62 TRANSPORTATION RESEARCH RECORD 1365 Left-Turn Adjustment Fators for Saturation Flow Rates of Shared Permissive Left-Turn Lanes FENG-BOR LIN For apaity ana lysi of signalized interseti11s, a left-turn adjustment fator is used in the 1985 Highway Capt1i1y Manual (H M) to aount for the effet of left turns on satmation flow rates. When hared permis ive left-turn lane. are the subjet of analysis, the HCM uses a theoretial model to determine the related adjustment fator. Qu tions onerning the reliability of this model have been raised, and a reent study sponsornd by the Federal Highway Administration ha sugge ted that the HCM model be teplaed by a set of new models. The new models, however, have erious flaws. An improved model that provides logial explanations of the ausal relationships b tween the leftturn adju tment fator and its ontributing fator is desribed. The ontributing fators inlude opposing flow rate, number of opposing lanes, flow rate in the lane adjaent to the shared lane, proportion of left turns in the shared lane proportion of left turn in opposing flow, proportion of opposing vehile arriving in red interval, yle length, green interval, and hange interval. A numerial exaniple is given to illustrate the appliations of the improved model. In the apaity analysis of signalized intersetion, the 1985 Highway Capaity Manual (HCM) (1) requires that the saturation flow rate of a lane or a lane group be determined from the following formula: where S = saturation flow rate (vphg) (vehiles per hour of effetive green interval); S = ideal saturation flow rate, taken to be 1,8 vphg per lane; N = number of lanes in a lane group; F = the produt of seven adjustment fators related respetively to lane width, heavy vehiles, approah grade, parking, bloking effets of loal buses, area type, and right turns; and f LT = adjustment fator for left turns. When an analysis involves shared permissive left-turn lanes, the determination of the left-turn adjustment fator beomes a rather diffiult problem. A shared permissive left-turn lane refers to a lane from whih opposed left-turn vehiles and vehiles of other diretional movements an move into the intersetion in a permissive left-turn signal phase. The presene of opposed left Department of Civil and Environmental Engineering, Clarkson University, Potsdam, N.Y. 13699-571. (1) turns in suh a lane disrupts vehiular movements and ompliates the determination off LT The 1985 HCM relies on a theoretial model to deal with this problem. The HCM onept in estimating f LT has been used by Levinson (2) to develop a model for estimating the apaity of shared left-turn lanes. Questions onerning the reliability of the HCM model have been raised, however, and a set of new models was reommended in a reent study sponsored by FHWA (3). The new models were developed from regression analysis of field data. They are easy to use but do not properly aount for the ausal relationships between flt and its ontributing fators. For example, flt an be expeted to vary with opposing flow rate, yet the model reommended for single lane approah assumes impliitly that f LT is independent of opposing flow rate. This flow may be partially responsible for the fat that the estimates obtained on the basis of that model have little orrelation with observed values (3). To provide an alternative, this paper desribes an improved model that explains logially the ausal relationships between!lt and its ontributing fators. This model deals with the leftturn adjustment fators for shared lane only (i.e., N = 1); it is developed on the basis of theoretial reasonings, field data, and omputer simulation. A numerial example is provided to illustrate the appliations of the model. RESEARCH APPROACH Equation 1 indiates that, if the saturation flow rate for a given F an be determined under a wide range of onditions, then a data base an be established for modelingflt Beause the vehiular movements in a shared left-turn lane an be interrupted by bloked left-turn vehiles, their saturation headway annot be meaningfully defined and measured in the field to estimate the orresponding saturation flow rate. This problem an be overome by taking advantage of the following relationship for N = 1 and for onditions represented by F = 1.: f -.._ - CQ,..... S G,S,, LT - - where Qm.. = apaity of a shared lane for F = 1. (level, 12-ftwide approah lane and ideal onditions for other fators related to F), C = yle length (se), and Ge = effetive green (se). (2)
Lin For a given ombination of C and G., Equation 2 shows that finding f LT is the same as finding Qm x Qmax an be determined by estimating the number of vehiles per yle that an move out of an intersetion. This expeted number of departures inludes the following omponents: early left turns, M 1 ; unbloked straight-through departures, M 2 ; departures in leftover green, M 3 ; and departures after green interval, M 4 Details of these omponents will be disussed later. The modeling of Qmax is divided into two parts. The first part onerns a basi flow pattern as shown in Figure 1 (top). The notations used in this figure are defined as follows: Q 1 = flow rate in the inside opposing lane (vph), Q 2 = flow rate in the outside opposing lane (vph), and Q. = flow rate in the lane adjaent to the shared lane (vph). The opposing lanes of the basi pattern do not ontain left-turn vehiles. Suh a situation may exist at a T-intersetion or at a four-leg intersetion where left turns from the inside opposing lane are prohibited. The maximum number of opposing lanes onsidered in this study is two. The seond part of the modeling effort involves the development of a mehanism to transform a flow pattern that ontains left turns in Q 1 into an equivalent basi flow pattern. This transformation involves the onversion of Q 1 into an equivalent straight-through opposing flow (Q 1 ), for the estimation of the apaity of a shared lane. The transformation proess is shown in Figure 1 (bottom). In this figure, P represents the proportion of left turns in the inside opposing lane. Qmax is onsidered to be a funtion of suh variables as Q 1, Q 2, Q., P, yle length C, green interval G, signal hange interval Y, proportion of left turns in shared lane P,, and so forth. This study employs a ombination of theoretial onsiderations, field data, and omputer simulation to identify the ausal relationships between Qmax and its ontributing variables. The simulation model used in this study is a mirosopi model developed at Clarkson University. This model an realistially simulate the stohasti movements of vehiles at intersetions ontrolled by a variety of traffi signals. Details of this model are desribed elsewhere (4). An example omparison of the simulation outputs and field observations is given in Table 1. One onern in modeling Qmax and its relatedfltis whether the effets of signal oordination should also be onsidered. A simulation analysis reveals that the apaity of a shared left-turn lane may be affeted by the presene of prominent yli platoons in the opposing lanes as a result of signal oordination. Nevertheless, as shown in Figure 2, the apaities of shared left-turn lanes when the arrivals are random tend to lie mostly within 5 vph of the values for oordinated signal operations. Suh disrepanies are not alarmingly large in the ontext of apaity analysis. To avoid unneessary ompliations, the arrivals will be assumed to be random for the purpose of modeling Q,,,.x and!lt To aount partially for the fat that arrivals are not neessarily random, the proportion of arrivals in red interval is inluded as a variable in the modeling proess. MODEL FOR BASIC FLOW PATTERN Given the four omponents of departures per yle, M 1, M 2, M 3, and M 4, the apaity of a shared lane for F = 1. an be determined as The modeling of eah of the departure omponents is disussed. Early Left Turns, M, Early left turns refer to those leading left-turn vehiles in various yles that turn in front of the leading opposing vehile shortly after green onset. Given the proportion of left turns in a shared left-turn lane (Ps) and the probability of early left turn a for a leading left-turn vehile, the expeted number of early left turns per yle an be estimated as M, = xp 5 (4) The values of ex are often less than.5. Therefore, when Psis muh smaller than 1., M, beomes negligibly small. 63 (3) 2 1 l k t shared Q lane Unbloked Straight-Through Departures, M 2 After the green onset, those straight-through vehiles ahead of the first left-turn vehile an move out without faing the possibility of being bloked. To failitate the estimation of suh departures, the diretional movements of the vehiles in a shared left-turn lane are lassified into a series of events as shown in Figure 3. In this figure, K 2 represents the expeted maximum number of straight-through vehiles that an move into the intersetion before the green interval G expires. The value of K 2 an be determined as G - Ls K1 = ----" HS (5) FIGURE 1 Basi flow pattern (top) and pattern transformation (bottom). where Ls is lost time due to starting delays (se), and Hs is saturation headway when only straight-through vehiles are present (se).
TABLE 1 Comparison of Observed and Simulated Signal Operations Observed Average Green, se Case Phase Mean s.. 1 Mean A 1 6.5 2.6 5.1 2 32.4 25.2 3.3 B 1 33.8 18.2 31.9 2 5.4 2.1 5.1 3 24.. 24. 1 27.8 12.8 27.6 2 12.4 6.3 13. D 1 18.8 9.2 17.5 2 1.7 5.8 9.5 E 1 29.3 7.6 3.2 2 2.5.7 2.2 F 1 12.9 3.9 12.6 2 9.2 4.1 9.4 3 33.6 7.3 32.1 G 1 1.8 6. 12.1 2 3..8 3. 3 19.1.5 19.2 Simulated S.D. 2.1 24.6 17.6 2.3. 12.1 6.3 8.1 5.2 9.1.9 3.5 4. 6.7 3.7.. Average Stopped Delay, se/veh Observed Simulated 7.7 2 6.6 11.1 3 1.7 14.7 3 14.9 Not available Not available 42.3 4 43.7 14. 3 13.5 3.1 5 2.7 25.1 6 25.6 H 1 13.4 3.4 13.4 2 3..2 3. 3 19.1.7 19.9 1 S.D. = Standard deviation 2 singlo-lane flow with right turns and left turns 3 exd usive lef\-turo flow 4 shared-permissive left tum flow (85% left turns) 5 exelusive right-turn fl ow with right-tum-on-red 6 shared-permissive left-tum flow (93% left turns) 7 shared-permissive left-tum flow (95% left turns) 3..1 2. 41.9 7 41.6 12. a. > 1!!I. 8-6 (")'. 4 :'= z. 2 '(j a.,, / v 1.1 ;ti u Capaity? I I/ l/' II ll '. t..mi -,_ It' :t!> JO 4 t:l' JU tllju 11.. u1 2 with Random Arrivals, vph FIGURE 2 Capaities with yli platoon arrivals in opposing lanes versus apaities with random arrivals. Event lo- G1 ---""--G2 --1- --- -- --- 2 S L 3 SSL!L-t:.!_ )L _ k1 +2 SSS... SSS SL k1+3 SSS... SSS SSL k2 S S S... S S S S S S... S S L l Set B k2+1 s s s... s s s s s... s s s ------ --- ---- -- Note: S straight-through L left-tum G = green Interval G 1 = bloked green FIGURE 3 Classifiation of arrival sequenes.
Lin The expeted number of unbloked straight-through departures per yle an be estimated as K 2 if P, = (6a) and Q ' (1 - R ) C qo1 = G + y (9) 65 K1 -I M2 = n.?o n(l - Ps)"P, + K 2 (1 \ (1 - P, - (1 - Ps)K 2 ]f Ps if PS > (6b) If there are x 1 queueing vehiles in the ith opposing lane at the green onset, the portion of the green interval f; onsumed by these and subsequent queuing vehiles before they all ross the onfliting point may be estimated as o ifx 1 = (loa) r f Departures in Leftover Green, M 3 In Figure 3, the green interval G is divided into two omponents: G 1 and G 2 G 1 is the average portion of the green interval onsumed by the queueing vehiles in the opposing Janes before these vehiles ross the onfliting point. The onfliting point an be onsidered to be a representative loation at whih a leading left-turn vehile that is bloked would ome to a stop to wait for a suitable gap in the opposing flow. On the basis of G 1, the events shown in the figure are grouped into Set A and Set B. Set A events allow a maximum of K 1 straight-through vehiles to move into the intersetion before a left-turn vehile beomes the leading vehile in the shared lane. The number of straight-through vehiles ahead of the first left-turn vehile in Set B ranges from K 1 + 1 to K 2. The value of K 2 is determined from Equation 5, and K1 an be determined in a similar manner as After the departures of unbloked straight-through vehiles, a mix of left-turn and straight-through vehiles an move out by using leftover green intervals. For eah of the Set A events, the leftover green interval is G 2 For the Set B events, the leftover green intervals depend on the portion of the green interval already onsumed by the unbloked straight-through vehiles. To failitate the estimation of G 1 and the leftover green intervals, let us define the following additional variables: i = inside opposing lane (i = 1) or outside opposing lane (i = 2), S ; = saturation flow rate of the ith opposing lane (vph), x 1 = number of queueing vehiles in the ith opposing lane at green onset, m 1 = average number of queueing vehiles in the ith opposing lane at green onset, q 1 = arrival rate in the ith opposing lane during green and signal hange intervals (vph), q 12 = q 1 + q 2 = sum of arrival rates q 1 and q 2 (vph), R = proportion of arrivals in red in opposing lanes, and = time required for queueing vehiles to go from the stop line until they lear the onfliting point (se). On the basis of these definitions, m 1 and q 1 an be determined as - Qo;RoC mo; - 3,6 (8) (7) { t; = 3,6x1 + L,q1 +Ls + :S G So; - qo; if X; > (lob) Therefore, if 1 2 :s 1 1, the queuing vehiles in the inside opposing lane would govern the time required to disharge all queueing vehiles in a given yle. Otherwise, the queueing vehiles in the outside lane would govern. In other words, the queuing vehiles in the inside lane would govern if the following inequality holds: 3,6x 1 + L,q 1 > 3,6x2 + Lqo2 Soi - qo1 - So2 - q o2 This inequality an be rewritten as (11) < -1- [ 3,6x, + L,qm(S _ )- L J (12) Xz - 3 6 S _ 2 q 2 sq 2, 1 q OI Let X be the largest integer of x 2 that satisfies Equation 12 and P(x;) be the probability of having x 1 queueing vehiles in Lane i at the green onset. For a given x 1, the portion of the green interval onsumed by opposing queueing vehiles would be t 1 if x 2 has a value equal to or less than X. On the other hand, the portion of the green interval onsumed by opposing queueing vehiles would be 1 2 if x 2 has a value larger than X. Therefore, the expeted value of G 1 an be estimated as If the arrivals in red are random, the P(x;) in Equation 13a an be determined from the following Poisson distribution: m;}e-"'n p (x;) - I x,. (13b) When only one opposing lane is present and arrivals are random, Equation 13a an be redued to 3 61111 Soi - qo1 + ( L,qo1 + L, + )(l _ e- mo1) :s G (14) Sm - qo1 If two opposing lanes are present, Equation 13a an be approximated by a simplified equation. Let QH be the larger one of Q 1 and Q 2. And, if q 1 1Sa 1 2: q zfs2, let mh = mo1,
66 qh = qoi> SH = SOI> SL = s2> and ql = qo2 Otherwise, let m.h = m.2, qh = qo2, SH = So2, SL = Soi, and ql = q1. Then the simplified equation an be written as where (.42 +.1R,,) Q 11 C 'Y 2 = 3,6 and (15a) (15b) (15) "'(3 = e 3,6-1 (15d) Equation 15a is the same as Equation 14 when only one opposing lane is present. Twelve samples of field data were olleted from three intersetions to test Equations 13a and 15a. The observed values of G 1 and the estimates obtained from these equations are given in Table 2. The disrepanies between the observed and the estimated values were small. Two other models have been reommended for estimating G 1 in the HCM and in the FHWA study (2). For omparison, the estimated values of G 1 obtained from these models are also given in Table 2. The model reommended in the FHW A study measures G 1 in terms of the time required for the front end of the last queueing vehile to reah the stop line after green onset. Therefore, the estimates obtained from this model TRANSPORTATION RESEARCH RECORD 1365 are inreased by an amount equal to The referene line used in the model given in the HCM is unknown, and therefore no adjustments are made. Given G 1, the leftover green interval for Set A events is G 2 = G - G 1 To failitate further analysis, this leftover green interval is modified into an effetive leftover green T, where (16a) (16b) This equation implies that if G 1 is greater than or equal to the lost time L,, there is no need to adjust for starting delays beause the lost time is ompletely aounted for by G 1 When G 1 is smaller than L,, however, G 1 may not fully aount for the starting delays assoiated with the vehiles in the shared lane. Equation 16b provides an adjustment for suh a situation. For the determination of the average leftover green interval for Set B events, one may proeed with the determination of the probability of an event being in Set B. This probability is (1 - P,)K 1 + 1, where P, is the proportion of left turns in the shared lane. Therefore, for the Set B events and < P < 1 the average number of straight-through vehiles Kb tht a move out before a left-turn vehile beomes the leading vehile in the shared lane an be determined as j { [ K2-1 ] Kb = (l _ p ) K I L n(l - P,)nP, s n=k1+l (17) TABLE 2 Observed and Estimated Values of G, Case Qo1 Qo2 1 323 266 81.8 2 278 246 81.8 3 35 36 81.8 4 388 7.6 5 415 72.2 6 52 71.2 7 494 72.3 8 44 68.9 9 356 8. 1 274 13.7 11 274 82.8 12 218 84. G Ro 23..79 23..88 23..75 3..67 3..68 3..62 3..71 3..52 22.4.8 19.2.92 17.2.72 18.2.78 GI Atual Eq. 13a Eq. 15a HCM FHWA 2. 2. 2.2 13. 23. 19.3 19.2 2.4 11.3 23. 2.4 21.6 23. 16.4 23. 19.1-18.9 11.2 19.2 19.4-2.6 12.6 2.3 22.4-23.5 15.9 21. 24.7. 24.4 16. 24.6 19.9-18.4 12.6 16.2 21.5 21. 14.2 22.2 19.2-19.2 15.2 19.2 17.2 17.2 11.8 17.2 13.4-14.8 9.1 16.4 Note: L, = 2. se (assumed value) = 2.2 to 3.8 se Saturation flow rate: 15 to 175 vphg Lost time per phase = 4 se (assumed for HCM model)
Lin The orresponding values of Kb for P, = and P, = 1 are if P, = if P, = 1 (18a) (18b) The _i!verage portion of the gree interval onsumed by these Kb vehiles is approximately KbHs + L,. The orresponding effetive leftover green interval Tb for the Set B events beomes (19) With the exeption of the last event shown in Figure 3, a leftturn vehile beomes the leading vehile in the shared lane after unbloked straight-through vehiles have moved out. This leading vehile has to use the gaps in the opposing flow to move out. Assuming that a waiting left-turn driver will only aept those gaps longer than -r se, the average number of gaps J that will be rejeted before a gap is aepted an be estimated as.. J = 2: nz(h s -r)"[l - Z(h s -r)] (2) n= O where Z(h s -r) is the probability that a gap h in the opposing flow is less than or equal to T. It an be shown that, for random arrivals, the average number of rejeted gaps an be approximated as '/1iT l=ej.6-1 (21) The value of -r an be onsidered to be equal to the median of the lengths of aepted gaps. Typial values of -r are between 4.5 and 5.5 se. For suh values of -r, the average length of eah rejeted gap an be approximated as -r/2 without inurring signifiant errors in estimating the apaity of a shared left-tum lane. On the basis of this approximation, a waiting left-tum driver will wait an average of lt/2 se before aepting a gap. After a deision is made to aept a gap, it will take an additional 8 se for the left-turn vehile to ross the onfliting point and for the next vehile to move up. Typial values of 8 are between 2. and 2.5 se. Let Hx represent the expeted portion of the green interval onsumed by the first left-turn vehile. Then, Hx an be determined as where H. is saturation headway only when unopposed leftturn vehiles are present and H, is saturation headway when only straight-through vehiles are present. When opposed left turns exist, the average departure headway an be expeted to exeed H and to inrease with the proportion of left turns in the shared lane P, and the opposing flow rate q 12 A logial model haraterizing this relationship between HY, P,, and q 12 is 67 (24a) For random arrivals, simulation reveals that the oeffiients A and B in this equation an be estimated as A =.181"; 68 and B = 1.2P,- O 15 (24b) (24) The first left-tum vehile in a Set A event onsumes Hx se of the effetive leftover green Ta, and the subsequent vehiles onsume an average of HY se eah. Thus, the expeted number of departures W" related to Set A events is r if T. s Hx (25a) Hx w. = 1. + T,, - Hx if T. > Hx (25b) H, Similarly, the expeted number of departures Wb related to Set B events is r if Tt s H, (26a) Hx Wh = 1. + Tu - H:. if Tb > Hx (26b) H,. Set A events and Set B events aount for a total probability of 1 - (1 - P,)K 1 +1 and (1 - P,)K 1 1, respetively. Therefore, the total expeted departures during the effetive leftover green in a yle is (27) (22) After the first left-turn vehile has moved out, the vehile following an be either a straight-through or a left-tum vehile. The expeted time HY needed by either of suh vehiles to move out is not amenable to simple analytial modeling. Nevertheless, when the opposing flow does not exist, the saturation headway H of the vehiles in the shared lane an realistially be estimated as H = (1 - P,)H, + P,H. (23) Departures After Green Interval, M 4 For apaity analysis of signalized intersetions, the 1985 HCM assumes impliitly that two bloked vehiles an move out of the intersetion after the green interval expires. On the basis of this assumed ondition, simulation data generated in this study show that the following equation an provide reasonable estimates of M 4 : (28) This equation implies that M 4 varies from 1.3 to 2. vehiles. The value of M 4 is 1.3 vehiles when opposed left turns
68 do not exist (P, =. or q 12 =.), and it reahes 2. vehiles when bloked vehiles are present in virtually every yle after the green interval expires. STRAIGHT-THROUGH EQUIVALENT OF Q 1 The present of left-turn vehiles in the inside opposing lane an inrease the number and the size of the gaps usable to the drivers in a shared left-turn lane. Therefore, when the proportion of the left-turn vehiles in the opposing lane inreases, the apaity of a shared left-turn lane also inreases. The inrease in apaity an be affeted by several other fators. This phenomenon is shown in Figure 4 on the basis of simulation data. By omparing suh data as shown in this figure with data for an inside opposing flow Q 1 that ontains no left turns, one an determine the straight-through equivalent (Q 1 )e of Q 1 An example of (Q 1 )e expressed as a ratio to Q 1 is shown in Figure 5. TRANSPORTATION RESEARCH RECORD 1365 In general, the relationship between (Q 1 )e and Q 1 an be represented by (29a) where 13 1 an be treated as a onstant oeffiient with a value of.97 and 13 2 is a funtion of several variables. To identify the relationship between 13 2 and the influening variables, a very large number of simulation runs were performed to develop a data base. On the basis of these simu lation data, an analysis was arried out to isolate the effets of eah variable on 13 2. This effort produed a set of equations for determining 13 2 To failitate the determination of 13 2, let us define two funtions, 1 and 8 2, as follows: 1 = - ( e 1 ' 39 G : Y - 1) and P, (29b). a. >1 Q) 6 5. (/) 4... >.3 u [2 (.) 18.. Proportion Ps =.5 Oo =4 vph C = 9 se G = 42 se. in a 1 G+Y ) 2 = (.6 +.233-C- +.21P, Q Then, for Q 1 4 vph, the value of 13 2 is And, for Q 1 > 4 vph, the value of 13 2 is 1 - + 4 ( 4.5-3.6 -C- G + Y -.5 P, ) 4 (29) (29d) (29e) The straight-through equivalent of eah vehile in Q 1 (i.e., (Q 1 )jq 1 as determined from Equation 29a through 29e) has FIGURE 4 Effets of left turns in Q 1 on apaity of shared left-turn lanes. 1....---------- - oa.s.., "'"'.J>.6 &'... 'O.4 +;.2... Proportion Ps.5 a 4 vph C 9 se G 42 se 1. in a 1 FIGURE 5 Charateristi relationships between Q 1 and its straight-through equivalent (Q 1 ).- several harateristis that are worth noting. First, larger Q and P inrease the hane of an opposing left-tum vehile being bloked and, thus, allow more vehiles in a shared leftturn lane to move out. Under suh onditions eah opposing vehile beomes less of a fator affeting the apaity of a shared left-turn lane. This is the reason why, as shown in Figure 5, (Q 1 )jq 1 dereases with Q and P,,. An inrease in the opposing flow Q 1 has similar effets. In ontrast, (Q 1 )) Q 1 inreases with P, and (G + Y)IC. APPLICATIONS In general, the appliations of the analytial model desribed involves the transformation of Q 1 into a straight-through equivalent, the determination of Qmax for the basi flow pattern resulting from the transformation, and the use of Equation 2 to determine f LT A numerial example is given in Table 3 to illustrate the appliations of the model. The example involves a flow pattern that has Q = 4 vph, Q 1 = 2 vph with P =.2, Q 2 = 35 vph, R =.32, and P, =.8. The related signal ontrol has a yle length C of 5 se, a green interval G of 3 se for the permissive left-turn phase,
Lin 69 TABLE 3 Estimation of Qm x and/lr-an Example A. Transformation of 1 into ( 1 ) j3 1 =.97 (in Eq. 29a); o 1 = -1.26 (Eq. 29b); o 2 = 1.55 (Eq. 29); 13 2 =.77 (Eq. 29d); ( 1 ) = 138 vph (Eq. 29a) B. Estimation of Qmax [set 1 to ( 1 ) = 138 vph] Determination of M 1 M 1 =.16 veh/yle (Eq. 4) Determination of M 2 = 14 (Eq. 5); M 2 =.25 veh/yle (Eq. 6b) Determination of Ma m 1 =.61 (Eq. 8); 1Doz = 1.56 (Eq. 8); to 1 = 138 (Eq. 9); to 2 = 35 (Eq. 9l; q 12 = 488; In Eqs. 15a through 15d, Soi = S 2 = 1,8; mil = 1.56; 8tt = Si. = 1,8; qh = = 35; qi. = qol = 138; QB = 2 = 35; y 1 =.39; y 2 =. 2 ; Ya =.475; G 1 = 8.5 (Eq. 15a); Ta= 21.5 (Eq. 16a); K 1 = 3.3 (Eq. 7); Kb = 4.6 (Eq. 17); Tb = 18.8 (Eq. 19); H, = 5.5 (Eq. 22); H = 2.8 (Eq. 23); A =.155 (Eq. 24b); B = 1.55 (Eq. 24); H. = 4.7 (Eq. 24a); w. = 4.4 (Eq. 25b); wb = 3.83 (Eq. 26b); Ma = 4.4 veh/yle (Eq. 27) Determination of M 4 M 4 = 2.6 > 2. (Eq. 28); set M 4 = 2. veh/yle Determination of Qmax Qmax = 49 vph (Eq. 3) C. Estimation of flt flt =.45 (Eq. 2 with G = G = 3 se) and a hange interval of 4 se. The saturation flows for straightthrough movements and unopposed left turns are, respetively, 1,8 vphg (Hs = 2. se) and 1,7 vphg (H, = 2.1 se). In addition, the following parameters are used: -r = 5.5 se, o = 2.5 se, () = 2.5 se, ex =.2, and Ls = 2. se. The model gives an estimated Qm.x of 49 vph and a!lt of.45. In omparison, diret simulation gives a Qmax of 442 vph. Over a wide range of onditions, the values of Qmax estimated from the analytial model and those determined from diret simulation are mostly within 5 vph of eah other. This harateristi is shown in Figure 6. Figure 7 further shows the ability of the model to estimate Qmax and the related f LT when a lane is hanged from a straight-through-only lane (Ps =.) to a shared left-turn lane (. < P, < 1.) and, finally, to an exlusive left-turn lane (Ps = 1.).. > a;.s "t:i e (/)... :;. (.) 4 2 -. 4 Opposing Flow, vph FIGURE 6 Capaities estimated from analytial model versus apaities determined from diret simulation.. > 1 Q) BOO "t:i :::E 6 4 E... e 2 u. (.) transformed pattern o basi pattern : Simulated Capaity, FIGURE 7 Simulated apaities and estimates obtained from analytial model for basi now patterns with 9-se yle and 42- se green interval.
7 CONCLUSIONS The left-tum adjustment fator for a shared left-tum lane is a omplex funtion of a number of variables. Refleting this omplexity, the analytial model developed in this study is muh more ompliated than the HCM and FHWA models. The added sophistiation enables the resulting model to provide better explanations of the ausal relationships between the adjustment fator and its ontributing fators. In omparison with elaborate mirosopi simulation, the analytial model developed in this study an yield equally realisti estimates. Field data may be olleted in future studies to test and modify the onstant oeffiients assoiated with the model. ACKNOWLEDGMENTS This paper is based on work supported in part by the New York State Siene and Tehnology Foundation. The author TRANSPORTATION RESEARCH RECORD 1365 wishes to thank John F. Edwards, a graduate student at Clarkson University, for his assistane in olleting the data presented in this paper. REFERENCES 1. Speial Report 29: Highway Capaity Manual. TRB, National Researh Counil, Washington, D.C., 1985. 2. H. S. Levinson. Capaity of Shared Left-Turn Lanes-A Simplified Approah. In Transportation Researh Reord 1225, TRB, National Researh Counil, Washington, D.C., 1989, pp. 45-52. 3. R. P. Roess, J. M. Ulerio, and V. N. Papayannoulis. Modeling the Left-Turn Adjustment Fator for Permitted Left Turns Made from Shared Lane Groups. In Transportation Researh Reord 1287, TRB, National Researh Counil, Washington, D.C., 199, pp. 138-15. 4. F.-8. Lin. Knowledge Base on Semi-Atuated Signal Control. Journal of Transportation Engineering, ASCE, Vol. 117, No. 4, 1991, pp. 398-417. Publiation of this paper sponsored by Committee on Highway Capaity and Quality of Servie.