Appendixes to NCHRP Report 572: Roundabouts in the United States (2007)

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NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-1 APPENDIX B LITERATURE REVIEW OF OPERATIONAL MODELS This appendix presents a detailed literature review of the operational models used in this project. A reference list for this appendix is included at the end. Fundamental Capacity Methods While the focus of the research is on roundabouts, a number of fundamental methods applicable to two-way-stop-controlled and two-way-yield-controlled intersection capacity analysis serve as a foundation for roundabout operational performance. There are currently two methods that have been used to develop such models: • Gap acceptance • Linear or exponential empirical regression Gap Acceptance Models In a gap acceptance model, the driver on the minor (entering) stream is required to select an acceptable gap on the major (circulating) stream, to perform the desired maneuver. The “gap” is defined as the headway maintained between two consecutive vehicles in the conflicting stream. The minimum gap that is acceptable to the minor-stream driver is their critical headway, tc (historically referred to in the literature as critical gap). The critical headway is not a constant and is typically represented by a distribution of values based on the variation of driver behavior. Estimation procedures exist for critical headway that do not require sites with oversaturated conditions. The follow-on time (otherwise known as follow-up time), tf, is defined as the time headway between two consecutively entering vehicles, utilizing the same gap in the circulating stream. The follow-on time can be directly measured in the field without utilizing complicated mathematical equations. According to Tanner (B1), from the point of view of the traffic on the minor road, the traffic on the major road forms alternate “blocks” and “gaps”. Bunched vehicles, each of which is separated by a minimum gap tm, form a block. During such a block no vehicles can enter the major stream flow. When the gap after the last vehicle in the block is equal to or greater than the critical headway, vehicles are able to enter the major stream flow. Vehicles can enter the larger gaps with a follow-on time of tf. Based on the gap acceptance model, the capacity of the simple two-stream situation can be evaluated by elementary probability theory for the assumptions: a) constant tc and tf values b) exponential distribution for priority stream gaps c) constant traffic volumes for each traffic stream Harders (B2) developed one of the first models, which is used in the current Highway Capacity Manual (B3). These idealized assumptions are considered somewhat unrealistic; however, various evaluations have suggested that more realistic headway distributions are not significantly more accurate. Furthermore, the resulting generalized solutions are not easy to apply in practice. In addition to the concern related to realistic distributions of headways and other gap acceptance parameters, there are a number of other theoretical limitations. These are described below:

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-2 • Inconsistent gap acceptance occurs in practice and has not been accounted for in theory. These include (a) rejecting a large gap before accepting a smaller gap, (b) driver on the roundabout giving up the right of way, (c) forced right of way when the traffic is congested, and (d) different vehicle types accepting different gaps. • Estimation of the critical headway is difficult. Maximum likelihood was found to be one of the most consistent methods (B4, B5); however, the evaluation is quite complicated. • Geometric factors are not directly taken into account. In response to concerns related to gap acceptance, Troutbeck and Kako have developed a theory for incorporating a “limited priority” process, in which the major stream vehicle slows down to allow the minor street vehicle to enter the circulating stream (B6). Linear or exponential empirical regression models are based on traffic volumes at one- minute intervals observed during periods of oversaturation. A linear or exponential regression equation is then fitted to the data, as shown in Figure B-1. Variation in the data is often created by driver behavior and geometric design. A multivariate regression equation can also be developed to include the influence of geometric design. Limitations of this technique include the following: • Empirical regression models may have poor transferability to other countries or at other times (e.g., inexperienced US drivers versus experienced UK drivers). • Regression models provide no real understanding of the underlying traffic flow theory of determining and accepting gaps upon entering the intersection. • The models are typically based on driver behavior in oversaturated conditions, thus requiring sites with continuous queuing. • Each situation (traffic volume pattern and/or geometric conditions) must be observed in order to develop an appropriate model. This requires a large data collection effort. 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Figure B-1. Exponential and Linear Regression Model. Conflicting Traffic (veh/hr) E nt ry C ap ac ity , ( ve h/ hr )

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-3 Regression Capacity Models Simply capacity models can be developed using linear and exponential regression forms. These are given in Equations B-1a and B-1b as follows: ce qBAq ⋅−=max, (B-1a) )exp(max, ce qBAq ⋅⋅= (B-1b) where: qe,max = maximum entry flow (veh/hr) qc = circulating flow (veh/hr) A, B = intercept and slope constants Expression for the constants may also be developed, e.g. as a function of other parameters including the roundabout geometry. Survey of International Capacity Models U.S. Capacity Models There are two major methods currently found in United States literature: 1) the operational method cited in the FHWA Roundabout Guide (B7), and 2) a gap acceptance procedure in the Highway Capacity Manual (B3). FHWA Method. The FHWA Roundabout Guide (B7) presents three capacity formulas for estimating the performance of roundabouts. These were intended for use as provisional formulas until further research could be conducted with US data. The FHWA method for urban compact roundabouts is based on German research (B8) and is given as follows: ce qq 74.01218max, −= , for 0 ≤ qc ≤ 1646 (B-2) where: qe,max = maximum entry flow (veh/h) qc = traffic flow on the circulatory roadway (veh/h) The FHWA method for single-lane roundabouts is based on the UK’s Kimber equations (B9) with assumed default values for each of the geometric parameters. In addition, an upper cap to the entry plus circulating flow of 1800 veh/h was imposed. The resulting equation is given as follows: ⎥⎦ ⎤⎢⎣ ⎡ − −= c c e q q q 1800 5447.01212 minmax, , for 0 ≤ qc ≤ 1800 (B-3) where: qe,max = maximum entry flow (veh/h) qc = traffic flow on the circulatory roadway (veh/h) The FHWA method for double-lane roundabouts is also based on the Kimber equations with assumed default values for each of the geometric parameters. The resulting equation is given as follows:

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-4 ce qq 7159.02424max, −= , for qc ≥ 0 (B-4) where: qe,max = maximum entry flow (veh/h) qc = traffic flow on the circulatory roadway (veh/h) HCM 2000 Method. The HCM 2000 method for one-lane roundabouts, first introduced in 1997 with background provided by Troutbeck (B10), is described below: 3600 1 3600 max, /tq e /tqeqq fc cc c e −− − = (B-5) where: qe,max = maximum entry flow (veh/h) qc = conflicting flow (veh/h) tc = critical headway (s) tf = follow-up time (s) The critical headway has an upper and lower bound of 4.1 and 4.6 seconds respectively. The follow-up time has an upper and lower bounds of 2.6 and 3.1 seconds respectively. UK Capacity Models Kimber (B9) reports the capacity estimation procedure currently used for roundabouts in the UK. The capacity, qe,max, has a linear relationship to the circulating flow rate, qc (see Equation B-1a). The regression parameters depend on geometric details of the entry roadway and roundabout. This method has been incorporated into the software packages ARCADY (B11) and RODEL (B12). The capacity formula used in the UK for roundabout entries is given as follows: )(max, cce qfFkq ⋅−⋅= for Qe > 0 else Qe = 0 (B-6) where: qe,max = maximum entry flow (veh/h) qc = circulating flow (veh/h) F = 303x2 (veh/h) fc = 0.21TD (1 + 0.2x2) k = 1 – 0.00347 (φ – 30) – 0.978(1/r – 0.05) TD = ⎟⎠ ⎞⎜⎝ ⎛ −+ + 10 60exp1 5.01 D x2 = v + (e – v)/(1 + 2S) S = (e – v)/l’ e = entry width (m) v = approach half-width (m) l’ = effective flare length (m) r = entry radius (m) φ = entry angle (°) S = measure of the degree of the flaring D = inscribed circle diameter (m)

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-5 Australian Capacity Models Detailed capacity expressions have been published in Australia. These are most recently available in Akçelik et. al. (B13), and have been incorporated into the software aaSIDRA (B14). The capacity is calculated lane-by-lane. The Australian capacity formula published by Akçelik et al. (B13) and used in this study is as follows: ),max(max, mgode qqfq = (B-7a) ))(exp( 3600 5.0 3600 13600 ccccg qqq ∆−−⎟⎠ ⎞⎜⎝ ⎛ +∆−= αλβϕβ (B-7b) )60,min( mem nqq = (B-7c) )(1 cdqdqcod ppff −= (B-7d) where: qe,max = maximum entry flow for an entry lane (veh/h) qg = minimum entry flow (veh/h) qc = conflicting flow (veh/h) qe = entry arrival flow (veh/h) fod = o-d adjustment factor pcdpqd ≈ 0.5 to 0.8 (0.6 used) nm = minimum entry flow (veh/min) nc = number of lanes in conflicting flow ∆c = minimum headway in circulating traffic (s) = 2.0 for nc = 1 = 1.2 for nc = 2 λ = arrival headway distribution factor (veh/s) = ⎪⎪⎩ ⎪⎪⎨ ⎧ ∆ ∆≤∆− else qfor q q c c cc cc cc ϕ ϕ 49 /98.03600/ 3600/1 3600/ ϕc = proportion of unbunched conflicting vehicles = exp(-5.0qc/3600) for nc = 1 = exp(-3.0qc/3600) for nc = 2 β = follow-up headway (s) For the dominant entry lane (lane at a multi-lane roundabout with the largest entry flow): cd q 4 0 10*94.3' −−== βββ , subject to maxmin ' βββ ≤≤ d (B-7e) ceii nnDD 388.0395.010*889.00208.037.3' 24 0 +−+−= −β , subject to 8020 ≤≤ iD (B-7f) where: Di = inscribed diameter (m) ne = number of entry lanes βmin = 1.2 (s) βmax = 4.0 (s)

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-6 For the subdominant entry lane (lane at a multi-lane roundabout with the smallest entry flow): dsds r)8735.05135.0(149.2 −+== βββ , subject to maxβββ ≤≤ sd (B-7g) where: rds = ratio of dominant and subdominant flow in the entry = qd/qs α = critical headway (s) = ⎩⎨ ⎧ −− ≤−−− − elsenw qfornwq cL ccLc β β )2775.0339.02371.3( 1200)2775.0339.010*137.36135.3( 4 subject to 1/0.3 ≥≥ βα and maxmin ααα ≤≤ αmin = 2.2 (s) αmax = 8.0 (s) wL = average entry width (m) For nc = 1 fqc = 0.04 + 0.00015qc for qc < 600 = 0.0007qc – 0.29 for 600 ≤ qc ≤ 1200 = 0.55 for qc > 1200 For nc = 2 fqc = 0.04 + 0.00015qc for qc < 600 = 0.0035qc – 0.29 for 600 ≤ qc ≤ 1800 = 0.55 for qc > 1800 German Capacity Models The Tanner-Wu capacity equation has been introduced officially into the German Highway Capacity Manual (B15). The German capacity formula for roundabout entries is given by Wu (B16): ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∆−−⋅−⋅ ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ ⋅∆ −⋅⋅= 23600 exp360013600max, f c c n c c f ee t tq n q t nq c (B-8) where: qe.max = maximum entry flow (pcu/h) qc = conflicting flow (pcu/h) nc = number of conflicting lanes (1 or 2 with nc ≤ ne) ne = number of lanes in the entry tc = critical headway = 4.1 s

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-7 tf = follow-up time = 2.9 s ∆ = minimum headway of circulating traffic = 2.1 s More recent re-calibrations show some bias of this equation for two-lane entries. Thus, as a better approximation to German observation data a new set of parameters is presented here for comparison purposes: ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∆−−⋅−⋅ ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ ⋅∆ −⋅= + 2 exp36001 )1/( max, f cc n c c f nn e e t tq n q t nq c FF (B-9) where: tc = critical headway = 3.3 s tf = follow-up time = 3.1 s ∆ = minimum headway of circulating traffic = 1.8 s nF = short lane length = 1.4 veh French Capacity Models Three parallel modeling efforts have been reported in France (B17). The model considered most current, employed within the software implementation Girabase, is an exponential regression that takes into account a number of geometric parameters and the influence of exiting flow. The form of the Girabase model published in 1997 and used for this study is as follows: )exp(max, gBe qCAq ⋅−⋅= (B-10) where: qg = tecetici ac a aa kqkqqq q kq ⋅+⋅+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +−⋅⋅ 1 A = 8.0 5.3 3600 ⎟⎠ ⎞⎜⎝ ⎛ e f L t qe.max = maximum entry flow (pcu/h) qc = total conflicting flow (pcu/h) qci = conflicting flow on inner lane (default 0.4*qk) (pcu/h) qce = conflicting flow on outer lane (default 0.6*qk) (pcu/h) qa = exiting flow (pcu/h) CB = 3.525 for urban area = 3.625 for rural area tf = follow-up time = 2.05 s R = radius of the central island (m) Le = entry width (m) La = circulating width (m)

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-8 Li = width of the splitter island (m) Li,max = 2 55.4 aLR +⋅ ka = ⎪⎪⎩ ⎪⎪⎨ ⎧ <−+ else LLforL L LAR R iii i max,max, 0 Kt,i = ( ) ⎪⎪⎩ ⎪⎪⎨ ⎧ +⋅ 1 160 LARLA Min Kt,e = ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ ⎟⎠ ⎞⎜⎝ ⎛ +⋅ −− 1 )8(1 2 LAR R LA LA Min Swiss Capacity Models The Swiss model (B18) includes the influence of exiting flow and the width of the splitter island. This capacity model is described below: β⋅⎟⎠ ⎞⎜⎝ ⎛ ⋅−= be qq 9 81500max, (B-11) where: qb = ak qq ⋅+⋅ αγ (pcu/h) qe.max = maximum entry flow (pcu/h) qk = circulating flow (pcu/h) qa = exiting flow (pcu/h) γ = 0.9 to 1.0 for single circulating lane (default = 1.00) = 0.6 to 0.8 for double circulating lane (default = 0.66) = 0.5 to 0.6 for triple circulating lane (default = 0.55) β = 0.9 to 1.1 for single entry lane (default = 1.00) = 1.4 to 1.6 for double entry lane (default = 1.50) = 1.9 to 2.1 for triple entry lane (default = 2.00) b = taken from Figure B-2 (m)

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-9 α = ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ > ≤<−− ≤< ≤<−− ≤< 280 2827)27(*1.01.0 27211.0 219)9(*12/5.06.0 906.0 bfor bforb bfor bforb bfor C C’ b Figure B-2. Swiss Measure of the Parameter ‘b’ Operational Performance Measures In general, the performance of traffic operations at an intersection can be represented by the following measures of effectiveness: • Degree of saturation (volume/capacity) • Average delay • Average queue length • Distribution of delays • Distribution of queue lengths (i.e. number of vehicles queuing on the minor road) • Number of stopped vehicles • Acceleration or deceleration between stop and normal velocity Delay Authors such as Kremser (B19), Brilon (B20), and Yeo (B21) have developed average delay equations based on queuing theory. These models are only applicable to undersaturated conditions where the traffic is considered constant over time. Time-dependant delay solutions (those that consider oversaturated conditions) were developed by Kimber and Hollis (B22). These were later simplified by Akçelik and Troutbeck (B23) and are presented in the HCM. The simplified equations do not take into flow rates before or after the analysis period. The Kimber and Hollis method is preferred, though more complicated. The HCM control delay equation for a given stop-controlled movement is presented below (Equation 17-38, HCM 2000, B3):

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-10 5 450 3600 119003600 2 + ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎣ ⎡ ⋅ ⎟⎠ ⎞⎜⎝ ⎛⋅⎟⎠ ⎞⎜⎝ ⎛ +⎟⎠ ⎞⎜⎝ ⎛ −+−⋅⋅+= T c v c c v c vT c d (B-12) where: d = control delay (s/veh) T = analysis time period (T = 0.25 for a 15-min period) (h) c = capacity (veh/h) v = flow rate (veh/h) The first term in the equation, “3600/c”, represents the average service time, or the time spent in the first position in the queue. The last term, “+ 5”, represents additional time added to reflect deceleration to and acceleration from a stopped position. In FHWA’s Roundabouts: An Informational Guide (B7), this last term has been dropped from the equation to more accurately reflect a yield condition. Queue Length The average queue length is of limited practical value, however, the maximum queue length is useful for design. Maximum queue length (95th-percentile queue length) relationships have been developed by Wu (B24) in the form of graphs, and are presented in the HCM 2000. These are illustrated in Figure B-3. The graph is only valid where the volume-to-capacity ratio immediately before and after the study period is no greater than 0.85, such that the queue length is negligible. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 ∞ E xp ec te d M ax im um N um be r o f V eh ic le s in Q ue ue , Q 95 [v eh ] v/c Ratio [-] Figure B-3. 95th Percentile Queue Length

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-11 Effects of Pedestrians on Entry Capacity Consideration of the effect of pedestrian modes on entry capacity has been of limited concern in previous research efforts. Stuwe (B25) observed three roundabouts with heavy pedestrian flow and developed an empirical entry capacity equation for one-lane and two-lane roundabouts. The method is included in the FHWA Roundabout Guide (B7). Marlow and Maycock (B26) also developed entry capacity models with pedestrian considerations, based on queuing theory. The capacity of roundabout exits with significant pedestrian flows has not been investigated to date. In most states in the United States, pedestrians have the right-of-way at intersections whenever they are in or about to enter a crosswalk. This legal priority for pedestrians is not universally observed in the United States, nor is it the rule of the road throughout the world. In German law, for example, pedestrians have priority over vehicles leaving the roundabout, because this situation is handled like leaving a main road into a side street. These regulations take place even if there is no Zebra crossing. However, the pedestrian does not have priority over the vehicles that are entering the roundabout. Pedestrians only have priority at entries if there is a Zebra crossing. If pedestrians have priority, they can cause a capacity reduction for vehicular traffic at the entries and exits of a roundabout. The extent to which the pedestrians may affect the capacity depends on the volume of pedestrians. To take this into account, there are three methods, which will be briefly explained later. All these methods are only valid if pedestrians have right of way. However, in reality the traffic doesn’t exactly follow the rules. The awareness of these rules and their specific application to roundabouts doesn’t seem to be common. For example, the following behavior can be observed in Germany: • Zebra crossings: If there are Zebra crossings, they are normally situated at both entry and exit of a roundabout. Generally, under these conditions pedestrians assume the right-of-way. There are sites where pedestrians totally bring down the traffic flow. • Roundabouts without Zebra crossings: Pedestrians and vehicles arrange themselves. Pedestrians cross the road with care, and vehicles can use this to their advantage to gain priority. On the other hand vehicles, even with the legal right-of- way, sometimes give the pedestrians right-of-way. This behavior cannot be described by theoretical mathematical methods. The traffic engineer should therefore be careful with his calculations. As a result, in Germany it is advisable to use this calculation method of the influence of pedestrians, although there is no Zebra-striped crosswalk. The following calculation methods were all developed for the case that pedestrians on crosswalks have unrestricted right of way. The following two calculations lead to completely different results. The method by Marlow and Maycock (B26) was developed from purely theoretical methodology of the queuing theory; the method by Stuwe (B25), recommended for use in Germany, was developed from interpretation of observed traffic. It considers, contrary to the Marlow-Maycock method, that even when a queue reaches across the Zebra-striped crosswalk it can still be used by pedestrians without having negative effect on the traffic. Consideration of Pedestrians at the Entry of Roundabouts Using video cameras, Stuwe (B25) has observed a total of twelve entries at three German roundabouts with heavy pedestrian traffic (one in Münster and two in Lübeck). At all entries there were Zebra-striped crosswalks installed. The study analyzed the number of pedestrians, the number of vehicles entering the roundabout, and the number of vehicles circulating inside the

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-12 roundabout during continuous queuing. All values were transformed into passenger car units using German definitions. Here an influence by the pedestrians on the capacity of the entry could be observed. This influence was described more clearly by Brilon, Stuwe, and Drews (B27) with a formula that was based on the same set of data. The analysis of these relations by statistical methods led to the following equations. Using this method, the capacity qe,max of the entry is calculated without consideration of pedestrians. Then, capacity of the entry is reduced by a factor M to account for the influence of pedestrians on the capacity of the entry. Thus, the capacity considering pedestrians is: Mqq eFge *max,max,, = (B-13) For M the following regression equations are valid: One-lane entry: k FgkFgk q qqqq M ⋅− ⋅⋅+⋅−⋅−= 65.01069 00073.0644.0715.05.1119 (B-14) Two-lane entry: k Fgk q qq M ⋅− ⋅−⋅−= 50.01380 381.0329.06.1260 (B-15) where: M = entry capacity reduction factor qk = volume of circulating vehicles in front of the subject entry (pcu/h) qFg = volume of pedestrians (ped/h) This method is therefore a valid application of the empirical regression method on the problem of pedestrians at Zebra-striped crosswalks in the case of roundabouts. The reduction factor M is dependent on the volume of traffic on the roundabout and the volume of pedestrians, shown in Figure B-4 (two-lane entry) and Figure B-5 (one-lane entry), where qk is the traffic volume in the roundabout (in pcu/h). The volume of pedestrians qFg is shown in some curves. Interim values can be interpolated. It can be seen that the more traffic on the roundabout, the less are pedestrians influencing the traffic flow. Pedestrians do not have any more influence on the capacity of the entry, if the circulating traffic is 900 pcu/h or higher at one-lane entries (or 1600 pcu/h at two-lane entries). The result seems to be logical, because when there is a queue the pedestrians can use the crosswalk without interfering with traffic. This calculation is recommended for German circumstances and for others as well. The Marlow-Maycock Method (two mostly independent operating queuing-systems) does, however, not consider the before- stated case of pedestrians when there is a queue. The number of observations supporting the work by Stuwe and by Brilon, Stuwe, and Drews is limited. Besides, it is possible that the situation measured did not exactly reflect all possible parameters (e.g., traffic volume, lane width, and speed). Further comparable measurements are unlikely to be done in the near future due to the costs. Therefore, this method will remain the most meaningful calculation method for the near future. The formulas in Equations B-13 to B-15 lead, in boundary areas (in which they are not supported by data), to implausible results. For example, it is possible, at one-lane roundabouts and low pedestrians volume ( < 100 ped/h), that with a marginal rising volume of pedestrians qFg the capacity would rise too. This does not question the formula as it is, but it demands careful application.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-13 0.5 0.6 0.7 0.8 0.9 1 Fa ct or M 0 200 400 600 800 1000 Circling flow in roundabout (pcu/h) 800 ped/h 600 ped/h 300 ped/h 100 ped/h 200 ped/h 400 ped/h Figure B-4: Reduction Factor M for the Consideration of Pedestrians on the Entry at Two-Lane Roundabout Entries. Fa ct or M Circling flow in roundabout (pcu/h) 0,5 0,6 0,7 0,8 0,9 1 0 200 400 600 800 1000 800 pd/h 600 pd/h 300 pd/h 100 pd/h 200 pd/h 400 pd/h Figure B-5. Reduction Factor M for the Consideration of Pedestrians on the Entry at One-Lane Roundabout Entries. Marlow and Maycock (B26) investigated the effects of pedestrians, who cross the entry to a roundabout, by using mathematical methods from the queuing theory. They treat the crosswalk and the roundabout entry as two queuing systems in secession. First of all, the capacity of the two queuing systems (crosswalk and entry) is calculated. Following that the calculation of the total capacity is calculated. Under the condition that pedestrians have priority, the capacity of the crosswalk is calculated based on the formula by Griffiths (B28):

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-14 ( ) 3600)1(1 ⋅−⋅−+⋅= ⋅−⋅ βµαµβµ µ eecFGÜ (B-16) where: cFGÜ = capacity of the crosswalk for vehicles (veh/h) µ = volume of pedestrians = qFg / 3600 (ped/s) β = minimum time gap between two vehicles (s) when driving across the crosswalk. = 0 1 c c0 = capacity of one lane of the entry, at an otherwise empty roundabout (veh/s) α = time needed to cross the crosswalk by the pedestrians (s) = B / vFG B = width of road at crosswalk (m) vFG = walking speed of pedestrians at the crosswalk (m/s) The speed vFG is about 0.5 to 2.0 m/s. If no further information is available, a value of 1.4 m/s should be used. The parameter B is set for every entry individually. This value has to be set according to the given situation in every single entry, when using the Marlow-Maycock formula. The parameter β is set to the capacity (c0) for one lane of the entry at an otherwise empty roundabout. For the total capacity of the crosswalk-entry system, the relation of R is very significant. max,e FGÜ q c R = (B-17) where: cFGÜ = capacity of the crosswalk for vehicles (veh/h) qe,max = capacity of the entry neglecting pedestrian-traffic (veh/h) The total-capacity qe,max, Fg of the crosswalk-entry system is the according to Marlow and Maycock (B26): Mqq eFge *max,max,, = (B-18) where: M = 12 2 − − + + N N R RR qe,max = Capacity of the entry neglecting pedestrian traffic (veh/h) N = Number of vehicles that can queue between the area between crosswalk and entry The parameter N is to be set for every single entry. It is the number of queue spaces for cars that fit into the area between the crosswalk and the boundary of the roundabout (queue spaces). The value can be taken from a map. For the car length the value of 5 to 6 m can be taken. The queue spaces for cars have to be added up for all lanes of the entry (e.g., 2 lanes and a

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-15 5 m gap between crosswalk and the edge of the roundabout: N = 2). This value, when using the Marlow-Maycock formula, has to be set according to the given situation. Consideration of Pedestrians at the Exit of Roundabouts Exits of roundabouts have a limited capacity, too. However, this has not been investigated to date (at least there are no relevant publications known). Observations in Germany show that a capacity of, for example, 1800 veh/h cannot be reached. The absolute limit seems to be somewhere near cA = 1200 to 1400 pcu /h (per lane). Note that many European countries choose not to build two-lane exits for safety reasons. Heavy pedestrian traffic that crosses the exit can also reduce the capacity. The effects of this can be calculated by the Marlow-Maycock method (B26) with usage of Equation B-18. The length of the crosswalk is considered in Equation B-16; the longer the crosswalk, the less the capacity. The variable cA (see above) is replaced by qe, max in Equation B-17. However, recent research has shown that the influence of pedestrians on the capacity of the exit is overestimated in these equations. Because of the general lack of scientifically valid results, a different calculation method, which seems more realistic, can be used as follows. The capacity of the exit is calculated using the formula by Griffiths (Equation B-16); this assumes that pedestrians have always priority when using the crosswalk at the exit. The queue length can be calculated by the method described in the previous paragraph (Little-M/M/1). This method is the best for the given task. However, caution is still advised in the use of these methods, as further research is needed to validate the methods. Use of a Gap Acceptance Approach Versus a Regression Approach to Estimating Capacity This section, prepared by Rod Troutbeck, documents the process used in Australia to adopt a gap-acceptance approach in the development of their capacity models. This discussion is relevant to the present study, as the US faces a similar dilemma in the development of its capacity models. Background The analysis of roundabouts in Australia has been developed from the gap acceptance approach in the late 1970’s (B29, B30). In the early 1980’s, a national body for standards and practices, then known as the National Association of Australian State Road Authorities (NAASRA), commissioned the Australian Road Research Board to develop relationships between the geometry of the roundabout and its performance. There was a natural tendency for Australia to continue to use the gap acceptance approach as this technique was currently in use. The author reviewed the practices worldwide before making a final decision. This report describes the thinking and the process around the adoption of the gap- acceptance approach and some of the history of the empirical approach from the UK. It also describes the history of the Australian method in some detail. Discussion with TRL Staff on the Use of the Linear Regression Equations In August 1993, Troutbeck went to England, Scotland, and Europe to discuss the analysis of roundabouts with different research organizations and staff. The comments and conversations have been recorded by Troutbeck (B31). Extracts from this report have been reproduced in this section and shown in italics. Other relevant and recent comments are also given here. In 1983, Troutbeck reported the following major conclusions.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-16 ‘The British have spent a considerable amount of effort studying the capacity of roundabouts. The TRRL [Transport Research Road Laboratory] has produced equations for the prediction of capacity on roundabouts and these have been included in the current Department of Transport practices. … Other researchers, particularly at universities, have also attempted to develop predictive equations for roundabout capacity, but have based these on gap-acceptance techniques. However, since the publication of the TRRL formula, their efforts have been reduced. Ashworth and Laurence (B32) have shown that the capacity of the smaller roundabouts [is] better modeled by gap acceptance theory. There seems to be no reason for ARRB not to continue to develop capacity formula based on the gap-acceptance approach.’ ‘After reviewing the research on the capacity of roundabouts and talking with the researchers, I consider that the dominant variables will be the number of entry lanes, the number of circulatory lanes, and aspects of the circulation flow. It is, however, the last aspect [that] I believe has not been well handled in the UK. TRRL studies at a roundabout with light and heavy circulation flows, as observed in track experiments, have been considered together, when in fact the flow conditions are quite different. At high circulation flows, the speed of the circulating vehicles is low and the gap-acceptance characteristics of the entering drivers are likely to be quite different to their characteristics when the flow is low. I now consider the speed of the circulating vehicles to be an important parameter in determining the research of a roundabout.’ These conclusions were developed from a detailed discussion with a number of researchers. The major issues are discussed in the following sub-sections. Gap Acceptance or Linear Regression Approach? Troutbeck (B31) reported that on the 18th August, the author met with Mike Grimmer, the head of the Traffic Systems Division; Mike Taylor, who was head of the traffic capacity and delay group; and Marie Semmens, who was involved in the early roundabout work and developed ARCADY2 (B33). Troutbeck (B31) reports: ‘When I questioned Marie on the use of a linear rather than a gap-acceptance model, she pointed to the fact that the circulating drivers tend to adjust their speed to allow the entering drivers to merge. Marie considered this adjustment to be contrary to gap-acceptance models. However, it can be handled by gap- acceptance techniques if the critical gap is made circulation flow dependent or, more promisingly, speed dependent. Marie considered that since the gap- acceptance models tend to give a second order relationship between entry capacity and circulation flow, the significance of this second order term will indicate whether the gap-acceptance approach is reasonable. TRRL examined the significance of this second order term for all their data sets, and in general, found that it was not significant. … However, recent work by myself indicates that the gap-acceptance formulae can give almost linear relationship within the workable limits of the data and I was not surprised that the second order term was not significant.’

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-17 It has since been found that the limited priority merging system allows for the gap acceptance approach to give results that are even closer to a linear line. John Wardrop, University College London who developed the weaving equation for roundabouts was Golias’ supervisor. He also gave an insight into the TRL approach. He believed that it may be unprofitable to use a complicated model when a simple linear equation might be quite satisfactory. I agreed that no practicing engineer would be interested in using complicated models or equations if a more simplified one gave ‘reasonable’ estimates. However, I believe that the process of finding suitable simplications requires knowledge of the effect of many parameters and the behaviour of more complicated (and more realistic) models. John Tanner, from TRRL, developed the formulae for delays (and hence capacity) of uncontrolled intersections also offered advice on the TRL approach. His equations have been included in the 1979 NAASRA (B34) guide to urban intersection design and the updated 1982 roundabout guide (B35). ‘John was also defensive of the TRRL approach. He reiterated many times that this formulation was largely an academic exercise and that the more empirical model of Kimber was more appropriate for estimating capacity.’ Vitz and de Wijngaert (B36) compared the gap-acceptance technique with the TRRL method of estimating capacity. They concluded that PICADY significantly overestimated capacity. At the University of Sheffield, Troutbeck met Robert Ashworth and Chris Laurence who have been researching the capacity of roundabouts since the early 70's. Much of their work has been summarized in Troutbeck (B37). Troutbeck (B31) reports: ‘I asked Robert why he thought TRRL adopted the linear regression approach. His answer was that he thought that exponentials were too clumsy to incorporate into a design formula. Pocket calculators were not in general use at that time and the simpler linear approach was favoured.’ ‘A third report by Ashworth and Laurence (B32) documented the final stages of the study sponsored by TRRL. In the third stage data were collected at 21 other large (conventional); roundabouts to assess the accuracy and predictive ability of equations developed in phase II. They concluded that the exponential equation given above was the best predictor. The original aim of the third phase was to identify the principal factors [that] cause a small island roundabout to have increased capacity over the larger ones. Time precluded a detailed analysis and all that could be done was to determine the effective number of lanes. Ashworth and Laurence concluded that the doubling of the number of entry lanes increased capacity by up to 34 per cent. They felt that insufficient length for splayed lanes and drivers' reluctance to use them gave rise to the small increase in capacity. Intuitively, a similar conclusion would be expected from Australian data. They were not able to shed any more light on the effect of geometric parameters. ‘ ‘Robert Ashworth and Chris Laurence also gave me reprints of a number of published papers on the prediction of roundabout capacity (B38, B39, B40) In

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-18 these papers the predictive ability of the formulae given by TRRL, the University of Southampton and the University of Sheffield, were examined. Ashworth and Laurence concluded that 'no single formula was the best predictor for conventional, small or conventional with flared entry roundabouts' .The TRRL equations predicted the capacity of larger roundabouts better than the gap- acceptance equations However, the gap-acceptance equations performed better for the smaller roundabouts and the roundabouts with flared entries.’ These statements indicate that the linear approach adopted by TRRL was not universally accepted and the gap acceptance approach still had some credence in the UK. At the University of Glasgow, Troutbeck met Roy Hewitt who was the supervisor of Khayer whose doctoral thesis was 'Capacity and delays at roundabouts'. Troutbeck (B31) reports: Khayer, as did Robert Ashworth in Sheffield, found that the critical gap was dependent upon flows. Khayer introduced two lag times. The first is the time a driver needs to follow another minor stream vehicle through the intersection. The second is the time required for a driver to enter without stopping. The second time is similar to the critical gap and was timed the critical lag. Khayer found that the critical lag was [slightly] shorter than the critical gap. This finding causes the gap acceptance equations to have a more linear alignment. Accuracy of Models. At Southampton, Troutbeck met with Nick Hounsel and Richard Hall. Troutbeck (B31) reports that the research by Mike McDonald (B41) had found that: Data from Hounsdown roundabout Figure 11 [Figure B-6 here] indicated that observed capacities could be more than 30 per cent greater than the values predicted by Kimber (B9). Figure 11 also indicates that the gap-acceptance curves (solid line) [provide] a reasonable estimate of capacity. At the Redbridge roundabout, Mike McDonald found [that] entering capacities were about 50 per cent greater than those predicted by Kimber's equations for low circulating flows. This evidence suggests that estimates within 20 per cent should be considered to be satisfactory. Moreover, there would seem to be little point of including parameters [that] produce effects of less than 10 per cent. Effect of Exiting Vehicles at the Preceding Leg. TRRL has developed procedures that ignore the impact of exiting vehicles. This has been an issue that other researchers have considered important. When TRL was questioned about this the result was: ‘Marie said that the exiting flows did not affect the capacity implying that entering drivers can perceive the intentions of the exiters. Hence, the conflicting, circulation flow is that measured across the entrance and not mid-way between the previous exit. Kimber and Semmens (B42) found that the proportion of vehicles leaving at the previous exit, the proportion of entering vehicles turning left, and the proportion of circulating vehicles leaving at the next exit did not have a discernable effect on entry capacity'. I find it difficult to accept that the effect of the exiting vehicles is likely to be insignificant at all sites.’

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-19 SOURCE: (B41) Figure B-6. Entry Capacity from the Hounsdown Roundabout. Effect of Platooning. The effect of platooning in a priority stream does affect the capacity to some extent. Although extreme levels of platooning are not common. Troutbeck (B31) reports that TRRL considered that: ‘Kimber and Semmens (B42) experimented with the circulating traffic interrupted and released with a 20 s cycle time. Under these extreme conditions they found that the entry capacity was increased by 10 per cent for circulation flows of about 800 to 1000 veh/h (my estimate). Kimber and Semmens dismiss the level of platooning as a parameter which is likely to affect capacity.’ Effect of Circulating Vehicle Speeds. It has been identified that speed of traffic through a roundabout does affect the performance of the roundabout at the higher flows. TRRL reported that: ‘Marie Semmens had no idea of circulating vehicle speeds in track experiments. If these speeds were lower than on public roads then we have some idea of whether the drivers would be accommodating. To travel around a roundabout at a lower speed is in itself accommodating. Although speeds were used to determine geometric delays, they were not used in the evaluation of capacity or queuing delay. Marie said that TRRL have only made limited speed measurements and it was concluded (but not reported) that the effect of speed was marginal. See comments in Semmens (B43).’ Delays at Roundabouts. The TRRL method of calculating delays is based on a queuing process with random arrivals and random service times. In 1983, Troutbeck asked Marie Semmens if they have experimental data to support the assumed arrival times and service times. The response was that:

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-20 ‘She said that they did not have any data although the measured delays are not unlike the experimental delays. The random service time [function] seems to be intuitively unreasonable and I find it hard to accept. Mike McDonald of the University of Southampton has a similar opinion. It [may be], however, that random arrivals/random service time gives good estimates in spite of the seemingly irrational basis.’ Track Studies. Track experiments were an important part of the TRRL approach. These experiments provided data that were not available from the public road system. TRRL reported that: ’Since TRRL [has] relied on the track experiments to measure capacity, I questioned whether these experiments were representative. The fact that drivers are participating in an experiment generally results in a change of behaviour from the Hawthorne Effect. The drivers in the track experiment were asked to drive normally, and are paid according to the distance driven. This should cause drivers not to wait for excessively large gaps. On the other hand, the drivers were using their own cars and would not be too impatient. TRRL used a 'control' intersection to determine if drivers' behaviour had changed throughout the day and the week. Since, the control intersection capacity varied by less than 6 per cent throughout the week, Rod Kimber was satisfied that the drivers were behaving normally and consistently. He argued that if they were not driving normally they would not be as consistent. This conclusion would seem to be justified and the track experiments produced reasonable results in the usable range of flows.’ Further, TRRL reported that: ‘The track experiments taxed the resources of the group. About 35 researchers and technical staff members were required to control about 200 subject drivers.’ The Transport Research Group in Southampton found the TRRL test track experiments to be of considerable value to their study. The data from the test track experiments proved to be more consistent than the results from public roads, although not statistically significantly different. Concluding Remarks. Given the comments form researchers overseas, there are good arguments for the use of linear equations to describe the relationship between the circulating flow and entry capacity. There was not a strong statement for adopting either the linear model or the gap acceptance approach. The author has also sought to have approaches that explained the relationship so that any extrapolation could be useful. Secondly, there were insufficient sites in Australia that had continual queuing and were available for a linear regression approach. Consequently, it was reasonable that Australia adopted the gap-acceptance approach. The value of the empirical model was tested using a congested site (B44, B45). The results indicated that the gap acceptance technique tended to overestimate slightly, but this was not considered to be excessive. It should be pointed out that the linear regression approach has been considerably enhanced with the test track experiments. In fact, if only public road data was used it, is unlikely that all the geometric terms used in the TRRL model would have been statistically significant.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-21 Development of the Gap Acceptance Terms in the ARRB Procedure The ARRB procedure (B46) formed the basis of the Austroads guide (B47). The development of the ARRB procedures is described in this section. Much of this section is a direct quote from Troutbeck (B46). Site Selection. Roundabout entries which had significant, but not necessarily continuous, queuing during the peak period were selected for this study. The behavior of motorists at these entries was monitored using a video camera mounted on a mast attached to a small trailer (B48). The trailer was positioned off the road, without obscuring the view of motorists. Data were recorded at roundabouts in Melbourne, Sydney, Canberra and Brisbane during 1985–1986. Geometric Terms Considered. The UK terms (B9) to describe the entry geometry were considered and the following list was chosen for the study (B46): • Inscribed Diameter (D): the largest arc that can be drawn inside the curb line of the roundabout. • Conflict angle (φ): represents the change in direction a driver would need to make to be tangential to the central island after entry. • Entry radius (re): measured at the 'Give Way' (yield) line. A more useful term is the curvature of the entry, 1 / re. • Entry width (e): the width available to the entering drivers. • Approach width: the road width for this direction of travel upstream of the entry. • Circulatory roadway width (cw): the clear distance between the central island and the inscribed diameter measure. • Extra circulatory roadway width (ecw): extra width results from a slip lane (left turn in Australia, right turn in US). • Maximum circulatory roadway width: the sum of the circulatory lane width and the extra circulatory lane width. These parameters were based on those used in the UK study. Figure B-7 illustrates these measures. Gap Acceptance Parameters for the Different Lanes of an Approach The ARRB study found that drivers in different lanes of an approach behaved differently. For instance, when reviewing the follow-on times for the different lanes, the lane with the largest flow had the shorter follow-on time. This was captured in the ARRB procedure by labeling the lane with the largest flow as the ‘Dominant’ stream as the drivers tended to ‘take control’ at that entry. The other stream was termed the ‘Sub-Dominant stream’. Figure B-8 indicates that the follow-on time for the dominant stream is usually less than the follow-on time for the sub- dominant stream. The mean ratio between the follow-on time for sub-dominant streams to the dominant stream is 1.2. The behavior in these two types of entry lanes is analyzed separately. Dominant Stream Follow-On Time. The follow-on time for this dominant entry stream was evaluated with additional transformed geometric parameters as follows. The number of entry lanes, ne, is a broad classification of the entry width. The value of ne is 1 for entry widths less than 6 m, 2 for entry widths between 6 m and 10 m, and 3 for entry widths between 10 m and 15 m.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-22 φ D/2 cw ecw e re Figure B-7. Description of the Roundabout Entry Terms 0 1 2 3 4 5 0 1 2 3 4 5 Follow-on Follow-on time for the dominant stream (s) time for the sub-dominant stream (s) Figure B-8. Relationship between the Follow-On Times for the Dominant and Sub-Dominant Streams

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-23 The number of circulatory lanes nc is again a broad classification of circulatory road width. The value of nc is 1 for entry widths less than 10 m, 2 for entry widths between 10 m and 15 m, and 3 for entry widths between 15 m and 20 m. The follow-on times had a significant correlation with the entry width (r = –0.682), the number of entry lanes (r = –0.658), the circulating road width (r = –0.492), the number of circulating lanes (r = –0.488), the entry curvature (r = +0.345), and the circulating flow (r = – 0.505). A quadratic function of the diameter of the roundabout also had a significant effect on the dominant stream follow-on time. Troutbeck (B46) states: ‘A linear function of the entry widths implies that an increase in width from 3 to 4 m or from 5 to 6 m produces the same net result. In the first case the lane remains a single lane whereas in the second case the result enables the entry to change from a one lane to a two lane operation. A linear function of width would also encourage designers to use the relationships for quite detailed geometric design and to assume that the relationships offer a degree of accuracy above that recorded.’ The equation developed used the number of lanes and not the total road width. The equation for the dominant stream follow-up time was cecfdom nnDDQt 388.0395.00000889.00208.0000394.037.3 2 +−+−−= (B-19) where: Qc = is the circulatory flow D = the inscribed diameter ne = the number of entry lanes. nc = the number of circulatory lanes. This equation explained 64.5 percent of the sum of squares and the equation and its regression coefficients were significant at the five percent level. The standard error was 0.40 s. For all practical purposes, the coefficients for the number of entry lanes and for the number of circulating lanes are equal. Sites with an increased circulating flow have a shorter follow-on time. Similarly, roundabouts with a small inscribed diameter have a longer follow-on time than moderately sized roundabouts. Exceptionally large roundabouts may be expected to have a longer follow-on time because of the higher circulating vehicle speeds. Sub-Dominant Stream Follow-On Time. The drivers in the sub-dominant stream, generally have longer follow-on times than the drivers in the dominant stream as shown in Figure B-8. The follow-on time for the sub-dominate stream was significantly affected by the dominate stream follow-on time and by the ratio of dominant stream flow to the sub-dominate stream flow. The selected equation was sub dom sub dom domfsubf Q Q Q Qtt 8735.05135.049.21 ,, −+= (B-20) where: Qdom = the dominant entry lane flow Qsub = the sub dominant entry lane flow tfdom = the calculated dominant stream follow-on time

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-24 This equation explained 40 percent of the variance. All coefficients were significant at the 5 percent level and the standard error was 0.58 s. Equation B-20 demonstrates that the follow-on time of the sub-dominant stream increases as the follow-on time of the dominant stream increases and generally as the ratio of flows of the approach lanes increases. Critical Acceptance Gap. The critical gap for a driver is assumed to be such that all gaps greater than his critical gap are acceptable whereas the gaps less than his critical gap are unacceptable. For the population of drivers, it has been frequently assumed that the critical gaps have a log-normal distribution (B4, B46, B49, B50, B51). The 'critical acceptance gap' is the mean of drivers' critical gaps and is used to quantify the overall performance of the driving population. These terms are used in the equations for capacity and delay. For the ARRB study, the maximum likelihood technique was used to estimate the distribution of critical gaps (B37). Most other methods of evaluating the mean critical gap have considerable bias. Kimber's equations (B9) imply an almost constant ratio between the critical acceptance gap (or the mean critical gap) and the follow-on time. As a consequence the measured mean critical gap was related to the estimated follow-on time. It would be logical to think of the critical acceptance gap as being the primary term with the follow-on time being related to it. The follow- on time is a simple term to measure reasonably accurately. The critical acceptance gap is far more difficult to estimate and considered better to relate its measure to the follow-on time. The mean critical gap was found to have a significant correlation (at the five per cent level) with the expected follow-on time, the circulating flow, entry width and the number of entry lanes. The circulatory width and the number of circulatory lanes were significantly correlated with the mean critical gap at the 10 percent level. Figure B-9 is a plot of the mean critical acceptance gap divided by the follow-on time as a function of the circulating flow. Apart from four points towards the higher circulating flows, there is a general downward trend. 0 1000 2000 3000 0 1 2 3 4 Circulating flow (veh/h) critical gap divided by the expected time follow-on Mean Figure B-9. The Mean Critical Gap Divided by the Expected Follow-On Time

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-25 The chosen equation identified was ( ) fccc tneQt ⋅−−−= 2775.03390.00003137.06135.3 (B-21) where: tf = expected follow-on time e = average entry lane width (m) This equation explained 44.2 percent of the variance. The equation is significant at the 5 percent level, and the standard error is 1.09 s. Equation B-21 is essentially a ratio of tc / tf, which is related to the gradient in Kimber's equation (B9) and hence the ratio can also be considered to be the relative influence of an additional circulating vehicle. Equation B-21 implies that this influence reduces as circulating flow increases, as the number of circulating lanes increase and the average entry lane width increases. The effect of the average entry lane width can be expected if it is accepted that wider entry lanes put less demands on the driver. The ratio of tc / tf given in Equation B-21 is less than 1.0 for high circulating flows. If this ratio is less than 1, then the capacity equation would not give a monotonic decreasing function for capacity against circulating flow. This is intuitively unacceptable. The gap acceptance assumptions would also be suspect under these conditions. Accordingly the ratio should be restrained to be not less than 1.1. The field results (Figure B-9) indicate the mean critical gap to follow-on time ratio does decrease with flow, and the ratio was typically greater than 1.0 although there were a few exceptions. Using equations B-19 to B-21, both the follow-on times and the mean critical gap decrease with circulating flow. While the data in this paper cannot describe detailed driver behavior, the study does indicate that at sites with a low circulating flow, the gap acceptance parameters are larger. This could result from drivers being prepared to reject the smaller gaps in the knowledge that a larger gap will be available shortly. This more relaxed behavior is evident at some roundabouts in provincial towns. At high circulating flows the entering drivers cannot yield right of way all the time without incurring long delays. Troutbeck (B46) wrote, “At sites with high circulating flows, the entering drivers 'share' their priority with the circulating drivers. Troutbeck (B52) demonstrated that the total average delay due would be reduced with priority sharing without involving a significant delay to the 'major' stream. When the entering drivers 'share' priority, they accept shorter gaps and cause the circulating drivers to slow. It is a mutual arrangement; the entering drivers accept shorter gaps and the circulating drivers allow these drivers to enter.” This priority sharing phenomena has been identified in Troutbeck (B46), but it has only been more recently that a process has been developed for evaluating a limited priority system that includes priority sharing (B53, B54). Shorter gap acceptance parameters, recorded at sites with larger circulating flows, can be expected from slower circulating vehicle speeds. Characteristics of the Circulating Streams. In Troutbeck (B55), it was concluded that the entering drivers gave way to all circulating drivers. As a consequence, a single lane Cowan

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-26 M3 model was used to represent the headways in all circulating streams. Cowan's model assumes that (1 – α) vehicles are in bunches with a consistent intra-bunch headway of τ. The remaining vehicles have an exponential headway. The cumulative probability equation for Cowan's headway model is as follows (shown graphically in Figure B-10): [ ] otherwisetF tttF 0)( ,)(exp1)( = ≥−−⋅−= ττλα (B-22) where: t = time α = proportion of free vehicles (those not in bunches) τ = intra-bunch headway (the same for all vehicles in bunches) λ = decay rate which is related to the flow, q, by the equation q q τ αλ −= 1 (B-23) 0 15 0 1 Headway t Probability of a headway less 1−α ∆ than t Figure B-10. Cowan's Headway Model The characteristics of the circulating streams headways at the Australian roundabouts were quantified using a technique to estimate the parameters (α, τ and λ) for Cowan's M3 model (B56). This technique, described in Troutbeck (B45), minimized the deviation between the predicted and recorded cumulative distributions and provided satisfactory estimates of the cumulative probability functions for gaps greater than 4 s, say, but did not necessarily provide satisfactory estimates for the short headways. This is not considered to be important, as these headways are unacceptable and would be rejected. A Kolmogorov-Smirnov goodness-of-fit test was used to compare the fitted distribution to the data.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-27 Intra-Bunch Headway. The minimum headway between vehicles in bunches is likely to be dependent on flow. If traffic arrivals were regular then the minimum headway τ would be inversely proportional to flow. Increase the flow and τ would decrease. The proportion of free vehicles, α, is dependent on the proximity of the roundabout to signalized intersections or to other traffic devices that establish platoons. The parameter α is expected to be more significantly affected by the nearby road infrastructure than by the geometry of the roundabout. As the flow in the circulating lanes increases so the proportion of free vehicles decreases. The parameter λ is not independent as it related to q, α and τ by Equation B-23. The α and τ terms must be considered jointly. It is not important to accurately describe the distribution of small headways and these two terms can be used to locate the 'knee' in the curve shown in Figure B-10. It is then preferable for one of these terms to be chosen and the other related to the geometry and the flows. The headway between vehicles in bunches could be related to flow by a hyperbolic function (Figure B-11). However, this figure also indicates that values of about 1 and 2 predominate and it could be convenient to set τ to 1.0 or to 2.0 s. Note that the choice of these values for τ as arbitrary, and are roughly the same as values of 2.2 and 1.1 used by Avent and Taylor (B30). Single-lane roundabouts with wider circulatory lanes (8 to 10 m) and higher flows (greater than about 1000 veh/h) could also be considered to have two effective lanes and intra- bunch headways of 1 s. This occurs because drivers are prepared to adopt a staggered orientation as shown in Figure B-12, forming in effect two lanes with small intra-bunch headways. 0 1000 2000 3000 4000 0 1 2 3 4 Circulating flow (veh/h) Intra-bunch headway (s)∆ Figure B-11. The Influence of the Circulating Flow on the Calculated Intra-Bunch Headway

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-28 Figure B-12. Staggered Vehicles in a Single Wide Lane Proportion of Free Vehicles. Considering the case of roundabouts with a single circulating lane, the proportion of free vehicles decreases with flow as shown in Figure B-13. The regression equation is as follows: Q000386.0723.0 −=α (B-24) where: α = proportion of free vehicles Q = flow (veh/h). This equation explains 47 percent of the variance, and with 13 data points the coefficient is significant at the one percent level. No geometric term had a significant correlation with α (at the 1 percent level). 0 1000 2000 0.0 0.2 0.4 0.6 0.8 1.0 Circulating flow (veh/h) Proportion of free vehicles ∆ equal to 2 s eqn (7) Figure B-13. Relationship Between the Proportion of Free Vehicles and the Circulating Flow for Single Circulating Lane Roundabouts

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-29 For the multiple circulating lane roundabouts, a similar finding was obtained. The proportion of free vehicles was related to flow by Q000241.0754.0 −=α (B-25) This equation explains 50 percent of the variance and the coefficient was significant at the one percent level. Refer to Figure B-14. Again no geometric parameters were found to be significant at the one percent level. These expressions were be approximated to Q0005.08.0 −=α (B-26) for roundabouts with single circulating lanes and Q00025.08.0 −=α (B-27) for roundabouts with multiple circulating lanes. 0 1000 2000 3000 0.0 0.2 0.4 0.6 0.8 1.0 Circulating flow (veh/h) Proportion of free vehicles ∆ equal to 1 s eqn (8) Figure B-14. Relationship Between the Proportion of Free Vehicles and the Circulating Flow for Multiple Circulating Lane Roundabouts Estimating Capacity Using the ARRB Method The maximum entry capacity is estimated using the equation from Troutbeck (B57), as follows: [ ] )exp(1 )(exp max f c e t tqq λ τλα −− −−⋅= (B-28) where: α = the proportion of free vehicles in the circulating streams

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-30 q = the flow of vehicles in the circulating streams tc = the critical acceptance gap tf = the follow-on time τ = the minimum headway in the circulating streams, and these are related by Equation B-23, given above Single-Lane Roundabouts. For a roundabout with a single entry and a single circulating lane the maximum entry capacity is significantly less than the results from the 1986 NAASRA (B58) guide for low circulating flows (Figure B-15). For higher flows, the values are comparable with those from NAASRA (B58). Comparisons were made with the NAASRA approach, as this was the accepted standard of the time. Roundabouts with a larger inscribed diameter have a greater capacity, although the increase is not large. Similarly, roundabouts with a wider entry width have a greater capacity. 0 500 1000 1500 2000 0 500 1000 1500 2000 Circulating flow (veh/h) Entry Capacity (veh/h) NAASRA (1986) Diameter = 20m Diameter = 40m Figure B-15. Effect of the Inscribed Diameter of a Single Lane Roundabout Multilane Roundabouts. When analyzing roundabout legs with more than one entry lane, each entry lane is considered separately. The potential capacity of the entry is then approximately the sum of the capacity for each lane. The capacity for each lane is different, with the dominant stream having the greater capacity. Doubling the number of entry lanes does not give double the total entry capacity. As for single-lane roundabouts, the performance of multiple lane roundabouts will be improved if the inscribed diameter is increased (Figure B-16), and if the average entry lane width is increased (Figure B-17). This is because the drivers’ gap acceptance parameters are reduced. The curves for different average entry width are co-incident at larger circulating flows because of the restraint that the critical gap be greater than the follow-on time. As discussed above, driver behavior is expected to be different at higher circulating flows where priority sharing occurs. The

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-31 0 1000 2000 3000 4000 0 1000 2000 3000 Circulating flow (veh/h) Entry capacity Total (veh/h) Diameter = 80m 40m 60m Figure B-16. The Effect of the Inscribed Diameter on the Total Entry Capacity for a Roundabout with Two Entry Lanes 0 1000 2000 3000 4000 0 1000 2000 3000 Circulating flow (veh/h) Total Average entry width 3m entry capacity (veh/h) 5m 4m Figure B-17. The Effect of the Average Entry Lane Width on the Total Entry Capacity speed of the circulating vehicles is likely to be low and it is reasonable for the entering drivers to be less affected by the entry width. Most drivers will be stopped before entering the roundabout and the difficulty of driving in a narrow lane will be decreased. The results in Figure B-16 are thus considered to be reasonable.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-32 The discontinuity at a circulating flow of about 2000 veh/h is developed from the limiting value of the ratio of the follow-on time to the critical acceptance gap to be greater than 1.1. This discontinuity will be further discussed. Concluding Remarks. The ARRB research has been based on empirical data and was able to relate driver behavior attributes to the gap-acceptance terms and the geometry of the roundabout. This has provided the basis for techniques that can reflect the behavior of motorists. SIDRA Development In the early editions of SIDRA, there was a faithful representation of the ARRB research. SIDRA 4.1 introduced the effect of origin and destination and SIDRA 5 introduced the alternative approaches to the analysis and the prediction of headway characteristics. These changes have influenced the outcome and they will be discussed in this section. In order to identify the effect of changes from the ARRB procedure, each change will be identified using results for both single-lane roundabouts with an inscribed diameter of 20 m to 40 m and double lane roundabouts with an inscribed diameter of 40 m to 60 m. Gap Acceptance Parameters Adjusted for Low Circulating Flows and High Entry Flows. In the ARRB study, it was identified that the critical gap parameters were affected at sites were there has a high degree of saturation and when the circulating flows are low. The effect was small, and it was not considered necessary to include this effect, as it would have made the approach iterative and more complex than necessary. Nevertheless the phenomenon still exists. SIDRA 5 has included this effect, but without detailed research on the outcomes. On the other hand, the effect in SIDRA 5 is in the right direction. The SIDRA approach is to reduce the follow-on time if the circulating flow is less than a specified limit (currently 900 pcu/h). The reduction is proportional to the ratio of the entry flow to the circulating flow. This ratio has a limit of 3. The reduction is also proportional to the ratio of the circulating flow to the specified limiting circulating flow. The effect of this change only on the ARRB (B46) formulation is shown in Figure B-18. The SIDRA curves relate to conditions where the entry flow is three times the circulating lane flow. The average queue length is a function of the degree of saturation and the effect is only likely for degrees of saturation above 0.8. The SIDRA effect is significant below circulating flows of 900 veh/h. This effect is more than is expected from the ARRB data collection. There is no evidence to support the SIDRA level of this effect. Limits on the Gap Acceptance Parameters. SIDRA limits the critical gap parameters to avoid extreme values. The limits used are as shown in Table B-1. The effect of these is shown in Figure B-19. These limits did not affect the single-lane roundabouts and had only a marginal effect on the multilane roundabouts. Again these limits were arbitrary and are designed to decrease the capacity at higher circulating flows. Revised Circulating Flow Characteristics. The most significant effect has been the changes to the circulating flow characteristics for two-lane roundabouts. These changes were again arbitrary and were not done with the appreciation of the interconnectedness of the terms used to define the headway distributions. The effect of the change in means to determine the proportion of bunched vehicles is marginal for single-lane roundabouts, as the minimum headway was not changed. This is shown in Figure B-20.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-33 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 Circulating flow (veh.h En tr y C ap ac ity (v eh .h ) 20m ARRB 20m SIDRA 40 m ARRB 40 m SIDRA Figure B-18. Effect of the High Entry Flows at Low Circulating Flows Using the Sidra Routines for Single Lane Roundabouts. TABLE B-1. Limits Employed within SIDRA on Gap Acceptance Parameters Minimum Maximum tc 1.2 4.0 tf 4.0 8.0 tc/tf 1.1 3.0 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 4000 Circulating flow (veh/h) En tr y ca pa ci ty (v eh /h ) 40m ARRB 60m ARRB 40m SIDRA 40m SIDRA Figure B-19. Effect of the Change to the Critical Gap Parameter Limits at Two-Lane Roundabouts.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-34 0 200 400 600 800 1000 1200 1400 1600 0 500 1000 1500 2000 Circulating flow (veh.h En tr y C ap ac ity (v eh .h ) 20m ARRB 40 m ARRB 20 SIDRA 40m SIDRA Figure B-20. Effect of the Change to the Circulating Flow Characteristics. For the multilane case there are two effects. The first is change in the proportion of bunched vehicles, and the second is the change in the minimum headway. This latter change had the effect of significantly reducing the capacities at the higher flows, as shown in Figure B-21. The minimum headway variable should not be changed without good reason, as the small change in the headway distribution significantly affects the distribution of headways. For instance, for a flow of 2200 veh/h, the distribution of headways assumed from the different analysis techniques is shown in Figure B-22. The process developed by Troutbeck (B59) and reviewed by Luttinen (B60) sought the best fit to the empirical headway. The transformation used in SIDRA has significantly altered the relationship. 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 4000 Circulating flow (veh/h) En tr y ca pa ci ty (v eh /h ) 40m ARRB 60m ARRB 40m SIDRA 60m SIDRA Figure B-21. Effect of the Change to the Circulating Flow Characteristics for Two Lane Roundabouts.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-35 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 Headway (s) Cumulative Probabiity ARRB SIDRA Figure B-22. Effect of Changing the Circulating Flow Characteristics for Two Lane Roundabouts with a Diameter of 60m. Use of an Origin-Destination Pattern. SIDRA has recognized that the origin- destination (O-D) pattern may have an effect. O-Ds certainly influence the characteristics of the circulating stream as it passes the entry. A number of left turners will cause traffic in the circulating lane to be in a single lane. SIDRA uses relationships for the reduction of capacity, developed from the output of a simulation model. The reduction factor, fod, is in turn a function of the proportion of the total circulating stream flow that originated from the ‘dominant’ approach. The ‘dominant’ approach contributes the highest proportion of queued traffic to the circulating flow. The reduction factor is also a function the proportion queued for that part of the circulating stream that originated from the ‘dominant’ approach. In the extreme, both the parameters for the proportion of traffic from the dominant approach and the proportion queued from that approach, could be equal to 1.0. Under these conditions, the reduction in capacity is shown in Figure B-23 and Figure B-24. The maximum effect ranges from 4 to 55 percent. Hence the effect is significant. 0 200 400 600 800 1000 1200 1400 1600 0 500 1000 1500 2000 Circulating flow (veh.h En tr y C ap ac ity (v eh .h ) 20m ARRB 40 m ARRB 40m SIDRA 20m SIDRA Figure B-23. Maximum Effect of the Origin Destination Procedure in SIDRA for One Lane Roundabouts.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-36 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 4000 Circulating flow (veh/h) En tr y ca pa ci ty (v eh /h ) 40m ARRB 60m ARRB 40 SIDRA 60m SIDRA Figure B-24. Maximum Effect of the Origin Destination Procedure in SIDRA for Multi-Lane Roundabouts. The Combined Effect of Changes. The combined effect of all aspects can be developed. However, the reader needs to check the outcomes with output from SIDRA, as other minor effects can influence the results. The analysis presented here is a combination of all effects listed above and does not account for the smaller effects in SIDRA. Figure B-25 describes the combined effect for single lane roundabouts and assuming that there are high entry flows at low circulating flows. Figure B-26 illustrates the same effect for two-lane roundabouts. These figures illustrate only the broad combined effect of the parts presented earlier. Both of these figures indicate the widely different answers for capacity based on the proportion of traffic from the dominant approach and the proportion queued from that approach. For the upper-bound cases both parameters were set to 0.0 and for lower-bound cases these parameters were set to 1.0. Both conditions are extreme. The conclusion reached is that the SIDRA changes were to decrease the capacity of the higher flows where the ARRB technique has been found to be lacking (also refer to B13). 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 Circulating flow (veh.h En tr y C ap ac ity (v eh .h ) 20m ARRB 40 m ARRB 20m SIDRA Upper 20m SIDRA Lower 40m SIDRA Upper 40m SIDRA Lower Figure B-25. Combined Effect of Changes in SIDRA for Single-Lane Roundabouts

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-37 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Circulating flow (veh/h) En tr y ca pa ci ty (v eh /h ) 40m ARRB 60m ARRB 40m SIDRA Lower 40m SIDRA Upper 60m SIDRA Lower 60m SIDRA Upper Figure B-26. Combined Effect of Changes in SIDRA for Double-Lane Roundabouts The Limited Priority Process The limited priority process has been found to improve the capacity estimates significantly. This section describes the limited priority approach. Assumptions. Troutbeck (B54) states the assumptions of the limited priority process are as follows. • The opportunity for minor-stream drivers is defined as the time the last major-stream vehicle departed until the next major-stream vehicle is expected to arrive. • The minor-stream drivers will accept an opportunity that is greater than the critical gap, tc. • A number of minor-stream drivers will accept an opportunity if it is long. The headway between these minor-stream vehicles is the follow-up time, tf. • The minor-stream drivers are assumed to be both consistent and homogeneous. • After the merge, each minor-stream vehicle will have a headway of tf to the vehicle in front. Similarly, each major-stream vehicle will have a headway of τ to a minor- stream vehicle in front. • If an opportunity is not acceptable, less than ta, then the headway between the major- stream vehicles will be at least ψ. • The major-stream drivers will have a headway distribution, upstream of the merge, given by Cowan’s M3 headway distribution (B56). • Major-stream headways will maintain a minimum headway specified by Cowan’s M3 distribution. • The critical gap, tc, is less than the tf + τ. Using these assumptions a generalized equation for the capacity is developed. Description of the Process. A minor-stream driver contemplating entering the intersection will review the opportunity to enter. This will require drivers looking upstream at the expected arrival time of the next major-stream vehicle and comparing this time with the departure time of the last major-stream vehicle. Figure B-27 illustrates the time of passage of major-stream vehicles upstream. This time and expected speeds enable the minor-stream vehicles to predict the arrival times of the major-stream vehicles. This is shown as the oblique arrows on

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-38 Passage time Expected arrival time Actual departure time Upstream headways Merging opportunity Major stream delay Figure B-27. Definition of the “Merging Opportunity” the lower part of the figure. However, in fact the driver is delayed slightly by either major stream vehicles ahead or by minor-stream drivers entering. The opportunity for merging is the difference between expected time of arrival of the major-stream vehicle and the departure of the previous major-stream vehicle. The delay to the next major-stream is dependent on the duration of the merging opportunity, whether the opportunity allows a minor-stream vehicle to merge and the relative headway between the major-stream vehicles. Troutbeck (B53, B54) has provided the following equations for the delay to the major-stream vehicles and the capacity of the merge. This approach gives the following equation for capacity as: Capacity = [ ] )exp(1 )(exp f c t tqC λ ψλα −− −−⋅ (B-29) where: C = [ ] [ ] [ ])(exp)(exp1)exp()(exp 1)exp( ψτβλτψλλψτβλλβ λ +−−−+−++−⋅ − f f t t β = cf tt −+τ ψ = the minimum headway between the major-stream vehicles upstream. α = the proportion of non-bunched vehicles in the major-stream upstream tc = the critical gap tf = the follow-up time or headway and the headway in front of the minor- stream vehicles after the merge. τ = the headway in-front of the major-stream vehicles after the merge. q = the circulating flow λ = given by Equation B-23

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-39 The average delay to major-stream vehicles when the merge is at capacity is: [ ] [ ] [ ] [ ])(exp)(exp1)exp()(exp )(exp5.0 2 1,1 ψτβλτψλλψτβλλβ ψτβλλβ +−−−+−++−⋅ +−⋅= ft d (B-30) When Does Limited Priority Occur? Limited priority merging occurs when the relative speeds between vehicles are slow and drivers are confident that the consequences of drivers not slowing are insignificant. Troutbeck and Kako (B6) have reported the limited priority effect at roundabouts in Australia. They collected headway data for circulating vehicles just before and just after the merge area and found that the opportunities in which a minor-stream vehicle had entered had generally increased in size. This found that the circulating drivers were prepared to slow and to accommodate a minor-stream vehicle. Figure B-28 from Kako and Troutbeck illustrates this point. The statistics of the distribution of the differences between the passage times of major stream vehicles are shown in Table B-2 and demonstrate that the means are significantly different from zero. Troutbeck (B61) investigated the critical gaps used by heavy vehicles and those used by cars. The result was that the drivers of cars and trucks had longer average critical gaps when accepting a gap terminated by a truck rather than a car (Table B-3). The drivers were less inclined to assume that the truck drivers in the major stream would slow. This behavior is consistent with the limited priority behavior. Troutbeck (B61) recorded the behavior at a roundabout. Roundabout studies and their discussion (for instance B27, B46) have identified a unique issue with heavily trafficked roundabouts. Drivers’ critical acceptance gaps are only marginally greater than their follow-up times. This again is a demonstration of limited priority behavior. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 -1 0 1 2 Difference between headways upstream and downstream of the merge point (s) Cumulative proportion of headway differences All gaps Merged gaps High degrees of saturatuion SOURCE: (B63) Figure B-28. Cumulative Difference Between the Times of Passage of Major Stream Vehicles at a Roundabout

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-40 TABLE B-2. Statistics of the Differences Between Major Stream Vehicles Before and After the Merge (Downstream - Upstream) Data-set Sample size Mean [s] Standard deviation [s] t-value Low Site A 67 - 0.02 0.56 - 0.29 saturation Site B 63 - 0.08 0.39 - 1.63 period Site C 63 - 0.01 0.62 - 0.13 High Site A 76 0.24 0.59 3.55 saturation Site B 66 0.17 0.48 2.88 period Site C 83 0.23 0.58 3.61 SOURCE: (B54) TABLE B-3. Critical Acceptance Gaps Entering vehicle type Cars Heavy vehicles Accepted gaps terminated by a car 5.37 5.37 Accepted gaps terminated by a heavy vehicle 6.60 7.36 SOURCE: (B61) What Are the Implications? The implication of the limited priority process is that the delays are reduced. The merge system is more efficient with more vehicles using the merge area more effectively. Looking at Equation B-29, it would appear that the capacity is reduced, as the C term is less than 1. However, this must be read with the fact that the critical acceptance gap is also reduced. If short critical gap values were used in the more traditional absolute priority system, then the headways after the merge would be similarly short and unrealistic. If the limited priority process is applied to the results from Austroads (B47) then for a 60 m diameter roundabout with 4 m entry lanes, the capacity – circulating flow relationship is very similar to other relationships worldwide. The Austroads approach was based on absolute priority and did not include the limited priority process. The SIDRA 5 (B62) curve uses absolute priority relationships but adjusts the variables in the model to provide the relationship shown here. The parameter values used in SIDRA 5 have been chosen arbitrarily and this work provides an increased understanding and demonstrates that the parameters need not be altered to achieve the same end. The effect of the limited priority process is shown in Figure B-29. The average delays to the circulating stream were modest.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-41 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 3500 Circulating flow (veh/h) Entry capacity (veh/h) UK method AUSTROADS Limited priority SIDRA 5 Figure B-29. Entry Capacity Against the Circulating Flow for a 60-m Diameter Roundabout with Two 4-m Entry Lanes and Two Circulatory Lanes The implications of the merge are complicated. In using the approach given here, it is assumed that the headway between the major-stream vehicles can be shortened to ψ if the major- stream vehicles are delayed. The headways after the merge between the major-stream vehicles may not be able to close to ψ as assumed. An estimate of changing the value of ψ to ψ’ and by keeping all other attributes the same can be found by using an assumed distribution as specified in Troutbeck and Kako (B63). Here the minimum headway has been changed and the proportion of free vehicles adjusted to compensate. The relationship between the assumed proportion of free vehicles, α’, and the recorded proportion of free vehicles α is given by the equation: [ ])'(exp' ψψλαα −⋅= (B-31) The capacity is then given by: Capacity = [ ] )exp(1 )'(exp' f c t tCq λ ψλα −− −−⋅ (B-32) where: [ ] [ ] [ ])'(exp)'(exp1)exp()'(exp 1)exp( ψτβλτψλλψτβλλβ λ +−−−+−++−⋅ −= f f t t C (B-33) and β has the same definition as before. If Equations B-31 and B-32 are used with the condition that ψ’ is equal to τ, then this will give an estimate of the maximum average delay to the major

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-42 stream vehicles. Figure B-30 illustrates the effect and indicates that this approximation gives a similar curve shape but with slightly lower capacities. Concluding Remarks. The limited priority process has been demonstrated to exist at roundabouts. The consequences of the process are larger capacities and reduced delays. Using the limited priority assumption, the equations for the capacity and the average delay to the circulating stream vehicles have been developed for a range of generalized conditions. The headways of major-stream vehicles approaching the merge are assumed to have a Cowan distribution. The minor-stream drivers are assumed to accept any gap greater than the critical gap and to enter into the larger merging opportunities with headways equal to the follow-on time. The headway between a major-stream vehicle and a merging vehicle is assumed to be τ. These conditions will cause the major-stream vehicles to be delayed up to tf + τ – tc, with the average delay to the major-stream vehicles being considerably less. Conclusions Australia has had a long history of use gap acceptance techniques. When the Austroads Guide for roundabouts was produced it is understandable that a gap acceptance approach was favored. This paper describes the processes and discussion used to identify if there were any issues in using a gap acceptance approach rather than the regression approach before the roundabout study was conducted. The outcome was that the there were no significant issues to preclude the use of the gap acceptance technique. The ARRB study was founded on empirical evidence. The data have been shown here in a graphical form. The ARRB study was not as extensive as the UK study, but there were few options to either increase the size of the study or to use more heavily congested roundabouts. 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 3500 Circulating flow (veh/h) Entry capacity (veh/h) UK method AUSTROADS Limited priority SIDRA 5 SOURCE: (B63) Figure B-30. Entry Capacity against the Circulating Flow for a 60-m Diameter Roundabout with Two 4-m Entry Lanes and Two Circulatory Lanes and Using an Assumed Headway Distribution with ψ’ equal to τ

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-43 The ARRB study used only absolute priority queuing systems. For higher circulating flows the critical gap is not much larger that the follow-on time and the use of these absolute priority models is not appropriate as it causes an overestimation of capacity. SIDRA has a strong use and reputation in Australia and uses the ARRB roundabout study for the fundamentals of roundabout analysis. However, SIDRA has had a number of changes to account for the overestimation of capacity. The changes to the model were developed from a simulation model and with professional judgment by Akçelik. The consequences of the changes have been discussed here. Finally, the concept of a limited priority process has been discussed and shown to produce an improvement in the estimation of the capacity, particularly at the higher circulating flows. References B1. Tanner, J. C. A theoretical analysis of delays at an uncontrolled intersection. In Biometrica, No. 49, 1962, pp. 163–170. B2. Harders, J. Die Leistungsfaehigkeit nicht signalgeregelter staedtischer Verkehrsknoten (The capacity of unsignalized urban intersections). Schriftenreihe Strassenbau und Strassenverkehrstechnik, Vol. 76, 1968. B3. Highway Capacity Manual. TRB, National Research Council, Washington, DC, 2000. B4. Kyte, M.; Z. Tian, Z. Mir, Z. Hameedmansoor, W. Kittelson, M. Vandehey, B. Robinson, W. Brilon, L. Bondzio, N. Wu, and R. J. Troutbeck. NCHRP Web Document 5: Capacity and Level of Service at Unsignalized Intersections: Final Report Volume 1: Two-Way Stop Controlled Intersections. TRB, National Research Council, Washington, DC, 1996. B5. Brilon, W., R. Koenig, and R. J. Troutbeck. Useful Estimation Procedures for Critical Gaps. Proc., Third International Symposium on Intersections Without Traffic Signals, Portland, Oregon, TRB, National Research Council, Washington DC, July 1997, pp. 71–87. B6. Troutbeck, R. J., and S. Kako. Limited Priority Merge at Unsignalized Intersections. Proc., Third International Symposium on Intersections Without Traffic Signals (M. Kyte, ed.), Portland, Oregon, TRB, National Research Council, Washington DC, July 1997. B7. Robinson, B. W., L. Rodegerdts, W. Scarbrough, W. Kittelson, R. Troutbeck, W. Brilon, L. Bondzio, K. Courage, M. Kyte, J. Mason, A. Flannery, E. Myers, J. Bunker, and G. Jacquemart. Roundabouts: An Informational Guide. Report FHWA-RD-00-067. FHWA, U. S. Department of Transportation, June 2000. B8. Brilon, W., N. Wu, and L. Bondzio. Unsignalized Intersections in Germany—a State of the Art 1997. Proc., Third International Symposium on Intersections Without Traffic Signals (M. Kyte, ed.), Portland, Oregon, TRB, National Research Council, Washington DC, July 1997.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-44 B9. Kimber, R. M. The Traffic Capacity of Roundabouts. Laboratory Report LR 942. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1980. B10. Troutbeck, R. J. Background for HCM Section on Analysis of Performance of Roundabouts. Transportation Research Record 1646, Transportation Research Board, National Research Council, Washington, DC, 1998, pp. 54–62. B11. Binning, J. C. ARCADY 5 (International) User Guide. Application Guide AG 37. TRL Limited, Crowthorne, Berkshire, United Kingdom, 2000. B12. Rodel Software Ltd and Staffordshire County Council. RODEL. Staffordshire County Council, Cheadle, Stoke-on-Trent, United Kingdom, 2002. B13. Akçelik, R., E. Chung, and M. Besley. Roundabouts: Capacity and Performance Analysis. Research Report ARR No. 321, 2nd ed. ARRB Transport Research Ltd, Australia, 1999. B14. Akcelik and Associates Pty Ltd. aaSIDRA. Greythorn, Victoria, Australia, 1999–2004. B15. FGSV. Handbuch für die Bemessung von Straßenverkehrsanlagen (German Highway Capacity Manual). Forschungsgesellschaft für Straßen- und Verkehrswesen (Hrsg.), No. 299. FGSV Verlag GmbH, Köln, Germany (2001). B16. Wu. N. A Universal Procedure for Capacity Determination at Unsignalized (priority- controlled) Intersections. Transportation Research B, No. 35, Issue 3. Elsevier Science Ltd., New York, Tokyo, Oxford, 2001. B17. Louah, G. Panorama Critique des Modeles Francais de Capacite des Carrefours Giratoires. Proc., Roundabouts 92, Nantes, France, October 1992. SETRA, Bagneux, France, February 1993. B18. Bovy, H., K. Dietrich, and A. Harmann. Guide Suisse des Giratoires. Lausanne, Switzerland, February 1991, p. 75 (cf. summary: Straße und Verkehr (Road and traffic) No. 3, p. 137–139, March 1991). B19. Kremser, H. Wartezeiten und Warteschlangen bei Einfaedelung eines Poissonprozesses in einen anderen solchen Prozess (Delays and queues with one poisson process merging into another one). Oesterreichisches Ingenieur-Archiv, Vol. 18, 1964. B20. Brilon, W. Recent developments in calculation methods for unsignalized intersections in West Germany. Intersections without Traffic Signals (W. Brilon, ed.), Springer- Verlag, Berlin, Germany, 1988. B21. Yeo, G. F. Single-server queues with modified service mechanisms. Journal Australia Mathematics Society, Vol. 2, 1962, pp. 499–502.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-45 B22. Kimber, R. M., and E. M. Hollis. Traffic queues and delays at road junctions. Laboratory Report LR 909. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1979. B23. Akçelik, R., and R. J. Troutbeck. Implementation of the Australian roundabout analysis method in SIDRA. Highway Capacity and Level of Service: Proceedings of the International Symposium on Highway Capacity (U. Brannolte, ed.), Karlsruhe, Germany, 1991, Balkema Publisher, Rotterdam, Netherlands, 1991, pp. 17–34. B24. Wu, N. An Approximation for the Distribution of Queue Lengths at Unsignalized Intersections. Proc., 2nd International Symposium on Highway Capacity (R. Akçelik, ed.), Vol. 2, Sydney, Australia, Vol. 2, Australian Road Research Board, 1994. B25. Stuwe, B. (formerly Hartz, B.) Untersuchung der Leistungsfähigkeit und Verkehrssicherheit an deutschen Kreisverkehrsplätzen. Schriftenreihe des lehrstuhls für Verkehrswesen der Ruhr-universität Bochum, Heft 10, Ruhr-University Bochum, Bochum, Germany, 1992. B26. Marlow, M., and G. Maycock. The effect of zebra crossing on junction entry capacities. Special Report SR 724. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1982. B27. Brilon, W., B. Stuwe, and O. Drews. Sicherheit und Leistungsfaehigkeit von Kreisverkehrsplaetzen (Safety and capacity of roundabouts). Research Report. Ruhr- University Bochum, Bochum, Germany, 1993. B28. Griffiths, J. D. A mathematical model of a non signalized pedestrian crossing. Transportation Science, Vol. 15, No. 3, Sept. 1981, pp. 223–232. B29. Horman, C. H., and H. H. Turnbull. Design and analysis of roundabouts. Proc., 7th ARRB Conf., 7(4), 1974, pp. 58–82. B30. Avent, A. D., and R. A. Taylor. Roundabouts—Aspects of their design and operations. Institution of Eng. Aust. (Qld Div. tech. papers), Vol. 20(7), July 1979. B31. Troutbeck, R. J. Overseas Trip Report—August, September 1983. Internal Report AIR 393-3, Australian Road Research Board, December 1983. B32. Ashworth, R., and C. J. D. Laurence. Capacity of rotary intersections phase III and IV. Department of Civil and Structural Engineering, University of Sheffield, United Kingdom, 1977. B33. Semmens, M. C. ARCADY2: An enhanced program to model capacities, queues and delays at roundabouts. Research Report 35. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1985. B34. National Association of Australian State Road Authorities (NAASRA). Interim guide for the design of intersections at grade. NAASRA, Sydney, Australia, 1979.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-46 B35. National Association of Australian State Road Authorities (NAASRA). Roundabouts – A guide to application and design. NAASRA, Sydney, Australia, 1982. B36. Vits, A., and J. de Wijngaert. Comparison of some results obtained by using different methods to evaluate the capacity of priority junctions. PTRC meeting, 1983. B37. Troutbeck, R. J. Capacity and Delays at roundabouts—A Literature Review. Australian Road Research Board 14(4), Dec. 1984, pp. 205–216. B38. Ashworth, R., and C. J. D. Laurence. A new procedure for capacity calculations at conventional roundabouts. Proc., Institute of Civil Engineers, Part 2, Vol. 65, March 1978, pp. 1–16. B39. Ashworth, R., and C. J. D. Laurence. Methods of estimating capacity at offside priority roundabouts. Proc., Symposium on Identification and Control of Road Traffic, Technical University of Krakow, Poland, Sept. 13–15, 1979. B40. Laurence, C. J. D., and R. Ashworth. Roundabout capacity prediction—a review of recent developments. Presented at PTRC Summer Annual Meeting, Seminar J, July 11, 1979. B41. McDonald, M. Analytical methods of traffic at road junctions and on inter-urban transport networks. Unpublished PhD thesis, University of Southampton, United Kingdom, 1981. B42. Kimber, R., and M. Semmens. A track experiment on the entry capacities of offside priority roundabouts. Supplementary Report SR334. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1977. B43. Semmens, M. The capacity of some grade separated roundabout entries. Supplementary Report SR721. Transport and Road Research Laboratory, Crowthorne, Berkshire, United Kingdom, 1982. B44. Troutbeck, R. J. Does gap acceptance theory adequately predict the capacity of a roundabout? Proc., 12th ARRB Conf., 12(4), 1984, pp. 62–75. B45. Troutbeck, R. J. The use of different approaches to predict the capacity of a single lane roundabout. Internal Report AIR 393-5. Australian Road Research Board, 1984. B46. Troutbeck, R. J. Evaluating the performance of a roundabout. Special Report 45, Australian Road Research Board, 1989. B47. AustRoads. Guide to Traffic Engineering Practice Part 6—Roundabouts. AustRoads, Sydney, Australia, 1993. B48. Troutbeck, R. J., and J. S. Dods. Collecting Traffic Data Using the ARRB Video Analysis Data Acquisition System. Proc., 13th ARRB/5THREAAA Conf., July 13, 1986, pp. 183–195.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-47 B49. Miller, A. J. and R. L. Pretty. Overtaking on two-lane rural roads. Proc., 4th ARRB Conf., 4(1), 1968, pp. 583–594. B50. Troutbeck, R. J., N. Szwed, and A. J. Miller. Overtaking sight distances on a two-lane rural road. Proc., 6th ARRB Conf., Australian Road Research Board, 6(3), 1972, pp. 286–301. B51. Troutbeck, R. J. Overtaking behaviour on Australian two lane rural highways. Special Report SR 20. Australian Road Research Board, 1981. B52. Troutbeck, R. J. Current and future Australian practices for the design of unsignalised intersections. Intersections without Traffic Signals (W. Brilon, ed.), Springer-Verlag, Berlin, Germany, 1988, pp. 1–19. B53. Troutbeck, R. J. The Capacity of a Limited Priority Merge. Transportation Research Record: Journal of the Transportation Research Board, No. 1678, TRB, National Research Council, Washington, DC, 1999, pp 269–276. B54. Troutbeck, R. J. The performance of uncontrolled merges using a limited priority process. Transportation and Traffic Theory in the 21st Century: Proceedings of the Fifteenth International Symposium on Transportation and Traffic Theory (Pergamon M Taylor, ed.), Adelaide, Japan, 2002, pp. 463–482. B55. Troutbeck, R. J. Intersections, roundabouts and minis. Proc., 26th ARRB Regional Symposium, Bunbury, Australia, March 1988, pp. 45–65. B56. Cowan, R. J. Useful headway models. Transportation Research, Vol. 9, No. 6, 1975, pp. 371–375. B57. Troutbeck, R. J. Average Delay at an unsignalised intersection with two major streams each having a dichotomised headway distribution. Transportation Science, Vol. 20, No. 4, 1986, pp. 272–286. B58. National Association of Australian State Road Authorities (NAASRA). Roundabouts— A Design Guide. NAASRA, Sydney, New South Wales, Australia, 1986. B59. Troutbeck, R. J. A review of the process to estimate the Cowan M3 Headway distribution parameters. Traffic Engineering and Control, Vol. 38, No. 11, Nov. 1997, pp. 600–603. B60. Luttinen. R. T. Properties of Cowan’s M3 Headway Distribution. Transportation Research Record: Journal of the Transportation Research Board, No. 1768. TRB, National Research Council, Washington, DC, 1999, pp. 189–196. B61. Troutbeck, R. J. The capacity and design of roundabouts in Australia. Transportation Research Record 1398. TRB, National Research Council, Washington, DC, 1993.

NCHRP Web-Only Document 94: Appendixes to NCHRP Report 572: Roundabouts in the United States B-48 B62. Akcelik, R., and M. Besley. SIDRA 5: User Guide. ARRB Transport Research, Vermont South, Victoria, Australia, 1996. B63. Troutbeck, R. J., and S. Kako. Limited priority merge at unsignalised intersections. Transportation Research A, Vol. 33A (3/4), 1999, pp 291–304.