A = 1 obs, B = 2 obs, etc.) I (Each data point represents the average of 50 obs) 3.0 A 2.5 2.0 + 1.5 + A A A A A A A AB A AA C BGAFDACE C AAAAEAFFHECFBBABAB A A BRA C CIFDGB AA A A BB A AA A A 1.50 1.75 2.00 2.25 2.50 Measured Minimum Discharge Headway, sec/veh Figure C-S. Comparison of predicted arid measured" through movement minimum discharge headways. Several additional effects were also evaluated during the mode! calibration process. Specifically, the elect of junchon type, phase durah on, and downstream signal indication were also evaluated. This latter factor was considered because it was reasoned that drivers might discharge at a more efficient rate if the downstream signal indication was green (as opposed to red), particularly if there was no downstream queue. Based on this additional analysis, it was concluded that these factors did not significantly affect discharge headway after the effects of distance to queue and traffic pressure were removed. Interpretation of Mode! Statistics. Three stat~shcs are provided in Table C-3 to indicate the quality of fit of the calibrated model. First, the "t-staustic" is provided for each independent variable to test the hypothesis that its regression coefficient equals to zero. When the t-statistic exceeds T.96, the hypothesis is rejected and it can be concluded that the corresponding vanable has a significant elect on the dependent variable. In this situation, there is a 5 percent (or less) chance of this conclusion being in error. In some situations, the effect of a variable can be misrepresented when there is a large amount of data. This problem was avoided in this research by using an analysis of variance C-14

(ANOVA) approach. An ANOVA is much more robust than a regression analysis at detecting the true effect of an Independent vanable. In addition, a graphical examination was conducted for each independent vanable. This examination provides a practical means of confirming any relationship that may exist between the dependent and independent variables, it can also indicate Me trend between these variables (e.g., linear, log, etch Figures C-3 and C-4 are examples of this type graphical examination. The influence of the independent variables is clearly indicated In these figures; the t-statistic provided for each variable in Table C-3 is further confirmation of the significance of the observed trend. The second measure of quad of fit is the root mean square error. This statistic represents the standard deviation of the dependent vanable. Presumably, We error represented by this statistic is from random sources, however, there could also be some variation due to systematic effects. Knowledge oftypical values of the root mean square error for We dependent vanable can be a usefid gage to assess whether additional systematic error exists in the data. For example, We standard deviation of vehicle headways is rarely reported In the literature to be less than 0.45 sec. Therefore, as the root mean square error of 0.56 reported in Table C-3 exceeds 0.45, it is possible that there is some additional systematic error In the data that additional mode] variables could explain. The third measure of quality of fit is the R2 statistic. This statistic represents the portion of the variability explained by the mode} relative to the total variability In the data. As such, it cart range In value Dom 0.0 to I.0. In general, larger values of R2 indicate a good fit; however, it must be remembered that the value (or range of values) used to denote a "good" fit is dependent on the amount of random variability In the data. For example, the only way that an R2 of 1.0 can be achieved is when all of the variability In the data is due systematic sources (i.e., there is no random error) and We model properly includes an Independent variable for each systematic effect. To illustrate this relationship, the following relationship between the actual and "best-possible" R2 was denved using mathematical relationships between R2 and We partitioned variance structure: ~s2 R2 = 1 - (1 - R2) r best ac ~2 act where: R2`best = largest possible R2 value obtainable (i.e., all systematic error explained); R2aC, = R2 value obtained from a regression analysis; 02r= variance in data due to random sources; and 02aC' = variance obtained from a regression analysis (i.e., = root mean square errors. (C-6) To illustrate the implications of this equation, consider the "actual" R2 and o2 values provided in Table C-3. If it can be assumed Nat We variance due to random sources is about 0.20 (= o.452), Men the largest R2 value possible is 0.38 (= 1 - (1 - 0.04) 0.452 / 0.562 ); it is not the 1.0 that is traditionally assumed. Data sets with larger amounts of random variability w~11 yield even lower maximum values for R2. C-15

Other factors that have a significant effect on Me value of R2 are the eject ofthe independent variable (i.e. We absolute value of the corresponding variable coefficient) and its vanability. In general, the R2 wait be limited to relatively low values when Me independent vanable's coefficient or its variance are small (and Mere is some variability due to random sources). The following equation can be derived for computing R2 using mathematical relationships between R2 and the partitioned variance structure: b2 ~2 R2 _ ~ x act `~2 -I- b2 <~2 r 1 x where: Ax = variance in the independent vanable; and by = regression coefficient for Me independent vanable. (Cal To illustrate the implications of this equation, consider the equation H = ho + b,/D (this equation is similar in fom1 to Equation C-S). If the variance of the independent variable (1/D) is 0.000036, the value of be is ~ 5, and the variance due to random sources is 0.20, then the R2 ~l be about 0.04. These variable values were chosen to be consistent with the data used in calibrating Equation C-S arid serve to illustrate the effect of a low variance in the independent variable on the R2. Based on this computation, Me R2 value of 0.04 cited In Table C-3 is probably about as large a value as could be expected given the relatively small earl ce of Me independent vanable. It should be noted Mat Equation C-7 is generally Insensitive to the form of Me model. To illustrate this attribute, consider the equation H = ho + be D. Using the same data as for the preceding example, values of -0.002 and 2,025 are obtained for by and o2x, respectively. As before, the resulting R2 is about 0.04. Hence, the R2 obtained from either mode! is essentially the same. Moreover, this example illustrates the effect of a relatively "small" (but significant) regression coefficient on R2; that is, small regression coefficients inherently yield small R2 values. Thus, in a comparison oftwo regression models with identical root mean square errors and t-statistics, Equation C-7 guarantees Mat the mode} with the larger coefficient (i.e., sIope) will have the larger R2. Yet, the similarity in root mean square errors and t-statistics suggest that each mode} is equally good. In general, Me models developed in this and subsequent sections deal with traffic data representing the reactions of individual drivers to the roadway geometry, the traffic signal, arid over vehicles. As a result, the data have a relatively large random error component making it impossible to obtain an R2 near I.0, regardless of the model form. More importantly, the magnitude of the systematic eject (as represented by He Independent vanable coefficient) is generally small for this type of traffic data which limits the R2 to low values. In conclusion, the R2 statistic is a usefill measure of mode! quality of fit when used for relative comparisons among similar models and data sets. It is impossible to universally define one single R2 value as He "niinimum acceptable value." Sensitivity Analysis. The calibrated discharge headway mode} was used to examine He effect of distar~ce-to-queue, spilIback, and traffic pressure on He queue discharge rate. These ejects are shown in Figures C-6 and C-7 for the discharge headway and saturation flow rate, respectively. . C-16

3-0 2.0 1.5 Minimum Discharge Headway, sec/veh 3.5 Traffic Pressure: 3 vpcpl \\ 1 0 vpcpl 1` ~\\ With Spillback 2.5 ~\~ \~ W~ 7~ Without Spillback I I _L I o 60 120 180 240 300 Distance to the Downstream Queue, m Figure C-6. Effect of distance-to-queue, Spillback occurrence, and tragic pressure on through movement minimum discharge headway. 1 900 1 700 1 500 1 300 1100 900 700 Saturation Flow Rate per Lane, vphgpl Without Spillback ~ \~L - ~' --- it--~-- - - _ = ~ ~ ~With Spillback ~a/ - - - - -D- - - - - - - . . ~Traffic Pressure: / 3 vpcpl 1 - - - - - - - - - - - - - - - - . 1 0 vpcpl 1 i 1 1 o 60 120 180 240 300  Distance to the Downstream Queue, m Figure C-7. Elect of distance-to-queue, Spillback occurrence, and tragic pressure on through movement saturation flow rate. C-17

Observation of Figure C-6 indicates that the calibrated model has a trend of decreasing headway with increasing distance-to-queue which is consistent with the trend shown in Figure C-4. Phases that Incurred spillback tended to be associated with larger headways and lower saturation flow rates. Specifically, discharge headways are higher for those vehicles that are able to discharge prior to the first occurrence of spillback. Traffic pressure decreases the minimum discharge headway, although, the effect of traffic pressure is small relative to that of distance-to-queue or spillback. The calibrated model predicts a mirnTnum headway of ~ .94 sec/veh (= 1,856 pcphgpl) when the distance-to-queue is irdinitely long and there is no traffic pressure effect. Inclusion of a nominal traffic pressure of 5.0 vpcpl yields a magnum headway of ~ .90 sec/veh (= 1,900 pcphgpl); this value is consistent with the ideal saturation flow rate recommended by the HCM (19 for signalized intersections. The calibrated model also logically predicts infinitely long headways when the distance-to-queue is zero. Saturation Flow Rate Model. The calibrated Trough movement minimum discharge headway model was converted into an equivalent saturation flow rate model. The form ofthis model was patterned after that used in Chapter 9 of the HCM (1). Specifically, the saturation flow rate for a particular location is estimated as the product ofthe ideal saturation flow rate and the venous site- specific adjustment factors. In this context, the adjustment factors found in this research relate to the elect of distance-to-queue, spillback occurrence, and traffic pressure. The basic form of the mode! is: 5! = So XfD xf~ where: saturation flow rate per lane under prevailing conditions, vphgpl; so = saturation flow rate per lane under ideal conditions, pcphgpl; fD = adjustment factor for distance to downstream queue at green onset; and fV = adjustment factor for volume level (i.e., traffic pressure). (C-X) The ideal saturation flow rate represents Me saturation flow rate when not affected by any external environmental factors (i.e., grade), atypical vehicles (i.e., trucks), and constrained geometries (e.g., less there 3.6-meter lane widths, curved travel path). In this regard, the saturation flow rate would be equal to the ideal rate when all factor effects are optimum for efficient traffic flow and the corresponding adjustment factors are equal to I.0. Based on this definition, it was determined Mat an inflate distar~ce-to-queue under non-spilIback conditions and a traffic pressure of 1 5.0 vpcp! were representative of ideal conditions. Using this definition of ideal conditions and associated parametric values, the resulting ideal saturation flow rate arid adjustment factors were algebraically derived from Equation C-5 as: 3,600 So b (l-b 150) (C-9) = 1,990 pcphgp! C-~8

1 fD - : no spillback + 8.13 D 1 + 21.8 D f = : with spiliback 1 (1 - b3Vl)/(l - b315.U) 1.07 - 0.00486 v'' (C-IO) (C-~) where: D = effective distance to the back of downstream queue (or stop line if no queue) at the start of the subject (or upstream) phase, m, v,' = demand flow rate per larle (i.e., traffic pressure), vpcpl; and by b3 = calibration coefficients from Table C-3. The precision of the ideal saturation flow rate predicted by Equation C-9 is estimated as +12 pcphgpI, based on the root mear1 square error and number of observations shown in Table C-3. As the resulting range of possible true ideal values includes 2,000 pcphgpI, this value is recommended as the ideal saturation flow rate for Trough movements. The relationship between distance-to-queue and the corresponding saturation flow adjustment factored is shown in Figure Cue. The trends shown are similar to those noted for the saturation flow rates shown in Figure C-7. Specifically, the adjustment factor (and saturation flow rate) increase as the distance to the back of queue becomes longer. Phases that incur spilIback have a smaller factor value, for the same distance to queue, than phases that do not incur spilIback. C.~.3 Sah~raffon Flow Rate Mode! for LefI-Turn Movements With one exception, the left-tu~n movements included in this study represent left-turns at Interchange ramp terminals. The one exception was a left-turn movement at an adjacent signalized intersection. Of the two types of left-turn movements at ramp terminals (i.e., off-ramp and arterial), the majority of the data were collected for the off-ramp leD-turn movement. Nevertheless, it is believed that Me factors identified in this section are sufficiently general that they are applicable to off-ramp arid arsenal left-turn movements at interchanges and to left-turn movements at adjacent intersections. C-19

1 o-9 0.8 0.6 0.5 0.1 Saturation Flow Rate Adiusiment Factor 0.0 0 60 120 180 240 300 Distance to the Downstream Queue, m Figure C-~. Effect of distance-to-que?ve and spiliback occurrence on the through movement saturation flow rate adjustmentfactor. lathe left-turn movements studied rarely, if ever, expenenced queue spilIback during the study periods due to He nature of He signal phase coordination between the two interchange ramp terminals. Hence, in contrast to Be though movements studied, the vanability In left-turn headway s among sites cannot be explained by differences in the distance to He downstream queue. This restriction is a characteristic of the twelve sites studied, certainly, left-turn movements can be affected by downstream queuing conditions at over sites. In fact, it is likely Hat the effect wall be very similar to that fourth for the through movements. Factors Affecting Discharge Headway. A review of He literature on He topic of left-turn headways suggests that several site-specific factors exist that can have an elect on the left-turn discharge process. For example, Kimber et al 66J measured saturation flows on curves with radii ranging from 6 to 35 meters and developed an equation for predicting saturation flow rate as a function of turn radius. An equivalent relationship, as it relates to minimum discharge headway, is: Hit = 1.73 ~ 2.60 R where: H.' = left-turn movement minimum discharge headway, sec/veh, and R = radius of curvature of the left-turn travel path (at center of palm, m. (C-12) In a previous study of single-point urban intercharlges conducted by Messer et al (39, Borneson 62) found an effect of radius on headway consistent with that found by Kimber 669. The C-20

range of radii included In this study was ~ ~ to 84 meters. Bonneson also found Mat We number of vehicles served per cycle had art effect on left-burn headway. This latter elect was referred to as that of "traffic pressure" (as discussed in a preceding section). Bonneson recommended Me following equation for predicting the minimum discharge headway of a left-turn movement as a function of radius and traffic pressure: H = 1.58 + - - 0.0121 v' -It R 0.245 where: v`' = demand flow rate per lane (i.e., traffic pressure), vpcpl. (C-13) The HCM (~) describes many additional factors Mat can affect discharge headway. These factors include: lane width, vehicle classification, local bus frequency, parking activity, and approach grade. To avoid confounding the elect ofthese factors with those specifically being considered in this study (e.g., turn radius), several steps were taken to avoid or remove the aforementioned factors from the data collected for this project. Specifically, the study sites all had lane widths of about 3.6 meters, approach grades of less than +2.5 percent, no local busses, and no parking activity. In addition, all heavy vehicles (i.e., vehicles with more than two axIes) and all queued vehicles that followed heavy vehicles were removed from He data base. Analysis Approach. The analysis ofthe leD-turn headways focused on an examination of the headways for passenger cars in the fifth and higher queue positions at eleven study sites. Analysis of Variance (ANOVA) techniques were used to control for differences in sample size and to account for extraneous differences among otherwise similar sites. All significance tests were conducted at a 95 percent confidence level (i.e., a = 0.051. Once the influential factors were identified Dom the ANOVA, both linear and non-l~near regression techniques were used to calibrate the data (via these factors) to the proposed model. More details of the modeling approach are provided In a preceding section describing the through movement saturation flow model. Traffic Pressure Effect. The ANOVA analysis indicated that three factors had a significant influence on the headways at each queue position. These factors include: traffic pressure, signal timing, and tum radius. The first factor is described in this section, the latter two are described in the next two sections. The combined eject of distance to queue and spilIback were not considered in He left-turn movement analysis. This omission was a consequence ofthe signal coordination used at the study sites. This coordination was generally good for He left-turn movements such that they did not experience spilIback nor were Hey faced with lengthy dowr~sheam queues. The average headway recorded for each queue position at each site was found to decrease slightly as the number of vehicles served per cycle increased. This trend is consistent with He findings of Bonneson 629 and Stokes et al (49. The effect of traffic pressure on discharge headway is shown in Figure C-9. C-21

Minimum Discharge Headway, sec/veh 2.1 + 2.0 + 1.9 (Each data point represents the average of 35 obs) A A A A A A A A A A A A 1.8 + A A A A A 1.7 + A 5 10 15 20 Left-Turn Demand Flow Rate per Lane (Traffic Pressure), vpcpl a) Traffic pressure e~fectfor queue position seven at aR study sites. 2.2 + 2.0 + A A IA A A A B A AA Minimum Discharge Headway, sec/veh (Legend: A = 1 obs, B = 2 obs, etc.) (Each data po~nt represents the average of 75 obs) A A A ~ A A CA A B A B AA A A A A A A A A A 1.S ~A A A A A | A A A A A A I A A A A  1.6 + A ~ 10 15 20 25 Left-Turn Demand Flow Rate per Lane (Traffic Pressure), vpcpl bJ Tra~c pressure eff~ectfor aR queue positions at aR study sites. Figure C-9. Effect of traffic pressure on left-turn movement minimum discharge headway. C-22

Figure C-9 shows the elect of traffic pressure as it exists in one queue position at all sites and at all sites and queue positions combined. The analysis indicated that there were no statistically significant differences in Me effect of traffic pressure among sites and queue positions (~omn~ri~nn of Figures C-9a and C-9b support this finding. ----of ~ -- ~r~~~~~~ The data shown In Figure C-9 represent the average of several observations. The number of individual observations was so large that, when they were plotted, they tended to become a black mass of ink which obscured the examination of trend-wise effects. To overcome this problem, We data were first sorted by lane volume, segregated into contiguous groups of 35 (or 75) observations, and then used to compute average headways and lane volumes for each group. Signal Timing Effect. The ANOVA tests also indicated that Me left-sum movements were affected by the duration of the phase and the cycle length. The effects of these two factors were dissimilar in Me sense that drivers adopted shorter discharge headways dunng shorter phase durations or longer cycle lengths or both. A similar effect of phase duration on headway has been noted by Stokes et al 649. The Implication of these findings is that drivers wall adopt shorter headways to avoid Me additional delay associated with having to wait for the next phase, particularly when Me expected delay is large (due to a large cycle length). As delay is theoretically related to the effective-green-to-cycle-length ratio g/C, the two factors were combined into this ratio for further examination. The results of this examination are shown in Figure C-10. Figure C-10 shows the effect of signal timing as it exists in one queue position at all sites and at all sites and queue positions combined. This analysis indicated that there was no statistically significant difference In the effect of signal tuning among sites and queue positions. Companson of Figures C-IOa and C-IOb support this finding. Closer examination of Figure C-lOb indicates that Me effect of g/C ratio is not constant over the range of g/C ratios observed. Specifically, it appears that the effect of g/C ratio is [united to values less than about 0.27. Beyond this value, the data suggest that the effect of g/C ratio is negligible. As wad Figure C-9, the data shown in Figure C-IO represent the average of several observations. This averaging was performed to facilitate the examination of trends In Me data rather there its variability. Unlike the left-turn movements, the g/C ratio was not foment to have significant effect on the through movement discharge rate. This finding is likely due to the large g/C ratios experienced by the through movements, relative to the left-turn movements. The examination of the left-tu~n movements found Mat the elect of g/C ratio was limited to situations where the ratio was relatively small (i.e., less than 0.27~. As the average g/C ratio for Me Trough movements was about twice that of Me left-tu~n movements, there were rarely any Trough movement phases Mat had a giC ratio low enough to warrant driver anxiety about being delayed for an additional signal cycle. Therefore, the existence of an effect of g/C ratio on through headways may exist but it cannot be determined (as it was for lefts n headways) because there is Insufficient data Tom sites with through movements operating at low g/C ratios. C-23

Minimum Discharge Headway, sec/veh 2.0 + (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 35 obs) A A A 1.9 + A A A A A 1.8 + A A A A A AA A A A 0.1 0.2 0.3 0.4 0.5 Effective Green / Cycle Length Ratio (g/C) a) Signal timing e~ectfor queue positior' seven at aR study sites. Minimum Discharge Headway, sec/veh (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 75 obs) A 2.0 + 1. 8 + A AA AA AA A A A A A 1.6 + A A A A A A AAA A A A B AA A AB A A AAAA A AA A A  AA A _ _ + _ _ _ 0.1 A BA A A A A 0.2 0.3 0.4 Effective Green / Cycle Length Ratio (g/C) b) Signal timing e~ectfor all queue positions at all stucly sites. Fig~ure C-l O. Effect of signal timing on left-turn movement minimum dlischarge head~wcIy. C-24

Travel Path Radius Effect. The radius of He travel path was examined for its effect on He minimum discharge headway. To facilitate this examination, the minimum discharge headways were tabulated for each site and traffic lane studied. In addition, the minimum discharge headways of several sites studied during a previous NCHRP project (3) were included in the tabulation to increase its range of coverage. The results of this effort are summarized In Table C-4. Table . Leit Tum Lane Type Single Lane |DUa1 |Lane Cal. Left-turn movement minimum discharge headway database summary Outside Lane of Dual L=es2 v,' (vpcpl) Junction Type' AGI Parclo B . SPUI . CDI ~ SPUI TUDI  TUDIw/i Location ~City, State 1 SR 60 & Belcher Road 1 - 1 Clearwater, FL SR 60 & Belcher Road . TV u~= Balm aD . Wellborn Rd. & Old Main Dr Clearwater, FL College Station, TX College Station, 1 Matilda Ave & Moffitt Rd. Sunnyvale, CA I-880 & Stevenson Blvd Newark, CA SR 4 & Somersville Rd. US 19 & SR 694 US 19 & SR 694 St. Petersburg, FL US 19 & SR 694 St. Petersburg, FL _ US 19 & SR 60 US 19 & SR 60 Antioch, CA St. Petersburg, PL Clearwater, FL Clearwater, FL l Metcalf Ave. & I-435 75th Street & I-35 Maple Street & I~80 Indian School Rd. & SR-5 1 USl9&SR686 US 19 & SR 686 7~ Street & I-10 Overland Park, KS Overland Park, KS Omaha, NE Clearwater, FL Phoenix, AZ Largo, FL Largo, FL Phoenix, AZ Peoria Ave. & I-17 Arapaho Rd. & US-75 Towneast Blvd. & I-635 Phoenix, AZ Richardson TX Mesquite, TX Leflc- Turn Radius4 (m) . Single Lane orInside Lane of Dual Lanes2 Obs. 181 18 18 18 23 24 52  7 85 5 - 5~ ~3 18 1 15 20 58 76 7 7 91 1 23 1 17 20 45 ]? 229 268 74 42 458 488 410 1,899 1,086 1,451 1,316 657 76 73 499 584 367 2,318 392 51 _I 919 408 192 l s34 1 | Hlr | 1 (S/V) 1 1 1.891 1.93 1.87 1 1.98 1 l.g3 1 1.89 i 1.98 192 1.86 1.97 1.87 1.82 1.91 1.89 1.98 1.66 1.78  1.85 1.83 1.84 1.88 1.92 1.85 OverallAverage:l 43 1 599 1 1.88 1 (~v) &Ac Ratio3 l 1 0.10 ~ _ . 0.10 1 0.18 i 0.18 0.20 1 0.22 1 0.22 0.18 1 0.20 1 0.20 0.22 0.22 0.19 0.19 1 0.17 0.22 1 0.18 1 0.15 1 0.23 T 0.13 0.20 0.32 0.20 0.21 1 1 0.20 0.19 1 - AGI - at-grade intersection, SPUI - single-point urban interchange; CDI - compressed diamond interchange; TUDI - tight urban diamond interchange (no frontage road or U-tu~n lanes); TUDI w/f - tight urban diamond interchange (with frontage road and U-turn lanes); Parclo B - partial cloverleaf with off-ramps beyond overpass. - Hit= left-turn movement minimum discharge headway. v,' = demand flow rate per lane (i.e., traffic pressure). 3 - g/C = effective green duration g of the subject leR-turn phase divided by the cycle length C. 4 - Turn radius of the left-turn lane (inside lane of dual-lane bays), measured to the center of the travel paw. Radii represent the visual "best fit" approximation of the travel path of the turn movement along a single radius curve. C-25

A preliminary examination of the relationship between minimum discharge headway and radius was conducted using Me average headways shown in Table C-4. This relationship is shown in Figure C-] ~ . As suggested by the scatter in the data shown In this figure, the results of this analysis were somewhat inconclusive; however, Were appeared to be a slight trend toward shorter headways on travel paths of larger radius. 2.1 1.9 1 1.7 1.6 Minimum Discharge Headway, sec/veh 1 1 1 1 1 1 0 15 30 45 60 75 90 105 Radius of Curvature, m Figure C-~. Effect of travel path radius on left-turrl movement minimum discharge headway. As was noted In a previous examination of Table C-l, Me variability in minimum discharge headways among sites can often be largely explained by the bias due to queue position and frequency of observation. The ANOVA techniques descnbed previously were used to remove this bias arid, thereby, facilitate a closer examination of the elect of radius. The results of this analysis indicated that Me effect of radius was statistically significant and Mat headways ace shorter on travel paths wad larger radii. Dual vs. Single Left-Turn Lane Effect. The effect of dub versus single lane lefc-turn movements was also examined for this analysis. The average minimum discharge headways of bow designs were computed and are shown in Table C - . The averages reported in this table indicate Mat there is no significant difference between single and dual-lane left-turn movements. The averages also suggest that there is no difference between Me inside and outside lane of the dual-lane movements. An ANOVA analysis of Me individual headway observations confirmed that Mere was no significant difference among single and dual-lane movements in Me sites studied (after the effects of traffic pressure, signal timing, and radius had been removed). These findings are consistent with Nose reported by Bonneson (29. C-26

Mode! of M~n~mum Discharge Headway. Based on a review of model forms used by other researchers for predicting the minimum discharge headway for left-turn movements and the trend analysis described In the preceding sections, the following empirical model form was developed and fit to the headway data: H w~th, I bo ! 1 + 1 ( 2 l )( 3 g) (C-14) tg = ( C) Ig + go (1 g) (C-15) where: Hi' = left-turn movement minimum discharge headway, sec/veh, R = radius of curvature of the left-turn travel path (at center of path), m; v/' = demand flow rate per lane (i.e., traffic pressure), vpcpl; g = effective green time where platoon mohon ttlow) can occur, see; C= cycle length, see; tg = signalization variable (0.0 < tg < go); Ig = indicator variable (1.0 if g/C < gX , 0.0 otherwise); gX = maximum g/C ratio (larger g/C ratios have no additional effect on headway); and by b,, b2, b3 = calibration coefficients. Mode! Calibration. The field data were used to calibrate He left-tu~n movement minimum discharge headway model, as shown in Equation C-14. The details ofthe model calibration analysis and statistics describing the model's predictive performance are provided in Table C-5. As the statistics In Table C-5 Indicate, the calibrated model explains only about five percent ofthe variability In He headway data. The remaining 95 percent of the variability is primarily due to the inherent randomness in headway data (as described in a preceding section dealing with the saturation flow rate model for Trough movements). Nevertheless, He statistics in Table C-5 indicate a statistically significant relationship between minimum discharge headway, radius, traffic pressure, and g/C ratio. The minimum precision of He average headway estimate is about ~tO.007 secJveh. The quality of fit of He calibrated model to the headway data is shown in Figure C-12. As discussed previously, the plotted data points represent averages of 75 observations each. This averaging was performed to facilitate He examination of trends rather than variability. The data were first sorted by predicted headway, segregated into contiguous groups of 75 observations, and then average headways were computed for each group. It should be noted that He individual headway observations were used for He model calibration; averaged data were used for plotting purposes only. C-27

98 26 t-statistic Table C-S. Calibrated lefI-turn movement minimum discharge headway mode! . Model Statistics Value R2. 0.05 Root Mean Square Error: 0.44 sec/veh l Observations: | 4,153 Range of Model Variables Vanable Variable Name Units Minimum He Left-turn movement min. discharge headway sec/veh 0.83 Radius of curvature of travel paw meters 15 _ v,' Demand flow rate per lane (traffic pressure) vpcpl 5 g/C Effective green to cycle length ratio na 0.06 . Specified Parameter Values Variable Variable Name | Units | Value . gx Maximum g/C ratio na 0.27 I_ ~ Coeff. ~ Coefficient Definition Value Std. Dev. be Intercept 1.55 0.034 _ b, Effect of radius of travel path 1.71 0.29 _ b2 Effect of traffic pressure 0.00630 0.00090 . ~ b3 | Effectofg~ratio | 0.868 | 0.128 | '~ Maximum 3.5 0.55 45.6 5.9 7.0 6.8 Predicted Minimum Discharge Headway, sec/veh 2.2 + 2.0 + .8 + .6 + (Legend: A = 1 obs, B = 2 obs, etc. ) (Each data point represents the average of 75 obs) A A A A AA  A A B A A ~ B AC AC A A A A AABBAA A A AAB A APs AA A A A A A A A A 1.4 1.6 1.8 2.0 2.2 Measured Minimum Discharge Headway, sec/veh Figure C-12. Comparison of predicted and measured" leff-turn minimum discharge head~ways. C-28

The predictive ability of Me calibrated mode! Carl also be assessed by comparing it win the models developed by Bonneson (29 arid by Kimber et al 664. The Tree models are compared in Figure C-] 3 over a range of leR-turn radii. As this figure indicates, We model developed for this research is In good agreement why bow of these previously calibrated models. 2.1 2.0 1.9 1.3 ~ 7 1.6 Minimum Discharge Headway, sec/veh \~< Equation C-14 Kimber (6) ~ , ~ Bonneson (2) Left-tun, volume = 6.0 vpcpl; g/C = 0.20 1 1 1 1 1 1 1 0 15 30 45 60 75 Radius of Curvature, m 90 105 Figure C-13. Comparison of alternative models of left-turn minimum discharge headway. Sensitivity Analysis. The calibrated mode} was used to examine the effect of radius, traffic pressure, and g/C ratio on the minimum discharge headway and saturation flow rate of left-turn movements. This effect is shown in Figures C-14 and C-15 for Me respective characteristics. The trend tines shown in these figures reflect left-turn volumes of 3 and 10 vpcp} and g/C ratios of 0.16 and 0.27. These ranges were selected to be inclusive of about 90 percent of the observations In the database. In general, the mode] has a trend of decreasing headway with radius. The range in traffic pressure considered makes a difference of about 0.08 sec/veh in minimum discharge headway and about ~ 00 vphgp} in saturation flow rate. In contrast, the g/C ratio has almost twice Me effect as traffic pressure (i.e., a change of about 0.15 sec/veh and 160 vphgpl). Of course, g/C ratio has no effect when Me ratio increases beyond 0.27. C-29

2.2 2.1 1.9 1.8 1.7 Minimum Discharge Headway, sec/veh Traffic Pressure: 3 vpcpl 1 0 vpcpl )e g/C > 0.27' -I ~ g/C = 0.16' 0 15 30 45 60 75 90 105 Radius of Curvature, m Figure C-14 Effect of traffic pressure, signal timing, and radius on leff-turn movement minimum discharge headway. 2200 Saturation Flow Rate per Lane, vphgp! 2100 2000 1 900 1 800 1 700 1 600 1 1 1 0 15 30 45 Traffic Pressure: 3 vpcpl 1 0 vpcpl 60 75 90 105 Radius of Curvature, m Figure C-15. Effect of traffic pressure, signal timing, and radius on left-turn movement saturation flow rate. C-30

Saturation Flow Rate Model. The calibrated left-turn movement mirumum discharge headway mode! was converted into an equivalent saturation flow rate model. The fonn of this mode} was patterned after Mat used in Chapter 9 of the HCM Aid. Specifically, the saturation flow rate for a particular location is estimated as the product of the ideal saturation flow rate and the venous site- specific adjustment factors. In this context, the adjustment factors found In this research relate to the elect of traffic pressure, signal timing, and turn radius. The basic form of Me mode] is: I O fR fv fg~c where: so = saturation flow rate per larle urlder prevailing conditions, vphgpl; s0 = saturation flow rate per lane under ideal conditions, pcphgpl; fR = adjustment factor for the radius of the travel path; fV = adjustment factor for volume level (i.e., traffic pressure); and fg~c = adjustment factor for signal timing. ~ - · ~· ~ (C-16) The ideal saturation flow rate represents the saturation flow rate when not affected by any external environmental factors (i.e., grade), atypical vehicles (i.e., trucks), and constrained geometries (e.g., less than 3.6-meter lane widths, curved travel path. In this regard, the saturation flow rate would be equal to the ideal rate when all factor effects are optimum for efficient traffic flow and the corresponding adjustment factors are equal to I.0. Based on this definition, it was determined that an infinite radius, a traffic pressure of 10.0 vpcpI, and a g/C ratio greater than 0.27 were representative of ideal conditions. Using this definition of ideal conditions and associated parametric values, Me resulting ideal flow rate arid adjusunent factors were algebraically denved from Equation C-14 as: s = 3,600 ~_ . ° ho (1 - 10.0 b2) (1 + 0~27 b3) = 2,010 pcphgpl 1 do 1.7 1 R 1 + fV (1 -b2vl')1(1 -b210.0) 1.07 - 0.00672 is' C-31 (C-~ (Cars) (C-19)

1 f = ' glC (1 + b3lg)1(1 ~ b3O.27) 1 0.8 1 0 ~ 0.703 t g f = ( c) ~ + 0.27 (1 - I ) where: R = radius of curvature of the left-turn travel path (at center of path), m; al' = demand flow rate per lane (i.e., traffic pressure), vpcpl; g = elective green time where platoon motion (flow) can occur, see; C = cycle length, see, tg = signalization variable (0.0 ~ tg < 0.27~; Ig = indicator variable (~.0 if g/C < 0.27 , 0.0 otherwise); and by by, by = calibration coefficients from Table C-S. (C-20) (C-21) The precision of the ideal saturation flow rate predicted by Equation C-17 is estimated as +12 pcphgpI, based on the root mean square error and the number of observations shown In Table C-S. As the resulting range of possible true ideal values includes 2,000 pcph~l, Mat this latter value is recommended as the ideal saturation flow rate for left-turn movements. The relationship between turn radius and the corresponding saturation flow adjustment factor By is shown In Figure C-16. The trends shown are similar to those noted for the saturation flow rates shown In Figure C-15. This figure extends Figure C-15 by including the adjustment factor developed by Messer et al (3, p. 469. Inspection of the trend line corresponding to Messer's mode] indicates that it yields slightly higher values than those of Equation C-~. This difference is due primanly to the fact that Messer's mode! includes the effect of traffic pressure. C.2 START-UP LOST TIME The effective time "lost" at the start of a phase (i.e., the start-up lost tune) stems from Me fact that the headways of the vehicles In the first few queue positions are larger than those of vehicles . . · · ~1 1 1 ~ ~1 ~ , r t ~ e 1 ~ 1 In t ~e leg per queue positions. lice headways or these first tew queued vehicles are large because of the acceleration they are undergoing as they cross the stop line. Once the vehicles In these positions near the "desired" discharge speed, their headways converge to the minimum discharge headway. Thus, factors that influence this speed (e.g., distance to queue, radius, etc.) also affect minimum discharge headway and saturation flow rate. As a result, there is an inherent relationship between saturation flow rate and start-up lost tune. C-32

Saturation Flow Rate Adjustment Factor 1.0 0.9 0.S 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 An_ ,^ ~ 1 l -Proposed Model -Messer et al (3) 1 1 1 1 1 1 1 0 15 30 45 60 75 90 105 Radius of Curvature, m Figure C-16. Elect of turn radius on the saturation flow rate adjustmentfactor. This section describes We development and calibration of Me start-up lost time model. As wad Me saturation flow rate model, a start-up lost time mode! is developed for bow Me leR-turn arid the Trough movement. C.2.l Start-Up Lost Time Mode! for Through Movements Effect of Saturation Flow Rate on Start-Up Lost Time. Equations C-l, C-2, arid C-3, were used to compute the minimum discharge headway, saturation flow rate, and start-up lost time for the through movements studied. The relationship found between start-up lost time and saturation flow rate is shown in Figure C-17. The data shown represent the average of 50 observations. These observations were generated by first sorting the database by saturation flow rate, combining the sorted values into groups of 50, arid then computing the average saturation flow rate and start-up lost time for each group. This averaging was performed to better illustrate the trend-w~se relationship between the two variables in light of Me presentation problems associated with plotting hundreds of individual observations. As Figure C-17 indicates, there is a strong correlation between start-up lost time and saturation flow rate. Specifically, start-up lost time increases in a linear manner with increasing saturation flow rate. Stan-up lost times for the database range from about 0.5 to 4.0 seconds for saturation flow rates ranging from 1,400 to 2,300 vphgpl. C-33

Start-up Lost Time, sec. (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 50 obs) 4 + 2 + O + A A A A A. AA A AAAA. ABAAA LAB AAA A A A A 1250 lSOO 1750 2000 2250 2500 Saturation Flow Rate per Lane, vphgpl Figure C-] 7. Effect of through movement saturation flow rate on startup lost time. The trend In Figure C-17 supports the contention that Me effective amount of time lost at the start of a phase is dependent on the saturation flow rate. In particular, it suggests that more time is lost when We saturation flow rate (and corresponding desired speed) is higher. In a summary of cause and effect, start-up lost bme increases wad saturation flow rate because it takes more time for the discharging queue to attain the higher discharge speed associated win a higher saturation flow rate. Of course, any resulting loss In capacity due to the added lost time is offset by an Increased saturation flow rate. Mode! of Start-up Lost Time. Based on the trend in Me data shown in Figure C-17, the following model form was developed for Me start-up lost time model: Z = be ~ be so where: Is = start up lost time, see; and s' = saturation flow rate per lane under prevailing conditions; vphgpl. (C-22) Mode! Calibration. The details of Me model calibration analysis and statistics describing the predictive performance of the model are presented in Table C-6. As the statistics in this table indicate, the calibrated model explains about 41 percent of the variability in the data which is indicative of a strong correlation between start-up lost tune and saturation flow rate. Based on the root mean square error arid number of observations, the minimum precision of Me average start-up lost time estimate is about +0.02 sec. C-34

Table C-6. Calibrated through movement start-up lost time mode! , Model Statistics ~Value R2 0.41 Root Mean Square Error: | 1.07sec j Observations: ~ 1,927 Range of Model Variables Variable | Variable Name Units Minimum 15 Start-up lost time see -3.1 l | Saturationilow rate perlane | vphgpl | 1,257 ~l Calibrated Coefficient Values l Coeff. | Coefficient Definition | Value | Std. Dev. I ° 1 Percept I - .64 1 0.199 l b, | Effect ofsa:uration flow rate | 0.00373 | 0.00010 ; , Maximum 8.0 3,326 t-statistic -23.3 37.3 The quality of fit of the calibrated mode! to the data is shown In Figure C-] 8. For reasons discussed previously, Me plotted data points represent averages of 50 observations each. The data shown in Figure C-~8 confirm that We mode! is able to accurately predict the start up lost time over the range of saturation flow rates included In Me database. Predicted Start-up Lost Time, sec. (Legend: A = 1 obs, B = 2 obs, etc.) 4 + 2 + O + (Each data point represents the average of 50 obs) A A A A AA BB AA BA A ACA ARAB ABAA AA A AA A A A o 2 Measured Start-up Lost Time, sec.  Figure Cow. Comparison of predicted and measured through movement start-up lost times. C-35

Sensitivity Analysis. The calibrated model of start-up lost time was used to examine We sensitivity of through movement start-up lost time to saturation flow rate. This relationship is shown In Figure C-19. As this figure indicates, Me start up lost tunes for saturation flow rates of 1,800 and 1,900 vphgpt are about 2.0 and 2.5 seconds, respectively. These values are slightly larger than the .0 to 2.0 seconds recommended in Chapter 2 of Me HCM (1J. 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1400 Start-up Lost Time, see 1600 1800 2000 Saturation Flow Rate per Lane, vphgpl 2200 Figure C-19. Expected through movement start-up lost time as a function of saturation flow rate. C.2.2 Stari-up Lost Time Mode! for Left-Turn Movements Effect of Saturation Flow Rate on Start-up Lost Time. Equations C-l, C-2, and C-3, were used to compute Me minimum discharge headway, saturation flow rate, and soup lost time for the left-turn movements studied. These movements pnmanly include the off-ramp and arterial left-turns at interchange ramp terminals; however, the relationships found are also applicable to the left-turn movements at adjacent signalized intersections. The relationship found between start-up lost time and saturation flow rate is shown in Figure C-20. The data shown in Figure C-20 represent the average of 50 observations. This averaging was perfonned to better illustrate the relationship between the saturation flow rate and start-up lost time In light of the presentation problems associated with plotting hundreds of individual observations. These observations were generated by first sorting the database by saturation flow rate, combining the sorted values into groups of SO, and Men computing the average saturation flow rate and start-up lost time for each group. These averages were computed for graphical presentation purposes only, Me individual observations were analyzed during mode} calibration. C-36

Start-up Lost Time, sec. (Legend: A = 1 obs, B = 2 obs, etc.) 4 3 2 ~ 1 + A (Each data point represents the average of 50 obs) A A A A A A A A A A A A 1500 1750 2000 2250 2500 Saturation Flow Rate per Lane, vphggl Figure C-20. Effect of lefi-turn movement saturation flow rate on start-up lost time. As this figure indicates, there is a strong correlation between start-up lost time and saturation flow rate. Specifically, startup lost time increases In a linear manner with increasing saturation flow rate. S=t-up lost tunes for We database range from about I.0 to 4.0 seconds for saturation flow rates ranging from 1,550 to 2,300 vphgpl. Mode! of Start-up :Lost Time. Based on the trend In Me data shown In Figure C-20, Me following mode] form was developed for the left-turn movement start-up lost time model: Is =bo +b~s' where: Is = start-up lost tune, see; and s' = saturation flow rate per lane urlder prevailing conditions, vphgpl. (C-23) Mode! Calibration. The details of the mode] calibration analysis and statistics describing the predictive performance of the mode] are presented in Table C-7. As the statistics in this table indicate, the calibrated mode! explains about 34 percent of Me vanability In Me headway data which is indicative of a good correlation between start-up lost time and saturation flow rate. The minimum precision of the average start-up lost time estimate is about +0.04 sec. C-37

Table C-7. Calibrated left-turn movement start-up lost lime mode! . Model Statistics Value ~ . 1 Root Mean Square Error: 1.14 see _ _ _ . Observations: 714 Range of Model Variables Vanable ~ Vanable Name ~Units ~Minimum 15 Start-up lost time see -1.5 ahlration i low rate per lane ~pcphgpl ~1,339 , Calibrated Coefficient Values Coeff. Coefficient Definition Value Std. Dev. bo ~ Intercept ~.43 ~0.373 by | Effect of so :uration flow rate | 0.00362 | 0.00019 , ~ u Maximum 7.8 2,770 t-statistic -11.9 19.1 The quality of fit of Me calibrated mode} to the data is shown in Figure C-21 . For reasons discussed previously, the plotted data points represent averages of 50 observations each. The data shown in Figure C-21 confirm Mat the mode] is able to accurately predict the start-up lost hme over the range of saturation flow rates included in Me database. Predicted Start-up Lost Time, sec. (Legend: A = 1 obs, B = 2 obs, etc.) 6 ~ 4 + O + (Each data point represents the average of 50 obs) A A A A B A A A AA A A A IA  Measured Start-up Lost Time, sec. Figure C-21. Comparison of predicted and measured leff-turn movement start-up lost times. C-38

Sensitivity Analysis. The calibrated mode} of start-up lost time was used to examine the sensitivity of left-turn movement start-up lost time to saturation flow rate. This relationship is shown in Figure C-22. As this figure indicates, the start-up lost times for saturation flow rates of 1,800 and 1,900 pcphgp! are about 2.0 and 2.5 seconds, respectively. These values are SlightlY larger than the I.0 to 2.0 seconds recommended In Chapter 2 of Me HCM All. v , O The trend line for through movements found in the preceding section is also included In Figure C-22. Comparison of this line with that for the left-turn movement indicates very little difference between the two movements. Therefore, it appears reasonable to conclude Mat the effect of saturation flow rate on start-up lost time is independent of movement type. 4.0 3.5 3.0 2.0 1.0 05 Start-up Lost Time, see Movement: Through -Left-Tum 0.0 1400 1 1 1 1 1600 1 800 2000 Saturation Flow Rate per Lane, vphgpl 2200 Figure C-22. Expected lefi-turn movement start-up lost time as a function of saturation flow rate. C.3 CLEARANCE LOST TIME When Me yellow indication is presented at the end of a phase, drivers close to Me intersection generally continue on through Me intersection because stopping would be impossible or, at least, very uncomfortable. Thus, these "clearing" drivers tend to use the first few seconds of Me yellow interval. The remaining portion of Me yellow interval that is not used by the average clearing driver plus the red clearance interval is defined as clearance lost time. Based on this definition, the following equation cart be used to compute clearance lost time: I = Y ~ R - gy e c C-39 (C-24)

where: le = clearance lost time, see; Y= yellow interval, see; RC = red clearance interval, see; and gy= effective green extension into the yellow interval, sec. It was hypothesized that extent of driver encroachment into the yellow Interval could be affected by the clearing vehicle's speed, the width of Me Intersection, and the "cost" of not clearing (i.e., delay). Thus, the calibration activity for this model focused on defining a relationship between green extension, speed, intersection width, and signal timing (as a surrogate for delay). Green extension was quantified as the average amount of the yellow interval that was used dunug phases where it was observed to be used to some degree. Phases where the yellow interval was not used were excluded from the analysis because the reason for this lack of use was not determinable from Me available data. In general, the yellow interval was used in about 27 percent of the phases; although, this frequency varied widely among the twelve study sites. C.3.1 ModelDevelopment Based on the preceding discussion, it was reasoned that the prediction of clearance lost time would require a mode! of green extension. Thus, Me objective of the model development process was to determine the nature and magnitude of the eject of venous factors on the amount of the yellow interval used by drivers. Several potentially influential factors were identified from past research on lost time and from insights obtained dunng the field studies, these factors Include: I. Number of vehicles per cycle. 2. Yellow~ntervalduration. 3. Red clearance Interval duration. 4. Saturation flow rate for the respective movement. 5. Green interval duration. 6. Intersection width (clearance distance). 7. Speed ofthe cleanng vehicle. After an exploratory analysis, several of the factors listed above were found to have an insignificant correlation with green extension and were eliminated from further consideration. Specifically, the yellow, red clearance, arid green Interval durations as well as the intersection wade were not fourld to be correlated win green extension. In contrast, the number of vehicles per cycle, minimum discharge headway, and clearance speed were found to be correlated wad green extension. Further examination of the first two of these ejects suggested that the number of vehicles per cycle relative to the maximum number that can be served per phase (i.e., volume-to-capacity ratio) would be art appropriate means of examining their combined effect on green extension. Specifically, these vanables were combined as follows: v, X.= i C-40 (C-25)

where: Xi = volume-to-capacity ratio in lane i, i = 1, 2, , N; vi' = demand flow rate in lane i, vpcpl, ci ' = capacity of lane i (estimated as ci ' = G /Hi ), vpcpl, G= green signal interval, see, and Hi = minimum discharge headway in lane i, sec/veh. The volume-to-capacity ratio has an intuitive correlation with the amount of the yellow a driver would be willing to use. In particular, as the phase becomes busier (i.e., as X approaches 1 .0), drivers will increase the amount of the yellow interval they are wing to use to avoid lengthy delays In spite of reduced safety and an increased chance of receiving a citation. The relationship between He X-ratio and green extension is shown in Figure C-23. As this figure indicates, He effect increases exponentially wad ~ncre~cing values of X However, a closer examination indicated Hat a correlation between green extension arid X-ratio only existed (m a statishcaBy significant way) for X values above about 0.90. The data shown In this figure represent averages of 50 observations, sorted by X-ratio. This averaging was performed to facilitate He examination of Mends by eliminating some of the variability in He individual observations. Green Extension, sec. 5 + 4 3 ~ 2 + (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 50 obs) A A A A A ~ A A A A A A A A A A A A A A A A . _ _ _ _ _ _ I. _ 0.25 0.50 0.75 1.00 1.25 Volume-To-Capacity (X) Ratio Figure C-23. Elect of x-ratio on green extension.  In addition to X-rabo, clearance speed was also found to be correlated with green extension. Specifically, drivers used more of the yellow interval when they were traveling at higher speeds. This trend suggests that drivers may be more challenged to estimate the adequacy (and safety) of Heir stopping distar~ce at higher speeds and, as a consequence, err on the side of maintaining speed C-41

and proceeding through Me intersection on yellow. Although cleararlce speed is likely to be more strongly correlated wad green extension Wan any other speed measure, it represents art impractical variable because it is generally an unknown quantity at most intersections. As a more useful alternative, the approach speed limit was substituted in Me green extension mode! as a surrogate measure for clearance speed. Based on Me preceding discussion of vanable effects on preen extension the following form of the green extension mode} was chosen: ,, ~, . . is, gy = be + b,SL ~ b2 (X, - b3)IX (C-26) where: gy= effective green extension into the yellow interval, see; AL = approach speed limit, ninth, Xi = volume-to-capacity ratio in lane i, i = I, 2, ..., N. ~x= indicator variable (~.0 if Xi > b3 , 0.0 otherwise); and be b,, b2, b3 = calibration coefficients. C.3.2 Mode' Calibration The analysis considered 1,044 signal phases with observed driver use of the yellow (and in some cases, red clearance) interval. These phases were observed at twelve interchange ramp terminals and at twelve intersection approaches. The green extension data used in this analysis represent observations made for both left-turn and through movements. The left-b~ns at the interchanges were made from either the on-ramp or the arterial. Details of the mode! calibration analysis and the performance of Me calibrated mode] are presented in Table C-~. The R2 statistic in Table C-8 indicates that the calibrated mode! accounts for eleven percent of Me variability in the green extension data. The remaining variability is likely due to random sources; although, some of it may be due to differences among the study sites (that was not explained by speed limit arid volume-to-capacity ratio). Nevertheless, it is believed that the calibrated mode} provides a relatively good fit to the data and Cat it can be used to predict Me average green extension with reasonable precision (i.e., a minimum of ~ 0.04 sec.~. The quality of fit to the data is also shown In Figure C-24. C-42

Table C-~. Calibrated green extension mode! Model Statistics Value R2 1 0.11 Root Mean Square Error: | 1.33seconds Observations: 9,044 Range of Model Variables Vanable | Vanable Name |Units| Minimum gY Effective Teen extension into the yellow interval see0.02 _ SL Approach speed limit 56 . G Green interval duration see10 H | Minimum disco ge headway | see T 1.7 v, demand flow rate in lane i vpcpl 2 Volume-to-capacity ratio in lane i na 0.08 , Calibrated Parameter Values Vanable Definition Value Std. Dev. 1 . bo | Intercept l 1.48 | 0.42 1 hi ~Effectofspeedlimit T 0.014 | 0.0068 | b2 Effect of X ratio 6.40 0.87 b3 1 Threshold X ratio l 0.88 1 0.022 1 ~ u Predicted Green Extension, sec (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 25 obs) 3.25 + 3.00 + 2.75 + A A A AB A A A AA A A A A AB A 2.50 + A A A A A A A | A AA A AA A A AA 1 2.25 + A A A A A Measured Green Extension, sec Figure C-24. Comparison of predicted and measured green extension. C-43 Maximum 7.3 72 99 2.2  37 t-statistic 1.3 3.5 2.1 7.4 40.0

C.3.3 Sensitivity Analysis The calibrated mode} was used to examine the effect of speed limit and volume-to-capacity ratio on clearance lost time. Prior to conducting this examination, it was necessary to define Me duration of the yellow and red clearance intervals. Recognizing that the yellow interval duration is often dependent on the approach speed and that Were are a wide range of methods being used to determine yellow interval duration, it was decided to set the yellow Antennas equal to 0.062 times the approach speed (i.e., Y = 0. 062 * speed limit Chow. This anoroach Yields values that are ~eneralIv consistent with those obtained Tom other methods or policies. The red clearance interval was established as I.0 second. Using these values, the clearar~ce lost time was computed for a range of speed limits and volume-to-capacity ratios. The results of this examination are shown In Figure C- 25. 4.0 3.0 2.0 0.0 Clearance Lost Time, see x-ratio < 0.88 X-rated = 1 05 48 56 64 72 80 Approach Speed Limit, km/in Figure C-25. Effect of approach speed limit on clearance lost time. As Figure C-25 illustrates, clearance lost time increases wad approach speed limit and decreases with increasing volume-to-capacity ratio. In general, it ranges hom ~ .0 to 3.0 seconds for typical speed limits arid uncongested conditiorls. This range compares with the I.2 to 2.~-second range for clearance lost time suggested ~ Chapter 2 of the HEM Alp. Clearance lost time increases wad speed because of a corresponding Increase in the yellow interval, however, it should be noted that this effect is offset to some degree by Me increase in green extension associated with higher speeds. C-44

C.4 LANE UTILIZATION The quality of service provided by a signalized intersection is highly dependent on He volume-to-capacity ratio of the intersection and its associated signal phases. One consideration in determining demand volume for the phase is He distribution of traffic among the lanes it serves. More specifically, these would be the lanes available to a "lane group," as defined in Chapter 9 of the HCM All. Obviously, if the traffic for a given lane group is concentrated in only one of the several available lanes, then the phase duration would need to be long enough to serve traffic in this one large. Alternatively, if the phase duration is not increased, Den the capacity of the lane group is effectively reduced by the degree of underutilization of its lower volume lanes. Unfortunately, an even distnbution of traffic among the lanes in a lane group is rarely achieved at most intersections for a variety of reasons. One reason is the inherent probability that the number of vehicles arriving per cycle wall not be evenly divisible by the number of lanes serving the movement. When this event occurs, the number of vehicles in each through lane watt differ by at least one vehicle; perhaps more, if drivers do not distnbute themselves everdy among the available lanes. To illustrate this effect, consider six vehicles arriving to an intersection while He signal indication is red. If He subject lane group is provided two lanes, then presumably Free vehicles would queue in each lane. However, if seven vehicles active, then one large wall have four vehicles and one lane wall have three vehicles (as a minimums. It is important to note that He effect of unbalanced lane volumes is defined herein in the context of "arrivals per cycle." This definition is consistent with the analysis approach described in the HCM (~) and the methods contained therein Hat account for unequal lane use. Another reason for an uneven distribution of traffic among the available lanes is Giver desire to "preposition" for a turn maneuver at a downstream intersection. This activity con~nonly occurs on the arterial cross street at interchange ramp terminals and any associated closely-spaced intersections. Specifically, drivers in these areas typically concentrate in one through lane at He upstream ramp tensional or intersection in anticipation of a turn onto the freeway at the downstream ramp terminal. The uneven allocation of traffic to the available traffic lanes is typically quantified In terms of the lane group Lane Utilization Factor U. This utilization factor can be computed using the following equation, where the volumes used represent averages per signal cycle: v' N man U = ~(C-2 ,' ~ Hi where: U= lane utilization factor for He lane group; v',, = maximum demand flow rate in any of Nlanes, vpcpl; vi' = demand flow rate in lane i, i = I, 2, N. vpcpl; arid N = number of lanes in He lane group. C-45

The overall effect of number-of-lar~es and movement-type on lane use were previously shown in Table B-! O. Two trends were suggested by the data in this table. First, the magrutude of the utilization factor tends to increase with number of lanes. This increase is indirectly due to the probability of an uneven distribution of vehicles among Me available lanes each cycle (i.e., the arrival volume is not evenly divisible by We number of lanes). The potential for an uneven distnbution increases with increasing number of lanes; hence, the utilization factors should tend to increase with art increasing number of lanes. The second trend suggested by the data in Table B-IO is that the utilization factor for left-tu~n movements is higher there for Trough movements, given the same number of lanes. This trend is likely due to the lower volume ofthe left-turn movements relative to the Trough movements studied. The effect of volume level on lane utilization can be demonstrated by the following example. Consider a lane group with two traffic cartes where the drivers always distribute themselves as evenly as possible. When there are Tree arrivals per cycle, Equation C-27 predicts a lane utilization factor of 1.33 (= 2 veil * 2 lanes / 3 arrivals); however, when there are 13 arrivals per cycle, Equation C-27 predicts a factor of 1.OS (=7*2/131. Thus, a low-volume movement will generally have a larger lane utilization factor than a high-volume movement. C.4.! Mode! Development ,1 , V, The lane utilization mode] developed in this research is based on a quantitative description Of the two problems previously described: (~) drivers not distributing themselves as evenly as Possible. and (2) drivers Repositioning for a downstream turn. The first problem is more fundamental ~ nature and deals specifically wad the random nature of vehicle arrivals per cycle and the extent that Divers collectively can and wti! distribute themselves among available traffic lanes. Unbalanced lane use stemming from this problem would be found in any multi-lane lane group on an intersection approach. The second problem is of a site-specific nature as it relates to Me effects of downstream turn movements on a driver's lane choice at the upstream intersection. This problem would not necessarily be found In all multi-lane lane groups. Random Behavior Model. The first problem descnbed above can be modeled by identifying the boundary values of the "most" and the "least" even distribution possible arid Men determining where drivers naturally fall between these boundaries. The most even distribution possible does not always equate to art equal number of vehicles in each lane due to Me Integer nature of vehicles and available lanes. When the "most-evenly possible" distribution is achieved, the flow rate in Me higher-volume lane can be computed using Me probability that the arriving vehicles are (or are not) evenly divisible by the number of available lanes. This probability can then be used to compute an expected maximum lane flow rate v ',,~rmJ Consider a two-lane lane group why v arrivals per cycle. One-half of Me time, the arrivals would be evenly distributed among available lanes and ore-half office time there would be one more vehicle In the largest lane (assuming a "most-even" posture). Thus, the average (or expected) maximum lane flow rate can be computed as: C-46

E[v' ] = ~ ( v ) + ~ ( v -I + 1) (C-28) v' . 1 where: E[v '~7~ <mj] = expected maximum lane flow rate based on the "most-even" distribution possible, vpcpl; v' = demand flow rate for the lane group, vpc. A key assumption made In Me development of Equation C-28, Mat the "even" and "uneven" arrival cases will occur with 50 percent probability (i.e., Me "~/2" factor shown), is based on the reasonable presumption that total number of arrivals per cycle is a random variable with a mean arrival rate of v'. An empiric examination of the strength ofthis assumption was conducted using several types of arrival distnbutions. Based on this examination, it was concluded that the assumption is valid, provided Mat the distribution of arrivals has a nonzero variance and is s~ngle- moded (i.e., does not have a multi-modal distributions. This examination also indicated that the probability of an "even" distribution for any number of lanes N can be assumed to be equal to 1/N, provided Mat Me number of arrivals per cycle equals or exceeds Me number of lanes. Based on Me preceding discussion, the largest lane flow rate can computed for We general case of ~ lanes as: maxims N N-1 - i O VN i + 1 - = - ( 1 ~) N 2v' (C-29) This equation predicts the flow rate In the highest-volume lane trader the assumption that drivers will distribute themselves as evenly as possible upon their arrival. It forms a lower bourns on the maximum large volume. The counterpart to the "most-even" distribution assumption would be Me case where drivers remain In Me lane In which Hey arrived and, thereby, combine in a purely random manner to yield relatively large flow rates In He highest-volume lane. This "least-evenly distributed" scenario forms a reasonable upper bourns on He maximum lane volume (precluding the possibility of prepositioning). When drivers make no attempt to redistribute themselves as they arrive, then the flow rate in the highest-volume 1aIle is a random variable that follows He distribution of maximum values. This behavior represents an extreme case where the resulting large distribution is the most uneven possible (neglecting prepositioIiing). Using the Poisson distribution to model the arrival process, He maximum value distribution can be written as: C-47

v' mar f(v man) = N it -v' e ~ (vl') i! \ / N -1 e (vl ) (vl )! maJc where: TV',, = distribution of the maximum demand flow rate in any of N lanes, N= number of lanes In the lane group; and v,' = demand flow rate per lane, vpcpl. (C-30) Using this distribution, the expected flow rate in the highest-volume lane V',~,~V can be computed using the following equation: co E[V ntax(l)] ~V ma;` f(v man) (C-31) me where: E[v', ~7J] = expected maximum lane flow rate based on the "least-even" distribution possible, vpcpl. Unfortunately, the combination of Equations C-30 and C-31 does not yield a tractable, closed-form solution. Therefore, Equations C-30 and C-31 were used to develop the following empirical model for predicting v',na,`'v: E[V 'maxim] N ~ 4 ~ ~ (C-32) This equation was developed from an examination of Equations C-30 and C-31 and Me use of heuristic techniques. Other forms are possible, however, Equation C-32 was found to predict the theoretic maximum lane flow rates obtained from Equation C-3 ~ with a relatively high degree of accuracy (i.e., R2 = 0.85) for lane group flow rates in the range of 0.0 to 40 vpc. To demonstrate We predictive ability of Equation C-32, Equation C-3 ~ was used to compute the theoretic lane utilization factors for a range of flow rates arid traffic larches. These factors are plotted as data points in Figure C-26 along with the trend lines representing Equation C-32. As shown in this figure, Me empirical model predicts the lane utilization factors quite well for Me two and ~ree-lane cases while it overestimates Me factors for the four-lane case by about 4.0 percent. As Equations C-29 and C-32 represent boundary values for Me maximum lane volume, the actual maximum lane volume would fall somewhere between these two extremes. Hence, the expected magnum lane volume can be computed as the proportional combination of bow equations: C-48

E[v ] = - 1 + N (} _ f ~ + V' 3 N N f (C-33) N ( 2 V ) N ~ 4 2~i l where: E[v ',,~] = expected maximum demand flow rate in any of Nlarles, vpcpl; and flu = proportion of drivers that do not attempt to evenly distribute themselves. Lane Utilization Factor 2.2 2.0 1.8 1.6 1.4 0~Foints~om Eq. C:-~~ ~ ~ - ~ Trend lines from Eq. C-32 -~_- 4 Lanes - - - - - - - - - - - - - - - - - - 0 5 10 15 20 25 30 35 40 Demand Flow Rate for the Lane Group, vpc Figure C-26. Comparison of lane utilization factors obtainedpom Equations C-31 and C-32. Equations C-27 and C-33 can be combined to yield Me expected lane utilization factor for isolated intersections (i.e., no preposition~ng) U.: U. = ~ +~ 2 y' ~ (} -fu) 4 ~fu (C-34) The effect of number-of-lanes, flow rate, and driver propensity to distribute evenly on the lane utilization factor is shown In Figure C-27. As the trends shown In this figure indicate, the lane utilization factor decreases with increasing flow rate. It also decreases with a decreasing number of lanes. This latter trend is consistent with the data shown In Table B-!0. C-49

1.8 1.6 1.4 1.2 Lane Utilization Factor ~Most Evenly Distributed \ \\ ~Least Evenly Distributed -- Butanes 0 ~ 10 15 20 25 30 35 40 Demand Flow Rate for the Lane Group, vpc Figure C-27. Range of possible lane utilization factors as Affected by volume and lanes. Also shown In Figure C-27 is the range of possible values of the lane utilization factor, based on the drivers' tendency to distribute themselves on the intersection approach. In general, We range width Increases with number of lanes arid decreases with lane group flow rate. The utilization factors recommended by the HCM (~) (i.e., 1.05 at two through lanes, ~ . ~ O at three through lanes) fall within the range of expected lane utilization factors shown In Figure C-27 when the volume exceeds about ~ O vpc. The HCM factors do not vary with traffic demand even though the preceding theoretic development indicates Hat they should have some sensitivity to it. As a result, the HCM factors would appear to be too low when traffic demands are less than 10 vpc. Preposition~ng Model. Driver preposition~ng for a turn maneuver at a downstream intersection can affect large utilization at an upstream Intersection if the demand for this turn maneuver is high relative to the volume not Erring at He next Intersection. If the number of drivers not horning are sufficiently high as to main art even distnbution of traffic in the upstream lanes, then the lane utilization would follow Hat described for the "Random Behavior Model." On He other hand, if drivers not turning are relatively low such that they cannot maintain an even distribution of traffic, then the propositioned drivers will dominate one lane arid yield a large utilization factor significantly in excess of I.0. The mode} of this process begins with a test to determine which ofthe two cases described in He preceding paragraph will occur. This test determines if the number of drivers that will not be turning are sufficient in number to maintain an even large distribution. The test compares He proportion of propositioned traffic web He proportion of traffic in any lane when this traffic is C-50

evenly distnbuted (i.e., 1/N). When the proportion of prepositioned traffic exceeds 1/N, then preposition~ng watt dictate the magnum large volume. The test equation is: Maxfv'dl,y `) 1 > - : propositioning v' Max(v'dl, ~'d) < - : no preposzfzonzng v' (C-35) where: I've = flow rate in the lane group that will be turning left at the downstream intersection, vpc, v 'deter = flow rate in the lane group that will be turning right at the downstream intersection, vpc, N= number of lanes in the lane group, and Ma;'c~v'd~bv'd~r) = larger of v ~` and v fir. It should be noted that He test equation does not include Be distance to the downstream Intersection. Logically, the motivation to preposition is dependent on this distance, as drivers are not likely to preposition if the next intersection is quite distant. However, the test equation is believed to be applicable to interchange areas because the inter-signal distances at these locations are sufficiently short that preposition~ng is necessary (say, less than 300 meters) If preposition~ng dictates He lane volume distribution, Hen He largest lane flow rate will be: E[v'max] = MaX(V'&I'V'&r) Similarly, the large utilization factor for the preposition~ng case can be computed as: Max (v'dl' v'd~r) N U p (C-36) (C-3 Combined Models. Equations C-34 and C-37 were combined to yield the generalized lane utilization model. This model is applicable to interchanges, adjacent intersections, arid over intersections where propositioning may occur. The form of this model, including two empirical calibration constants, is: U = ~1 ~2 lo' ~ (l b0) + 4 ~ b (} -~ ~ + ~ dI' d,) b ~] (C-38) where: Ip = indicator variable (~.0 if March ',&l, v'`~rJ/v' > 1/N, 0.0 otherwise), arid be, by = calibration coefficients. C-S!

C.4.2 ModelCalibration The lane utilization model was calibrated using data for 97 five-minute intervals. The data in each interval represent the maximum lane flow rate and the lane group flow rate that was observed each cycle during the five-minute interval, these cyclic values were then averaged to obtain the five- minute interval data. The 97 intervals represent the observation of 10,585 through vehicles in two, three, or four-lane lane groups. The results ofthe model calibration are shown In Table C-9. As the statistics provided In this table indicate, the calibrated model provides a reasonably good fit to the data. The minimum precision ofthe average lane utilization factor estimate is about +0.0 l, based on the root mean square error and the number of observations. The R2 of 0.~8 is lower than values traditionally expected; however, it must be remembered that there is considerable random variability In the lane utilization factor. This variability stems Tom the fact that two random variables (i.e., v ',, and v ~ are being used In the computation of the lane utilization factor. Thus, the variability In this factor represents the combined variability of the two underlying random vanables. It should be noted that a regression analysis using Equations C-33 and C-36 to predict Me maximum observed lane volume v ',, yielded an R2 of 0.95. Table C-9. Calibrated lane utilization mode' Model Statistics | Value R2: 1 0.18 Root Mean Square Error: 0.1 1 Observations: 97 Range of Model Variables Variable Variable Definition | Units | Minimum U Lane utilization factor na 1.0 vim Maximum lane flow rate in any lane vpcpl 5.8 Member of lanes in the lane group na 2 v',` No. of vehicles boning left downstream vpc O v'd ~No. of vehicles turning right do~Ansheam vpc O v ' Demand flow rate for Me lane Coup vpc 12.4 Calibrated Coefficient Values | Coed. _ ~ _ Value Std. Dev. 1 | be Portion of drivers not evenly distributed 0.577 0.044 b, | Effect ofad/iitionalnon ~ning drivers | 1.05 | 0.030, Maximum 1.56 40.3 4 23.3 16.1 69.7 t-statistic 13.1 3s.0 Two observations cart be made based on the parameter coefficient values. First, the ho coefficient in the calibrated mode] indicates that about 58 percent of drivers at the sites studied do C-52

not attempt to evenly distribute themselves among Me available lanes. Second, We fact that the by coefficient exceeds t.0 suggests Mat additional "non-turning" Divers are choosing the same lane as the propositioning drivers. Specifically, lane groups that experience propositioning typically experience a five percent increase in the number of vehicles In the "maximum large" due to non- ~ning drivers. The calibrated model's quality of fit to the data is also shown in Figure C-28. The data points shown In this figure represent the average of three observations each. This averaging was done to facilitate an examination of mode] fit over the range of the measured data. This type of assessment can be made difficult when individual observations overlap and appear as only one data point. This problem is particularly evident when comparing predicted and measured values where Me tendency is for Me marry data points in agreement to overlap and appear as only one observation. Predi cted Lane Ut il i zat ion Factor 1.4 ~ 1.3 + 1.2 ~ 1.1 + 1.0 + (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 3 obs) A A A A A A A B A A A A A AA AA AGAPE AAA A A A A A A A A 1.0 1.1 1.2 1.3 1.4 Measured Lane Utilization Factor Figure C-28. Comparison of predicted and measured lane utilization factors. C.4.3 Sensitivity Analysis The calibrated lane utilization mode} can be used to examine the relationship between lane utilization, lane group flow rate, and number-of-lanes. Figure C-29 illustrates these relationships for the Random Behavior component of the mode} (i.e., no preposition~ng). As expected, the lane utilization factors increase with number of lanes and decrease with increasing volume. It should be noted Mat the predicted lane utilization factors exceed Me values recommended by Me HCM (~) (i.e., .05 at two through lanes, ~ .10 at three Trough lanes). C-53

1.8 1.6 1.4 1.2 1.0 Lane Utilization Factor \\ ~ \~ ~W TTI Model Proposed Model _ , 0 5 10 1 1 1 15 20 25 30 35 40 Demand Flow Rate for the Lane Group, vpc Figure C-29. Elect of pow rate and number of lanes on the lane utilization factor. Also shown in Figure C-29 are the lane utilization factors predicted by a mode} developed by Fambro et al (7, 89. This mode! is coined the "AT} Model" In reference to the authors' affiliation. It was calibrated to ten traffic movements at nine signalized intersections. In general, the rim ~ Mode} predictions compare favorably with those of the calibrated lane utilization model; however, the agreement is best at the higher flow rates. This agreement is partly due to the fact Mat both data bases had the majority of Weir observations In this higher range of flow rates. This agreement suggests Cat the calibrated lane utilization mode} may be applicable to all signalized Intersections. C.S CAPACITY ANALYSIS This section summarizes the models developed in the preceding sections in the context of a proposed procedure for capacity analysis. These models predict capacity-related traffic characteristics for a range of conditions common to Interchanges and closely-spaced signalized intersections. Initially, the models for predicting these characteristics are presented in a summary format Cat is consistent with the method of presentation used in the HCM All. Then, two topics Cat are related to capacity analysis are discussed. These two topics include lane utilization and queue spillback. Next, the ability of the proposed models to predict capacity is demonstrated. Finally, a sensitivity analysis is conducted to illustrate the effect of saturation flow rate and lost time on capacity.

C.5.! Introduction The capacity of lane group represents the maximum number of vehicles that can be served by the group's traffic lanes during the time allocated to it within the signal cycle. A lane group flowing at capacity is typically characterized by the continuous discharge of queued traffic for the duration of the signal phase that controls it. To this extent, lane group capacity is dependent on the discharge characteristics of the departing traffic queue. These characteristics include the time lost at the start of the phase due to driver reaction time and acceleration, the saturation flow rate, and the time lost at the end of the phase due to a necessary change interval. The following equation is commonly used to compute the capacity of a lane group: , _ gs c - 3,600 with, where: (C-39) g = G ~ Y + R - (I ~ I ) ~ CP (C-40) c ' = capacity of the lane group, vpc; g = effective green time where platoon motion (flow) can occur, see; s = saturation flow rate for the lane group under prevailing conditions, vphg; G = green signal interval, see; Y= yellowinterval,sec; RC = red clearance interval, see; Is = start-up lost time, see; le = clearance lost time, see; arid CP = clear period during cycle/phase when subject flow is unblocked (see Appendix D), sec. Many of the variables defined in Equations C-39 and C-40 represent the basic set of capacity characteristics. The proposed models for predicting these characteristics (i.e., saturation flow rate, start-up lost time, and clearance lost time) are described in the next three sections. C.5.2 Saturation Flow Rate The saturation flow rate for a lane group can be predicted using the following equation: s = NXSo XfW XfHrXfg XfpXfbbXf~TXf~TxfDXfV where: s = saturation flow rate for the lane group under prevailing conditions, vphg, so = saturation flow rate per lane under ideal conditions nc~h~l; fw = adjustment factor for lane wake; fHv= adjustment factor for heavy vehicles; (cads) --, r -~--~-7 C-55

fLT f[T fR fig = adjustment factor for approach grade; fp = adjustment factor for parking; Ebb = adjustment factor for bus blockage; fRT= adjustment factor for nght-turns In the lane croup: fit= adjustment factor for left-t~s in the lane group; fD = adjustment factor for distance to downstream queue at green onset; and fV = adjustment factor for volume level (i.e., tragic pressure). _ · ~· ~ ~ ~ A ~ ~ _ . · . ~ ~ The first seven adjustment factors listed in this equation represent those described in the HCM rim, Chapter 99. The last two factors were developed for this research arid are specifically applicable to interchanges arid closely-spaced signalized intersections. A Bird adjustment factory was developed for this research that quantifies the effect of turn radius on He saturation flow rate of a left or right-tutn movement. This factor is introduced by revising the adjustment factors for protected turn movements provided In the HCM. These revised right and left-turn adjustment factors are: fRT ~\ : Protected, Shared Right-T?'rn Lane 1 UP ~ - 1 RT {R (C-42) fRTfR : Protected, Exclusive Right-T?vrn Lane 1 PETER - 1 : Protected, Shared Left-Turn Lane : Protected, Exclusive Left-Turn Lane (C-43) where: fR = adjustment factor for He radius of the travel oath (based on the radius of the applicable left or r~ght-turn movement), PRT = portion of right-turns in the larle group, and PI T = portion of leD-turns in the lane group. Adjustment factors for permissive-ondy and protected-permissive phasing can also be developed using Equations C-42 and C-43 as a basis. The ideal saturation flow rate so represents the saturation flow rate of a large that is not a~ectedby any external environmental factors (e.g., graded, non-passenger-car vehicles (e.g., trucks), and constrained geometries (e.g., less than 3.6-meter lane wades, curved travel path). In this regard, the saturation flow rate would be equal to He ideal rate when all factor effects are optimum for C-56

efficient traffic flow arid the corresponding adjustment factors are equal to I.0. Based on this definition, it was deterTruned that ideal conditions were represented by an infinite distance-to-queue, non-spilIback conditions, a tangent alignment (i.e., an inflate radius), and a traffic pressure representing that typically experienced during peak traffic penods. Under these conditions, 2,000 pcphgp! is recommended as the ideal saturation flow rate for all traffic movements. The following sections descnbe the individual adjustment factors developed for this research. One potential factor that was not included in Equation C-] is Mat of phase duration. There is some evidence that Divers adopt larger saturation flow rates for phases of short duration than for those of long duration. The evidence was strongest for the left-turn movements and when the g/C ratio was less Wan 0.27. This effect was also found In the through movements studied, however, it was much smaller In magnitude and not statistically significant (due in part to the fact that the g/C ratios for most through movements were above 0.30). Nevertheless, the data suggest that saturation flow rates can increase by 5 to 10 percent when the g/C ratio for the subject movement is less than 0.20. However, more research wall be needed to verify the significance of this trend and its magnitude before it can be recommended for application. Distanc - to-Queue Adjustment Factor. The distar~ce-to-queue adjustment factor fD accounts for the adverse effect of downstream queues on the discharge rate of art upstream traffic movement. In genes, Me saturation flow rate is low for movements that have a downstream queue relatively near at the start of the phase, it is high for movements that are not faced with a downstream queue at the start of the phase. Thus, the distance-to-queue adjustment factor is based on the distance to the back of the downstream queue at the start of the subject phase. The variable ''distance to queue" is measured from the subject (or upstream) movement stop tine to the "effective" back of queue. The effective back of queue represents the location of the back of queue if all vehicles on the downstream street segment (moving or stopped) at the start of the phase were joined into a stopped queue. If Mere are no moving vehicles at the start of Me phase, then the effective and actual distance to queue are the same. If Mere are no vehicles on Me downstream segment at the start of the phase, then the effective distance to queue would equal the distance to the through movement stop line at the downstream intersection. The distar~ce-to-queue is computed as: wit, n D = L - S L (C-44) L = (1 ~ PHV) LPC PHV HV (C-45) where: D = effective distance to the back of downstream queue (or stop line if no queue) at the start of the subject (or upstream) phase' m; C-57

~ = distance between Me subject and downstream intersection stop lines (i.e., link length), m; nS = number of vehicles on the downstream sheet segment (moving or queued) at the start of the subject phase, vein, N~ = number of through lanes on the downstream segment, lanes; [v = average lane length occupied by a queued vehicle, mlveh, [pc = lane length occupied by a queued passenger car (= 7.0 mlpc), m/pc, THE= lane length occupied by a queued heavy vehicle (a 13 m/veh), mlveh; and Par = portion of heavy vehicles in the traffic stream. The lane length occupied by a queued passenger car vehicle was measured for this research. The summary data reported in Table B-1 1 indicate that the average values is 7.0 m/pc. The lane length occupied by a queued heavy vehicle can vary, based on the types of heavy vehicles in the subject traffic stream. An approximate value that can be used when field measurements are not available is 13 m/vein. This value is representative of most buses, recreational vehicles, arid inter-city trucks. The effect of distance-to-queue is also dependent on whether spillback occurs during the subject phase. Spillback is characterized by the backward propagation of a downstream queue into the upstream intersection such that one or more of the upstream intersection movements are affectively blockedirom discharging during some oral1 oftheir respective signal phase. If spillback occurs during the phase, the saturation flow rate prior to the occurrence of the spillback is much lower than it would be if there were no spillback. Thus, the magnitude of the adjustment to We saturation flow rate is dependent on whether spillback occurs during Me subject phase. A procedure for detenn~rung if queue spillback occurs is provided in a later section. The distance-to-queue adjustment factor is tabulated In Table C-IO. It can also be computed using the following equation: 1 fD no spillback + 8.13 D 1 : with spillback 21.8 1 + D (C-46) where: fD = adjustment factor for distance to downstream queue at green onset; arid D = effective distance to the back of downstream queue (or stop line if no queue) at the start of the subject (or upstream) phase, m. C-58

Table C-IO. Adjustment factor for distance-to-queue ~D) Distance to Back of Spillback Condition Queue at Start of Subject Phase, m No Spillback With Spillback 15 ~~0.649 0.408 30 0.787 0.579 . 60 0.881 0.734 _ . _ 120 0.937 0.846 180 0.957 0.892 240 0.967 0.917 300 0.974 0.932 360 0.978 0.943 l Turn Radius Adjustment Factor. Traffic movements Mat discharge along a curved have! paw do so at rates lower Cart Rose of through movements. The effect of travel paw radius is tabulated In Table C-1 I. It cart also be computed using We following equation: f = R 1 1 + 1.71 R where: fR = adjustment factor for Me radius of Me travel path, arid R = radius of curvature of Me left-turn travel paw (at center of path), m. Table C-11. Adjustment factor for turn radius CAR) l Radius of the Travel Movement Type Path, m ~Left & Right-Turn Through 8 _ 0.824 1.000 15 0.898 1.000 30 0.946 1.000 . . . 45 0.963 1.000 60 0.972 1.000 75 0 ~1 1.000 90 0.981 1.000 . . 105 0.984 1.000 C-59 (C-4~

This factor was calibrated for left-turn traffic movements; however, a comparison of this adjustment factor to that developed by others for right-tu~n movements indicates close agreement. Therefore, this factor is recommended for use with bow left and right-tu~n movements. Traffic Pressure Adjustment Factor. Saturation flow rates are generally found to be higher during peak traffic demand periods than during off-peak periods. This trend is explained by the effect of "traffic pressure." In general, it is believed that traffic pressure reflects the presence of a large number of aggressive drivers (e.g., commuters) wing high-volume traffic conditions. They demonstrate this aggressive behavior by accepting shorter headways du ing queue discharge than would less-aggressive drivers. As these aggressive drivers are typically traveling during the moming and evening peak traffic periods, they are found to represent a significant portion of the traffic demand associated with these periods. The elect of traffic pressure was found to vary by traffic movement. Specifically, the left- turn movements tended to be more affected by pressure as their saturation flow rates varied more widely then those ofthe through movements for similar conditions. It is possible that this difference between movements stems from the longer delays typically associated with left-tu~n movements. Based on the preceding definition, it is logical that the effect of traffic pressure is strongly correlated with the demand flow rate In the subject lane group. Thus, the effect of traffic pressure, as represented by traffic volume, is tabulated in Table C-12 for each movement type. It cart also be computed using the following equation: 1 fit = where: 1.07 - 0.00672 v'' 1.07 - 0.00486 v`' : left-turn through or right-t2`rn fV = adjustment factor for volume level (i.e., traffic pressure), and v,' = demand flow rate per lane (i.e., traffic pressure), vpcpl. (C - ~) As noted previously, ideal conditions were defined to include a traffic pressure elect representative of peak traffic periods. In this regard, traffic pressures under ideal conditions were defined as 10 arid 15 vpcp} for Me lefts n and through movements, respectively. These flow rates are conservative in Weir representation of higher volume conditions as they exceed 80 to 90 percent of all traffic demar~ds Mat were observed at Me field study sites. One consequence of this approach to defining ideal conditions is that it is possible for the traffic pressure adjust~nent factor to have values above ~ .0 when the flow rate is extremely high. C-60

Table C-12. Adjustment factor for volume level (i.e., traffic pressure) ~fvJ Traffic volume, vpcpl Left-Turn 3] 3.0 0.953 0.947 6.0 0.971 0 9.0 1 0.991 1 0.974 12.0 1.011 0 918 15.0 1.032 I 003 18.0 1.054 l.OlS 21.0 1.077 I ~3 . 24.0 1.100 1.049 . C.5.3 Start-up Lost Time The start-up lost hme associated with a discharging traffic queue varies with its saturation flow rate. More specifically, start-up lost time increases with saturation flow rate because it takes more time for the discharging queue to attain the higher speed associated with the higher saturation flow rate. The recommended start-up lost tunes corresponding to a range of saturation flow rates are provided in Table C-13. These values can also be computed using the following equation: 1 = -4.54 + 0.0036Ss 2 0.0 s where: Is = S~-Up lost time, see, and 51 = saturation flow rate per lane under prevailing conditions (= sin), vphgpl. (C-49) The recommended start-up lost times are applicable to led, through, arid right-turn movements. Table C-13. Start-up lost time (IJ . Saturation Flow Rate, vphgp! 1,400 1,500 1,600 1.35 1 ,700 Lo I ,800 2 os 1,900 2.4s 2,000 2.82 2,100 3.18 C-61

C.5.4 Clearance Lost Time The dine lost at the end of the phase represents the portion of the charade interval that is unavailable for traffic service. This time is intended to provide a small time separation between the traffic movements associated with successive signal phases and, thereby, promote a safe change in right-of-way allocation. Clearance lost time can be computed as: ~ = Y ~ R - gy e c where: le = clearance lost time, see; Y= yellow~nterval,sec; RC = red clearance interval, see; and go= effective green extension into the yellow interval, sec. (C-SO) Green extension was found to be dependent on the volume-to-capacity ratio of Me subject movement arid the approach speed Apt. Specifically, at high vol~rne-to-capacity ratios, drivers tend to use more of the yellow tears at lower ratios. This trend suggests that drivers are more likely to enter the intersection on the yellow when demands fared corresponding delays) are high. ~ addition, drivers on higher speed approaches were also found to use slightly more of the yellow interval than those on lower speed approaches. This trend is likely due to We undesirably high decelerations often associated with quick stops from higher speeds. The amount of green extension for a given approach can be estimated as: go = 1.48 ~ 0.0145L + 6.40 (X-0.88)IX where: go= effective green extension into the yellow integral, see; SL = approach speed limit, lunch; X= volume-to-capacity ratio for the lane group; and fx = indicator variable (~.0 if X> 0.~8, 0.0 otherwise). (C-51) For speeds in the range of 64 to 76 km/in and volume-to-capacity ratios of 0.~8 or less, the average green extension is 2.5 seconds. This average value is recommended for use in most capacity analysis. More appropriate values can be computed from Equation C-51 when the speeds are outside the range of 64 to 76 km/in or when X for the analysis period exceeds 0.~. C.5.5 Lane Utilization Drivers do not distribute ~emseIves evenly among Me traffic lanes available to a large group. As a consequence, the lane of highest demand has a higher volume-to-capacity ratio and the possibility of more delay Man the other lanes in Me group. The HCM (~) recognizes this phenomena C-62

and offers the use of a lane utilization factor Uto adjust the lane group volume such that it represents the flow rate in the lane of highest demand. The adjusted lane group volume can be computed as: v' = ~ ~ U g where: v' = demand flow rate for the lane group, vpc, vg' = unadjusted demand flow rate for the lane group, vpc, and U= lane utilization factor for the lane group. (C-52) There are two reasons for an uneven distnbution of traffic volume among the available lanes. One reason is the ir herent randomness in the number of vehicles in each lane. While drivers may prefer the lesser-used lanes because of the potential for reduced travel time, they are not always successful in getting to them for a wide variety of reasons (e.g., unable to ascertain which lane is truly lowest in volume, lane change prevented by vehicle in adjacent lane, driver is not motivated to change lanes, etc.~. Thus, it is almost a certainty that one lane of a multi-lane lane group Will have more vehicles In it than the other lanes, during any given cycle. The second reason for an uneven distribution of traffic volume among the available lanes is driver desire to "preposition" for a turn maneuver at a downstream intersection. This activity commonly occurs on the arsenal cross street at interchange ramp terminals and at any associated closely-spaced intersections. This prepositional causes drivers to concentrate in one lane at the upstream ramp terminal or intersection In anticipation of a turn onto the Deeway at the downstream ramp tenninal. Random Behavior Model. The lane utilization factor for an intersection lane group Mat does not experience prepositiomng (e.g., at an isolated intersection) is dependent on the degree of randomness in the group's collective lane-choice decisions. In addition, it is also based on the group's traffic volume on a "per cycle" basis. In general, lane utilization is more uneven for groups with low demands or a high degree of randomness or both. The recommended lane utilization factors for random lane-choice decisions are provided in Table C-14. These values can also be computed using We following equation: "7 ~ U = 1 + 0.423 {N 1l +0.433N 2 al ) \ \ where: Ur = lane utilization factor for random lane-choice decisions; v' = demand flow rate for the lane group, vpc; and N= number of lanes In the lane group. N - 1 2 v' (C-53) The recommended lane utilization factors are applicable to left, ~rough, arid right-turn lane groups. C-63

Table C-14. Lane utilization factors for lane groups with random lane choice (U.) Lane Group Flow | Number of Lanes in the Lane Group Rate, vpc ~2 ~3 5.0 ~1.00 1.32 1.67 0.0 1.00 1.22 1.45 15.0 1.00 1.17 1.36 20.0 1.00 1.15 1.3 25.0 1.00 1.13 1.28 30.0 1 1.00 1.12 ~1.25 35.0 1.00 1.1 t -- 1.23 40.0 I l.oo 1 1.10 1 1.22 4 2.08 .74 .59 .51 .45 .41 .38 1.35 Preposition~ng Model. The large utilization factor for interchanges arid associated closely- spaced intersections can be strongly influenced by drivers propositioning for downstream turns. The magnitude of the effect is largely a site-specific characteristic, depending on the number of vehicles mining left or right at the downstream intersection (or ramp terminals. If this information is available (such as hom a turn movement count where downstream destination is also recorded), We lane utilization factor can be computed as: U = p Max (v 'ill ~ V 'd) ~ Y' (C-54) where: Up = lane utilization factor for propositioning; vie= number of vehicles In the subject lane group that will be mining left at the downstream intersection, vpc; vies = number of vehicles in Me subject lane group Cat will be turning right at Me downstream Intersection, vpc; aIld Max(v 'fib V'dJ = luger of v a, "d v dr. Inane Utilization Factor. The possibility of propositioning must be evaluated to determine whether to use Equation C-53 or C-54 to estimate the large utilization factor U. This possibility can be determined Mom Me following test: Max(v','`, v'`~) V' - : prepositioning Max f v'd', v'd,) ~ < _ v' N no preposifioning C-64 (C-SS)

Based on the outcome of this test, the lane utilization factor is computed as: U= U : propositioning U r . · · - no preposltlonzng (C-56) The distance to the downstream intersection could effect the propensity of drivers to preposition. In recognition of this effect, the test equation is recommend only for intersections in interchange areas or other closely-spaced Intersections where the ~nter-signal distances are less than 300 meters. C.5.6 Predicting Queue SpiBback Two conditions must occur to precipitate spillback during a saturated flow state. First, the discharging stream from the subject traffic movement must arrive at the back of a stopped downstream queue. Second, the subject phase must be sufficiently long as to permit the discharge of enough vehicles to fill the available downstream distance under jam density conditions. Occurrence of the first condition can be predicted by comparison of the actual and ideal signal offset between the subject phase and that of the clown stream through movement. In general, if the difference between the actual and ideal offsets is positive, then there is a possibility of spillback. A negative difference indicates that spillback is unlikely. This offset difference can be computed as: ~ = 095a Hi with, where: 0// = L u s nS 3,600 h N s c d n (C-5 (C-58) ~ = difference between the actual and ideal signal offset, see; ok = actual offset between the subject phase and that ofthe downstream through movement (phase start time downstream minus phase start time upstream), see, off= ideal offset to ensure progression without speed disruption, see; us= speed at saturation flow, m/s (say, 14 m/s); he = clearance headway between Me last queued vehicle and We first arriving vehicle, see (say, 2.0 see); Sn= saturation flow rate for the subject lane under prevailing conditions assuming the "no spillback'' condition, vphgpl; C-65

L = distance between the subject and downstream intersection stop lines (i.e., link length), m; nS = number of vehicles on the downstream street segment (moving or queued) at the start of the subject phase, vein; and Nit = number of through lanes on the downstream segment, cartes. Occurrence of Me second condition is dependent on the first condition. If We first condition occurs (i.e., the offset difference is positive), spillback we occur if the phase is long enough to discharge enough vehicles such that Me resulting queue exceeds that of the downstream street segment. If the phase is short, such that the discharged vehicles are Insufficient in number to fill the downstream distar~ce-to-queue, then spillback will not occur. The corresponding maximum green interval duration that, if exceeded, would lead to spilIback is computed as: G = D ~ 3,600) ~ max v ~w J s (C-59) where: G,~'a, = maximum green signal Interval duration for the subject (or upstream) phase that is allowable without spiliback during saturated flows, see; D = elective distance to the back of downstream queue (or stop line if no queue) at the start of the subject (or upstream) phase, m; so= saturation flow rate for the subject lane under prevailing conditions assuming the "w~th- spiliback" condition, vphgpl; [, = average lane length occupied by a queued vehicle (see Equation C - 5), m/vein; and Is = start-up lost fume, sec. Equations C-58 and C-59 were used to develop two figures that can be used to predict the likely occurrence of spilIback for the subject phase. Specifically, the relationship between distance- to-queue D and ideal offset ok (i.e., Equation C-58) is shown in Figure C-30. The relationship between distance-to-queue D and the maximum green interval duration G,~ (i.e., Equation C-59) is shown in Figure C-31 . As Figure C-30 Indicates, the ideal offset is often negative for most Intersection pairs when there is a downstream queue. A negative value indicates that the downstream through phase must start prior to the subject upstream phase to provide smooth traffic progression. Figure C-30 can be used instead of Equation C-58 to determine whether Me first condition is satisfied. Specifically, Me actual offset is compared with Me ideal offset for a given sheet segment length arid distance-to-queue. If Me Intersection of the actual offset (entered Dom the y-axis) and the known distance-to-queue (entered from the x-axis) falls below the line corresponding to the segment length, then Me difference In offset wall be negative and spiliback is not likely to occur. If Me intersection occurs above the appropriate segment length line, then the difference In offset wit} be positive and spiliback is likely to occur during saturated flow conditions. In this latter case, Me second condition should be checked to venty whether spiliback watt occur. C-66

40 20 Ideal Signal Offset, see Ideal offset with no queue 40 -60 -80 -100 -120 · 200~ ~ 400-m segment length No spillback for actual offset & distance combinations tbaLfaU beloYv low. -140 ~ ~ ~ ~ ~ i 15 45 75 105 135 165 195 225 255 285 315 345 375 Distance to the Downstream Queue, m Figure C-30. Relationship between distance-to-queue and ideal signal onset. 100 90 80 70 60 50 40 30 10 lo Maximum Green Interval, see ~ ~spillb~ckfor~tual yr~errS dista, ,ct:-=mbir~ations- - ~ ~ ~ - ~ ~ ~ that fall below line. ! 1 1 1 1 1 1 1 ! 1 1 1 15 45 75 105 135 165 195 225 255 285 315 345 375 Distance to the Downstream Queue, m Figure C-31. Relationship between distance-to-queue and maximum upstream green. C-67

Figure C-3 ~ indicates whether the second condition occurs. Specifically, it Indicates the maximum green interval duration that would not lead to spilIback; green intervals in excess of this maximum would likely lead to spilIback. In general, this figure indicates that the maximum green interval increases linearly with increasing distance-to-queue. In fact, a first-order estimate of the max~murn green duration is about one second of green for every three meters of distance. If the offset difference (USA ~ Off) is determined to be positive, then Figure C-3 ~ can be consulted to determine if spiliback will occur for saturated flow conditions arid the known green interval duration. If the intersection of the actual green interval (entered from the y-ax~s) and the known distar~ce-to-queue (entered from the x-axis) falls below the line, then the green Interval is sufficiently short as to prevent spillback. If the intersection occurs above the trend line, then the green interval is sufficiently long as to precipitate spillback whenever traffic demands are high enough to fully utilize the signal phase. Therefore, when the intersection point is above the trend line and demands are expected to fully use the phase, the "with spillback" form of the distance-to- queue adjustment factor ED should be used. The following example is presented to illustrate the use of Figures C-30 and C-31 to determine if spillback is likely to occur under saturated flow conditions. Consider a 400-meter street segment bounded by signalized Intersections. The average distance-to-queue to queue at the start ofthe upstream Intersection through phase is 135 meters. This phase has 50 seconds of green. The Offset to the downstream through movement phase is -10 seconds (i.e., Me downstream through phase starts 10 seconds before the subject phase). Consultation wad Figure C-30 indicates that an ideal offset for this distar~ce-to-queue is about -45 seconds and, more importantly, that the actual offset-dist~ce combination intersects above the "400-meter" line. This point of Intersection indicates that the difference in offset is positive arid Mat the first condition is satisfied. Therefore, spilIback is possible, depending on the duration of upstream green. Consultation with Figure C-3 ~ checks the second condition. Specifically, it mbicates that the maximum green interval duration is about 41 seconds (or about 45 seconds using the ~ :3 rule) and, more unportantly, that Me actual green-distar~ce combination intersects above the trend line ~ndicat~g Me strong likelihood of spilIback during saturated flow conditions. This finding indicates that the "win spilIback" fob of the distance-to-queue adjustment factor should be used to compute the capacity of the phase based on the assumption that traffic demands are sufficiently high as to fully use the signal phase. C.5.7 Predictive Ability of the Proposed Models As a verification of the calibrated saturation flow rate and start-up lost time models, the predicted and measured discharge times of several last-in-queue vehicles were compared. The predicted discharge time was computed using the following equation: T = J H + I (C-60) where: C-68

Tow = discharge time of He J~ queued vehicle, see; J= last queue position to discharge; H= miIiimum discharge headway, sec/veh; arid Is = start-up lost the, sec. For this verification, Equation C-60 was used (with the saturation flow rates predicted by Equation C41 arid the start-up lost times predicted by Equation C49) to predict He discharge time of the last queued vehicle observed dunng several hur~dred signal cycles. evaluation are shown In Figure C-32. Predicted Discharge Time, sec (Legend: A = 1 obs, B = 2 obs, etc.) 60 40 ~ 20 + A A A AA A AAA ~ JBAI BABAA AABBBA AGACA ~7 The results of this (Each data point represents the average of 25 offs.) A 10 20 30 40 50 Measured Discharge Time, sec Figure C-32. Comparison of predicted and measured discharge times. As the trends In Figure C-32 indicate, the recommended models are able to accurately predict the discharge time of the last queued vehicle. Moreover, the fit is equally good over the range of discharge times, indicating no time-based bias in the prediction process. The R2 for the fit of the predicted to measured discharge times is 0.95. It should be noted Hat He database used for this evaluation was the same as that used for the mode] calibration, thus, this evaluation is a mode} verification rather than a model validation. Nevertheless, it still provides strong evidence ofthe accuracy of the recommended models. It should be noted that the data points shown in Figure C-32 represent the average of 25 observations each. This averaging was ~d~ to facilitate the graphical presentation of the model's trend-wise fit to the data, the ~ndividu~ data points were used in He assessment of model fit. C-69

C.5.S Effect of Saturation Flow Rate and Lost Time on Capacity The ideal saturation flow rate recommended in Chapter 9 the 1985 Highway Capacity Manual 699 was 1,800 pcphgpl. Chapter 2 of this manual recommended a start-up lost time of 2.0 seconds. More recently, the ~ 994 HCM (~) included a recommendation of 1,900 pcphgp! for Me ideal saturation flow rate but made no recommendation for a specific start-up lost time value Comer than to note that appropriate values for this lost time were in a range of i.0 to 2.0 seconds). The ideal saturation flow rate recommended In this report is 2,000 pcphgpl which represents art increase beyond He current HCM value. However, the "cost" associated with this increase is an increase In the starbup lost time. Fortunately, the increase in start-up lost time is generally small and does not totally offset the gains in capacity obtained from the higher flow rates. The relative unpact of alternative combinations of saturation flow rate and start-up lost time on capacity are shown In Figure C-33. The capacity shown In this figure was computed using Equation C-39 with a 90-second cycle length and a green extension of 2.5 seconds. In general, Here is an ~ ~ percent increase In capacity Implied by the increase in saturation flow rate Tom I,SOO to 2,000 pcphgpI, for the same I.S-second start up lost time. However, if a more accurate start-up lost time of 2.82 seconds is used with He 2,000 pcphgp! flow rate, then the increase is reduced by about 20 to 40 percent (corresponding to a true increase in capacity of only about 6 to 9 percent). The point to be made here is that start-up lost time is not a true "lost time;" rawer, it is a "cost" of associated wad accelerating to the speed at saturation. Hence, the higher the saturation flow rate, He higher the speed wait be at saturation flow and He longer He acceleration (or staxt-up lost) time. 1600 1400 1200 1 000 800 600 Capacity, vphpl r Sat. Flow / Lost Time _1,800 /1.50 +2,000 /1.50 . `2,000 / 2.82 an_ _ ~_ ,~ ~ I- - - - - - - - - - - - - - - - - - - gO~ec. cyctelengtr - - - - - - - - - - 400 20 - - - - - - - - - - - - - ~ - - - ~ 1 - 30 40 50 60 70 Phase Duration, see Figure C-33. Elect of saturation flow rate and start-up lost time on capacity. C-70

C.S.9 Conclusions and Recommendations This section summarizes the conclusion and recommendations resulting from the analysis of capacity characteristics associated with signalized interchange ramp terminals and closely-spaced intersections. The ideal saturation flow rate recommended In this report is 2,000 pcphgpl. In Me context ofthe factors studied for this research, this ideal flow rate applies to through traffic movements that have art infinite distance to the back of downstream queue, operate under non-spilIback conditions, and have traffic volumes that are relatively high (reflecting those fond during peak demand periods). Based on this research, it is concluded that the distance to the downstream queue has a significant effect on the discharge rate of upstream traffic movements. This effect is amplified when the signal timing relationship between the two intersections allows queue spiliback to occur. As the distance-to-queue variable is bounded to a maximum value equaling the length of the downstream street segment, the effect of distance-to-queue also includes the effect of spacing between Interchange ramp terminals or between a ramp terminal and a closely-spaced intersection. Turn radius has a significant effect on the discharge rate of a turn-related traffic movement. Saturation flow rates were much lower for turn movements with small radii than they were for An movements with large radii. In the context of junction type (e.g., s~ngle-point urban diamond, at- grade intersection, etc.), the saturation flow rates for the left-turn movements at s~ngle-po~nt urban interchanges are more nearly equal to those of through movements because of We large turn radii associated with this interchange type. Traffic pressure, as quantified by traffic volu~ne per cycle per lane, has a significant effect on saturation flow rate. The saturation flow rates of lower volume movements are much lower Can those of higher volume movements. Other factors were examined for Heir potential eject on saturation flow rate. These factors include: g/C ratio, junction type, downstream signal indication at the start ofthe upstream phase, and dual versus single left-turn lane. Of these factors, only g/C ratio was found to be correlated wad saturation flow rate in a statistically significant manner. Specifically, Me saturation flow rate for lefc-turn movements with low g/C ratios was Soured to be higher than the rates of similar movements with larger TIC ratios. This effect was also found in the Trough movements studied, however, it was much smaller In magnitude and not statistically significant. Therefore, it was determined that more research is needed to verify Me significance of this trend and its magnitude before an adjusunent factor for this effect can be recommended. Start-up lost time was found to Increase with saturation flow rate. This increase is due to Me increased time it takes for the discharging queue to attain the higher speed associated win a higher saturation flow rate. Predicted start-up lost tunes range from 0.61 to 3.IS seconds for prevailing saturation flow rates of 1,400 to 2,100 pcphgpl. C-71

The average amount of Me yellow interval used by drivers is termed "green extension." This characteristic can be used to compute clearance lost torte and the effective green tune. The recommended green extension value is 2.5 seconds. Other values are possible if the approach speed is outside the range of 64 to 76 lymph or when the volume-to-capacity ratio for the analysis period is larger Cart 0.~. An equation is provided for these situations. Lane use is almost always uneven (or unbalanced) in intersection lane groups. The degree of this imbalance is expressed In terms of the lane utilization factor. The lane utilization factor vanes depending on the nature of drivers' lane-choice decisions (i.e., to minimize travel time or to preposition for a downstream turn). Lane utilization factors based on have} time minimization tend to be subject to randomness in the lane-choice decision process. The factors stemming from this process range from I.] to 2.0, depending on the number of lanes in the larle group and its corresponding traffic volume. Lane utilization factors based on Diver desire to preposition can vary widely, depending on the volume of traffic Mat is preposition~ng In the subject lane group. Capacity is dependent on the prevailing saturation flow rate, start-up lost dine, and clearance lost time. The equations provided in this appendix should be used to estimate the saturation flow rate, as affected by distance to downstream queue, tum radius, and traffic pressure. Start-up lost time is not a constant value; rather, it is dependent on the prevailing saturation flow rate. Thus, We equation provided in this appendix should be used to estimate the start-up lost time that corresponds to the prevailing saturation flow rate. C.6 APPENDIX C REFERENCES 1. Special Report 209: Highway Capacity Manual. TRB, National Research Council, Washington, D.C. (1994). 2. Bonneson, I.A. "Study of Headway and Lost Time at Single-Po~nt Urban Interchanges." In Transportation Research Record 1365. TRB, National Research Council, Washington, D.C. (1992) pp. 30 - 39. 3. Messer, C.~., Bonneson, I.A., Anderson, S.D., and McFarland, W.F. NCHRP Report 345: Single Point Urban Interchange Design anal Operations Analysis. TRB, National Research Council, Washington, D.C. (1992). 4. Stokes, R.W., Messer, C.~., and Stover, V.G. "Saturation Flows of Exclusive Double Lefi-Turn Lanes." In Transportation Research Record 1091. TRB, National Research Council, Washington, D.C. (1986) pp. 86-95. 5. SAS/STAT User's Guide. Version 6, 4th ed. SAS Institute, Inc., Cary, Norm Carolina (1990~. 6. Kimber, R.M., McDonald, M., alla Hounsell, N.B. TRRE Research Report RR67: The Prediction of Saturation Flowsfor Road Junctions Controlled by Tragic Signals. Transport and Road Research Laboratory, Department of Transport, Berkshire, England (1986~. C-72

7. Fambro, D.B., Messer, C.J., and Andersen, D.A. "Estimation of Unprotected Lef~c-Tu~n Capacity at Signalized Intersections." In Transportation Research Recorc! 644. TRB, National Research Council, Washington, D.C. (1977) pp. ~ 13-] 19. 8. Messer, C.J. and Fambro, D.B. "Critical Lane Analysis for Intersection Design." In Transportation Research Record 644. TRB, National Research Council, Washington, D.C. (1977) pp. 26-35. 9. Special Report 209: Highway Capacity Manual. TRB, National Research Council, Washington, D.C.'(1985~. C-73