**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

**Suggested Citation:**"Chapter 4 - Model Details." National Academies of Sciences, Engineering, and Medicine. 2023.

*Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide*. Washington, DC: The National Academies Press. doi: 10.17226/27131.

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CHAPTER 4 Model Details 4.1âIntroduction This chapter provides a summary of the models for RTOR volume and capacity used in the spreadsheet tool, as well as an additional discussion of the results for estimating delay. This chapter provides a summary of the developed models. Additional details on the model develÂ opment are provided in NCHRP Web-Only Document 368. 4.2â Volume Estimation Methodology Statistical modeling was used to develop models for RTOR volumes. Several different models were created considering different right-turn lane configurations and also considering different variables about site conditions. Many different statistical forms of these models were also tested in the course of this research. The complete details of this analysis are included in NCHRP Web-Only Document 368. The forms of the equations are shown here. Ultimately, four different models were prepared: â¢ Model 1 employed the largest number of statistically significant variables possible, using the zero-inflated negative binomial form, which had superior performance compared to other model forms. Two versions of this model were produced. Model 1A included all variables collected during the study, which included certain variables that are not easily obtained in the field. Model 1B was restricted to only variables that would typically be obtained from field count studies. â¢ Model 2 was restricted to only more easily obtained data, similar to Model 1B, except that the simpler negative binomial equation form was used. â¢ Model 3 was a simple model (using logistic regression) using only the total right-turn flow rate and the red-to-cycle ratio. To explain the terms used in the models, it is necessary to reference various movements at the intersection. FigureÂ 14 shows a diagram of an intersection with a subject RTOR movement. Labels are provided for three conflicting vehicle movements: V1, the conflicting through, V2, the conflicting left turn, and V3, the conflicting U-turn. In addition, the shadowed left turn, LT, is indicated as well as two pedestrian movements. Movement P1 conflicts with the RTOR movement. 32

Model Detailsââ 33Â Â V2 V3 V1 RTOR LT P1 P2 Figure 14.ââ Movements at a typical intersection that conflict with RTOR. The equations for the models are as follows. For single right-turn lanes, Model 1A: R V q RTOR = SS- 0.167 + 5.020 a r C k + 0.01037q RWW T X R V S2.923 + 1.389 a r C k - ` 1.290 # 10 -4 j q 1, r + ` 2.489 # 10 -3 j q L, rW S W S W # exp S + ` 3.360 # 10 -3 j q R - ` 2.517 # 10 -3 j q ped, r - 0.06377d PLCW W (4) S W S W S - 0.1024d 1RCL + 0.1291d SLT W T X For single right-turn lanes, Model 1B: R V q RTOR = SS- 0.167 + 5.020 a r C k + 0.01037q RWW T X R V S 2.793 + 1.486 a r C k - ` 2.069 # 10 -4 j q 1 - ` 3.069 # 10 -4 j q 2 W S W S W # exp S+ ` 6.990 # 10 -4 j q SL + ` 3.558 # 10 -3 j q R - ` 2.233 # 10 -3 j q pedW (5) S W S W S - 0.05420d 1RCL W T X For single right-turn lanes, Model 2: R V S 2.497 + 1.743 a r g k - ` 2.025 # 10 -4 j q 1 - ` 4.152 # 10 -4 j q 2 W S W q RTOR = exp S W (6) SS+ ` 9.084 # 10 -4 j q SL + ` 3.869 # 10 -3 j q R - ` 2.302 # 10 -3 j q pedWW T X

34ââ Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitionerâs Guide For single right-turn lanes, Model 3: exp ;- 2.321 + 3.470 a r C kE q RTOR = R V (7) 1 + exp SS- 2.321 + 3.470 a r C kWW T X For shared right-turn lanes, Model 1A: R V q RTOR = SS- 0.458 + 2.734 a r C k + 0.01406q RWW T X R V S2.670 + 1.438 a r C k - ` 2.870 # 10 -4 j q 1, r - ` 9.837 # 10 -4 j q 2, r W S W S W # exp S- ` 2.733 # 10 -3 j q 3, r - ` 1.939 # 10 -3 j q ped, r + ` 3.692 # 10 -3 j q RW (8) S W SS WW - 0.1871d CBL - 0.2827d 1RCL T X For shared right-turn lanes, Model 1B: R V q RTOR = SS- 0.459 + 2.728 a r C k + 0.01223q RWW T X R V S 2.678 + 1.262 a r C k - ` 1.941 # 10 -4 j q 1 - ` 9.304 # 10 -4 j q 2 W S W S W # exp S+ ` 1.523 # 10 -4 j q SL + ` 3.607 # 10 -3 j q R - ` 2.088 # 10 -3 j q pedW (9) S W SS WW - 0.04132d 1RCL T X For shared right-turn lanes, Model 2: R V q RTOR = exp SS2.013 + 1.725 a r C k - ` 1.180 # 10 -3 j q 2 + ` 4.441 # 10 -3 j q R - ` 1.200 # 10 -3 j q pedWW T X (10) For shared right-turn lanes, Model 3: exp ;- 2.462 + 2.844 a r C kE q RTOR = R V (11) 1 + exp SS- 2.462 + 2.844 a r C kWW T X For dual right-turn lanes, Model 1A: R V q RTOR = SS0.245 + 5.160 a r C k + 0.02175q RWW T X R V S2.390 - 0.2293d 2L + 0.1343d I + 1.334 a r C k - ` 2.461 # 10 -4 j q 2, rW S W S W # exp S- ` 2.428 # 10 -3 j q plped, r - ` 2.224 # 10 -3 j q ped, r + ` 5.260 # 10 -3 j q R W (12) S W SS WW - 0.04242d PLCW T X

Model Detailsââ 35Â Â For dual right-turn lanes, Model 1B: R V q RTOR = SS0.245 + 5.160 a r C k + 0.02168q RWW T X R V S2.351 - 0.2079d 2L + 0.1410d I + 1.467 a r C k - ` 2.235 # 10 -4 j q 1W S W S W # exp S - ` 3.373 # 10 -4 j q 2 + ` 3.348 # 10 -5 j q LT + ` 5.281 # 10 -3 j q R W (13) S W - ` 2.670 # 10 j q ped S -3 W S W T X For dual right-turn lanes, Model 2: R V S1.530 + 0.4177d I + 2.470 a r C k - ` 2.539 # 10 -3 j q 2W S W q RTOR = exp S W (14) S + ` 3.582 # 10 -3 j q R - ` 1.736 # 10 -3 j q ped W T X For dual right-turn lanes, Model 3: R V exp SS- 2.293 + 0.4159d I + 2.851 a r C kWW q RTOR = TR X V (15) 1 + exp SS- 2.293 + 0.4159d I + 2.851 a r C kWW T X where: qRTOR = RTOR flow rate, qR = total right turn flow rate, r/C = red-to-cycle ratio, q1 = total conflicting thru flow rate, q1,r = conflicting thru flow rate during red, qL,r = shadowed left turn flow rate during red, q2 = total opposing left-turn flow rate, q2,r = opposing left-turn flow rate during red, qSL = total shadowed left-turn flow rate, qped = total conflicting pedestrian flow rate, qped,r = conflicting pedestrian flow rate during red, qplped,r = parallel pedestrian flow rate during red, qLT = total shadowed left turn flow rate, Î´2L = indicator variable for the presence of two or more right-turn lanes on the approach, Î´PLCW = indicator variable for the presence of a parallel pedestrian crosswalk, Î´1RCL = indicator variable for the presence of one receiving lane, Î´SLT = indicator variable for the presence of a shadowed left turn, Î´CBL = indicator variable for the presence of a conflicting bicycle lane, and Î´I = indicator variable for whether the subject approach is an interchange ramp. For all indicator variables, the value is equal to 1 if the condition is true and equal to 0 if the condition is not true. 4.3â Capacity Estimation Methodology In addition to RTOR volume estimation, models of right-turn capacity considering RTOR were also conducted. To develop forms of the equations, it was necessary to obtain data from right-turn movements with continuous high demand. However, very few field data were available

36ââ Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitionerâs Guide for which such conditions existed. Thus, simulation modeling was done to generate data to suggest forms of the equations. Further details are included in NCHRP Web-Only Document 368. The starting point of capacity estimation is the signal timing. FigureÂ 15 breaks down the effective red time for a right-turn movement into three intervals that typically exist at most intersections. â¢ RTOR Interval 1 occurs when the shadowed left turn is active. If a right-turn overlap exists at the intersection, this interval would be part of the green time. If no overlap exists, drivers may turn right and typically do not have conflicting vehicle volumes unless U-turns exist from the shadowed left-turn movement or pedestrian overlaps exist. â¢ RTOR Interval 2 occurs when the signal is green for the cross street through movements. The main conflicting movements are V1 and P1 (FigureÂ 14). â¢ RTOR Interval 3 occurs when the signal is green for the opposing left turn. The main conflict- ing movement is V2. 4.3.1â Single Right-Turn Lane Begin by considering movement V1. During its red interval, queues accumulate on the approach for V1, and this queue is discharged during green. First, the queue service time for V1 (rQ,V1) is estimated from: R V q V1 S C ` 1 - PV1 j W S W rQ, V1 = max S0, 3600 (16) S s V1 q V1 CPV1WW S - 3600 3600 g V1 W T X where: C = cycle length, qV1 = flow rate, PV1 = proportion on green, gV1 = green time, and sV1 = saturation flow rate for movement V1. The remaining unsaturated red time, rU,V1, on the RTOR movement is given by: rU, V1 = rV1 - rQ,V1 (17) where rV1 is the red time associated with movement V1. The unsaturated red time is the time period in which gaps are available and the RTOR movement can take place. Figure 15.ââ Intervals within the red light interval for a right-turn movement.

Model Detailsââ 37Â Â The capacity contributed by the unsaturated red time associated with movement V1, CV1, is given by: R J V rU, V1 3600 S K t c - 1 NO W c V1 = exp S- K q (18) C tf S 500 O V1WW T L P X where: tf = follow-up time and tc = critical gap for the right-turn maneuver. In brief, the RTOR movement is analyzed as a two-way stop-control intersection during this interval. The same procedure is done to calculate the capacity contributed by the unsaturated red time associated with movement V2, cV2, by using the above three equations, substituting values for movement V2 in place of V1. The capacity contributed by the shadowed left-turn movement, crLT, is given by: g LT 3600 c LT = (19) C tf where gLT is the green time of the opposing left turn and other terms are as previously described. Finally, the total RTOR capacity is given by: c RTOR = c LT + c V1 + c V2 (20) 4.3.2â Dual Right-Turn Lane The capacity, cRTOR, of the dual right-turn lane group is similar to that of the single right-turn lane. However, instead of performing one calculation for the single lane, two calculations are performed with one for the inner lane and one for the outer lane. Thus, the total RTOR capacity would be given by: c RTOR = c I + c O c I = c I, LT + c I, V1 + c I, V2 (21) c O = c O, LT + c O, V1 + c O, V2 where the symbols I and O refer to the inside and outside lane, respectively. Altogether, six instead of three capacity calculations are worked out before adding up the total RTOR capacity. Different critical gap and follow-up time values are suggested for each lane. 4.3.3â Shared Right-Turn Lane The estimation method is modified for the shared right-turn lane scenario because of the need to consider the possibility of blocking by through traffic. The first step is to calculate the proportion of shared lane traffic turning right, PR, from: qR PR = R V S ` q R + q T jW q T + max S0, W (22) S NL W T X

38ââ Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitionerâs Guide where: qR = right-turn flow rate, qT = through flow rate, and NL = number of lanes on the approach. The capacity associated with movement V1 is given by: R V rU, V1 3600 S ` 4PR + 0.3t c - 1 j W c V1 = â¢ 0.01 exp 84.3PRB â¢ exp S- q V1W (23) C tf S 1000 W T X All the terms are defined the same as previously for the single-lane scenario and calculated for unsaturated time, rU,V1, using Equations 10 and 11. Similarly, the same calculation would be done for movement V2 by swapping out the appropriate variables. The capacity contributed by the shadowed left-turn movement is given by: g LT 3600 c LT = â¢ 0.01 exp 84.3PRB (24) C tf The total RTOR capacity is calculated using Equation 14. 4.4â Estimation of Delay There is a limit to the effect that subtracting out RTOR volumes can have on the delay. To explain, consider the expression for the d1 term (uniform delay): CJ g N2 KK1 - OO 2L CP d1 = (25) 1 - min 81, XB g C where: X = volume-to-capacity ratio, g = green time of the movement of interest, and C = cycle length. If X is set to zero, the resulting delay represents a minimum value that is a function of the signal timing. Thus, even if all the right-turn traffic is served during red (with presumably negligible amounts of delay), the uniform delay for the movement would reach a minimal value that is greater than zero. While delay has other components, the uniform delay is often the largest during undersaturated conditions. For this reason, users of the spreadsheet will observe that changes to the volume do not result in intuitive changes to the delay and LOS. For example, removing all the conflicting volume does not make a right-turn movement behave as if it were green all of the time. This is because of the g/C term in the delay equation that is based on the given green time. An adjustment to green could make this occur: if g/C is increased, the delay will be reduced. To employ the estimated RTOR capacity in a delay estimate for a right-turn movement, the HCM delay equation may not be the appropriate equation form, because it is based on the assumptions of certain traffic behaviors occurring in the red and green intervals, which for a RTOR movement do not follow those behaviors. NCHRP Web-Only Document 368 discusses this in more detail.

Model Detailsââ 39Â Â 4.5â Suggested Signal Timing The spreadsheet tool includes a macro that will generate a cycle length and splits for purposes of analyzing a location for which signal timing data are not known. The starting point is the calculation of the volume-to-saturation (v/s) ratios for each phase and the sum of these for the critical path, Y: Y = max 8y 1 + y 2, y 5 + y 6B + max 8y 3 + y 4, y 7 + y 8B (26) here, yi is the volume-to-saturation ratio (equal to the volume divided by the saturation flow rate) for phase i. The two max functions in the formula identify the critical path (phases {1234, 1278, 5634, or 5678}). Based on which critical path is selected, the corresponding lost times are also summed for those phases belonging to the critical path; their sum is L. For example, if y1 + y2 > y5 + y6 and y3 + y4 > y7 + y8, the critical path is 1234, and L = ,1 + ,2 + ,3 + ,4, where ,i is the lost time of phase i (yellow change and red clearance intervals). Three options are included for selecting cycle length. Besides the use of a user-defined cycle length, one option is to use Websterâs formula, 1.5L + 5 C= (27) 1-Y while the other is based on the degree of intersection saturation: LXCl C= (28) XCl - Y In this formula, Xâ²C is the target degree of intersection saturation. After the cycle length has been determined, the splits are then set. First, the splits are set using the following rule: C-L gi = yi (29) Y This gives the green time; the split is obtained by adding the lost time. Splits can never be shorter than the minimum split. It is assumed that the minimum green time is five seconds for purposes of this analysis. Initially, green times are determined for the critical path phases. Next, the non-critical path phase splits are balanced so that the total splits in each ring and concurrency group add up properly. 4.6âConclusion This chapter provided a description of the equations used for the volume and capacity adjust- ments contained in the spreadsheet tool. Complete methodological details, including descriptive statistics of dependent variables, other alternative models considered, and model validation are included in NCHRP Web-Only Document 368.