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Review ArticleReview of Roundabout Capacity Based on Gap Acceptance

Ruijun Guo Leilei Liu and Wanxiang Wang

School of Traffic and Transportation Dalian Jiaotong University Dalian 116028 China

Correspondence should be addressed to Wanxiang Wang wangwanxiang4163com

Received 27 August 2018 Revised 12 December 2018 Accepted 15 January 2019 Published 4 February 2019

Academic Editor Peter Furth

Copyright copy 2019 Ruijun Guo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Circulating vehicles have priority at modern roundabouts Entrance vehicles can enter the roundabout when there is a time gaplarger than the critical gap otherwise the vehicles need to wait until there is a large enough gapThe gap acceptance theory was usedto analyze the entrance capacity of roundabouts which can be derived by queuing theory involving two vehicle streamsThe paperintroduces the main styles of headway distribution which are named as bunched exponential distribution or M3 distribution Thecalculationmodel of free stream ratio is also introducedThe entrance capacitymodels can be classified by different entrance vehicletypes which are piecewise function or linear function or by different critical gap types which are constant or stochastic functionFor each form the typical capacity expressions are given The calculation values show a very small difference between these kindsof models The capacity value based on the critical gap of stochastic function is more realistic and more complex in function styleSome conclusions were derived that driversrsquo nonhomogeneous and inconsistent character is more realistic than the fixed criticalgap and following gapThe calculation results of capacity are similar to the field capacity under the assumption of homogeneity andcontinuance with only a minor percent deviation Finally the paper points out additional problems and the suggested research incapacity of roundabouts

1 Introduction

Modern roundabouts have some obvious characteristics suchas small diameter of the central island entrance vehiclesyield to circulating vehicles deviation of entrance vehiclesand split island between entrance and exit The bottleneckappears at the entrance of the roundabout because vehi-cles need time to judge whether to enter the roundaboutor to wait for another lager gap Therefore the researchemphasis of modern roundabouts is always the entrancecapacity

Vehicles on the entrance lanes should yield to vehicleson the circulating lanes because of the entrance yield ruleThe entrance lanes are regarded as the minor road andthe circulating lanes as the major road The roundaboutcan be regarded as the typical priority-controlled intersec-tion The entrance capacity delay and queue length canbe calculated by using the gap acceptance theory Thegap acceptance theory was well developed in Germany [12] The base theory was proposed by Major Buckley andTanner et al (refer to [3 4]) The capacity model had

been developed based on different signal timing differ-ent lane numbers and different vehicle traffic characteris-tics

Brilon [5] defined the full capacity as the sum of arrivalflow rates when the saturated degree is 1 on the lanes It istoo difficult to find the field status because the arrival flowrate of every entrance is equal to entrance capacity basedon the definition Some research [6 7] regarded the sum ofentrance capacities as the full capacity when an entrance issaturated with the increase of flow rate of all entries by acertain proportion

Some methods advised 3000 vehicles per hour as thecapacity value of a single lane roundabout Weaving theorywas used to describe the traffic characteristics of round-abouts such as Claytonrsquos method the equation of DOE(Department of Environment in UK) andWardroprsquos methodwhich was proposed by Transport and Road Research Labo-ratory (TRRL) in UK

After the 1970s Wardroprsquos method could not be used tocalculate the weaving capacity of roundabouts because theentrance yield rule was applied in the field The regression

HindawiJournal of Advanced TransportationVolume 2019 Article ID 4971479 11 pageshttpsdoiorg10115520194971479

2 Journal of Advanced Transportation

method and gap acceptance method were since developed(Ashworth 1989) [8 9]

Philbrick proposed the linear regression method (referto [10]) which is mainly used in England The exponentialregression methods (Brilon and Stuwe 1993) were based onnumerous survey data among the saturated entrance flow rateand conflicting flow rate geometry etc

The linear regression models in England were developedfrom traditional roundabouts which had the large-diameterisland design and entrance priority The capacity such as inthe models proposed by Kimber [11] McNell and Smith [12]and Hollis and Semmens et al [13] was regressed withgeometry parameters of roundabouts and the total circulatingtraffic volume [14] regardless of traffic status of every lane

In gap acceptance theory some parameters should bedetermined including headway distribution of circulatingvehicles the critical gap and following gap although theyare variable with different geometry and traffic conditionsof roundabouts Vehicles enter the roundabout by use ofacceptable gaps of circulating flow and the capacity is mainlydetermined by circulating flow rate and headway distribu-tion

Some countries such as USA Germany Australia UKJapan France and Russia have built complete capacitymethods which are suitable for their own traffic conditionsincluding Highway Capacity Manual in USA aaSIDRAAUSTROADS andNAASRA in Australia Swedish CAPCALSETRA method and CETUR method in Germany

The gap acceptance can be regarded as the signal timingprocess which was proposed by Achelik and Chung (1994)Both the gap acceptance theory and regression method canbe used to calculate the delay and queue length The gapacceptance models are identical among roundabouts two-way stop and all-way stop intersections

The capacity models and traffic control methods ofroundabouts were studied in many countries Al-Madani[15] collected the data of 13 large roundabouts includingcirculating and exiting flows the number of lanes andlateral position of vehicles in Bahrain The developed modelmatched the field data reasonably well and fell well in othermethods Khoo and Tang [16] proposed a control strategywhich was effective in reducing system travel time andincreasing volume especially duringmedium to high levels ofdemand A case study of a two-lane roundabout in Malaysiawas developed in a microscopic simulation environmentto study the roundabout system properties and to test theeffectiveness of the proposed control strategy Macioszek(2016) applied the capacity method of HCM 2010 in the fieldroundabouts in PolandThe calculated values were similar asthe experience values

Biel andWong [17] proposed the entrance capacitymodelof circulating multilanes roundabout based on the regressionmodel and HCM model Under the limited priority condi-tion Qu et al [18] calculated the entrance capacity basedon Raff rsquos critical gap model and the maximum likelihoodmethod Yap and Gibson et al [19] proposed two regressionmethods and compared variousmethods based on the surveydata of roundabouts in UK The capacity model of negativeexponential distribution showed a coincidence with the

reality than the regression model under both the high flowrate and low flow rate By using of the Germany capacityprogramsMauro andBranco [20] compared the capacity anddelay model between circulating multilane roundabout andTurbo roundabout

Weaving theory was used to describe the traffic charac-teristics including weaving capacity in weaving sections It isapplicable in the traditional roundabouts where the centralisland is large and entry flow stream has the priority But theregression method and gap acceptance theory can be used tocalculate entrance capacity in modern roundabouts

Linear regression method is mainly used in EnglandThe entrance capacity is regressed with conflicting flowrate geometry etc regardless of traffic status of every laneRegression method usually can obtain satisfied results aftera large number of data survey of roundabouts under thecondition of saturated traffic flow

Gap acceptance characters are related to traffic conditionsof roundabouts including the headway distribution of circu-lating vehicles circulating volume critical gap and followinggap Queuing theory was used to describe the gap acceptanceprocess and to deduce the capacity equation Gap acceptancetheory and regression method can be used to calculate thedelay and queue length too By choosing the reasonable trafficparameters such as critical gap value or its distribution gapacceptance theory can obtain the consistent capacity resultsas the field data Gap acceptance theory was usually appliedin the science research of traffic flow whereas the regressionmethod was more appropriate in the traffic engineering field

Based on the gap acceptance theory the basic assump-tions were induced the headway distributions were intro-duced and the relevant parameters were summed up Mostentrance capacity models were classified according to thedifferent critical gap types and the g(t) functions Theirapplicable conditions and current application fields wereconcluded

2 The Assumptions of Gap Acceptance

The headway is the time interval that elapses between thearrival of the leading vehicle and the following vehicle at thedesignated test point Gap is a little different from headway inthat gap is the measure of the time between the rear bumperof the first vehicle and the front bumper of the second vehiclerather than front-to-front time

Gap acceptance usually happens in unsignalized intersec-tions that are controlled by priority The entrance vehicleswill enter the intersection when there are no vehicles on themajor road or wait for the acceptable gap at the stop lineTheentrance vehicle will enter the intersection if the gap on themajor road is larger than or equal to the critical gap otherwisethe entrance vehicle will decelerate or stop so as to wait forthe next gap when it is larger than the critical gapThe criticalgap can be regarded as the driverrsquos judgment threshold whichdetermines whether the entrance vehicle has enough timeto enter the intersection safely or not Driversrsquo behaviorsare different from each other in reality so the critical gapsof various drivers are different The critical gap is usually

Journal of Advanced Transportation 3

regarded to follow a certain distribution described by averagevalue and variance

Passing behavior hardly happened on the circulatinglanes of a roundabout It was supposed that the headway ofcirculating vehicles followed a certain distribution such asnegative exponential distribution M3 distribution or Erlangdistribution In addition some assumptionswere given in gapacceptance theory [21ndash23]

(1) Gap of vehicles is regarded as headway for ease of datacollection

(2) All gaps of circulating vehicles can be combined intosingle-lane traffic flow

(3) Vehicles on the entrance arrive stochastically at theroundabout

(4) Vehicles on the circulating lanes do not change theirrunning behaviors when vehicles enter into the roundabout

(5) Drivers on the entrance can recognize the vehiclesgoing to exit roundabout before the conflict spot

In reality the above assumptions cannot exactly reflect allthe driving operations in a roundabout More complicatedpriority types such as limited priority and priority conver-sation reasonably accord with the reality The interactionsamong vehicles are more complicated under these condi-tions

Some influence factors are important to gap acceptanceincluding queue length on the minor road traffic volume onthe major road number of lanes on the major road and theminor road the exiting vehicles the geometry of entrancelanes and the velocity of major road vehicles

3 Traffic Character of Major Flow

Headway is important to analyze the gap acceptance processand calculate capacity and signal timing of intersectionsThe headway distribution is especially indispensable totraffic flow simulation The regular headway distributionsinclude negative exponential distribution also named asM1 shifted negative exponential distribution named as M2and bunched exponential distribution named as M3 dis-tribution M1 and M2 distribution can be regarded as thespecial types of M3 distribution We just give the probabilitydensity function and cumulative probability function ofM3 distribution which is a dichotomized headway modelErlang distribution log-normal distribution and mixed dis-tribution (refer to [24 25]) were also usually used in thefield

Critical gap 119905119888 gap 119905119891 and minimum headway 119905119898 arealso important traffic parameters They can be constantor stochastic distribution depending on driversrsquo behaviorcharacterThe empirical values were given in HCM and sometraffic engineering manuals There are a lot of methods tocalculate 11990511988831 M3 Distribution of Major Stream Cowan [26] proposedthe M3 distribution in which some vehicles run as a fleet andother vehicles run under the free flow status The probabilitydensity function f(t) and cumulative distribution functionF(t) are as follows

119891 (119905) = 120572120582119890minus120582(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (1)

119865 (119905) = 1 minus 120572119890minus120582(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (2)

where 119902 is the flow rate on the major road (vehs) 120572 is theratio of free flow 120582 is the decay constant (1s) 119905119898 is theminimum headway The estimation of 120582 can be found byusing (3) (Troutbeck 1997)

and120582= 1205721199021 minus 119902119905119898 (3)

The estimated value of 119905119898 is as followsand119905119898 = min 1199051 1199052 sdot sdot sdot 119905119899 (4)

Some extremely small values probably arise becauseindividual drivers violate the priority rule Hidden perils existin these driving operations and such data is unreasonable Sothe average headway of the vehicle fleet should be determinedas 119905119898 in the application of (4)

32 Calculate the Ratio of Free Flow The parameters of M3distribution are predicted by using least squaremethod120582 canbe calculated by the following equation [27]

120582 = 1(1119899)sum119899119894=1 119905119894 minus 120577 (5)

where 120577 is a headway value at which the vehicles areassumed to be free (s) 120577 is accepted as 3 or 4 seconds(Troutbeck 1997 Hagring 1998) (1119899)sum119899119894=1 119905119894 is the averageof headways which are greater than 120577

The ratio of free flow can be estimated to satisfy thecondition that the mean headway must be equal to thereciprocal of the flow rate

Jocabs (1980 refer to [2]) proposed the relation betweentraffic volume 119902 and the ratio as follows

120572 = 119890minus119896119902 (6)

Amore complicated equation (7) was given in the SIDRA20 and the early model (Akcelik Chung 1994)

120572 = 119890minus119887119902119905119898 (7)

where 119896 and 119887 are constantsTanner [28] proposed the linear relation between the ratio

of free flow and traffic volume (refer to [4] Ashworth 1970)

120572 = 1 minus 119902119905119898 (8)

The analogous relation model was proposed in theRoundabout Manual of AUSTROADS [29]

120572 = 075 (1 minus 119902119905119898) (9)


Table 1 Parameter estimation of free flow ratio

Lanes numbers 1 2 gt2Non-blocking traffic flow

119905119898 18 09 063600119905119898 2000 4000 6000119887 05 03 07119896119889 02 03 03

Circulating flow in roundabout

119905119898 20 10 083600119905119898 1800 3600 4500119887 25 25 25119896119889 22 22 22

Table 2 The ratio of free flow using different equation underdifferent flow rates on circulating vehicles

q(vehs) 005 01 015 02 025 03 035 04 045Eq (7) 078 061 047 037 029 022 017 014 011Eq (8) 090 080 070 060 050 040 030 020 010Eq (9) 068 060 053 045 038 030 023 015 008Eq (10) 080 065 051 041 031 023 016 010 005

The model in SIDRA 20 (refer to Akcelik 2007) can beapplied in nonblocking traffic flow which is deduced fromthe traffic volume model between the velocity and the flowrate and the headway distribution model

120572 = 1 minus 1199021199051198981 minus (1 minus 119896119889) 119902119905119898 120572 ge 0001 (10)

where 119896119889 is the bunched delay constant for convenience0001 is the minimum value of 120572

The constants in (7) and (10) are as listed in Table 1The ratio of free flow is 0 when 119902 = 1119905119898 except in the

exponential modelFor the circulating flow in roundabout 119905119898 = 2119904 119887 =25 119896119889 = 22 we can calculate the ratio of free flow under

different circulating flow rates as Table 2 and Figure 1Equations (8) and (9) are linear models There are relativelysmall difference value among (7) (9) and (10) The values of(8) are always 075 times of (9)

33 Critical Gap and following Gap Calculations of criticalgap 119905119888 and following gap 119905119891 are also important in gapacceptance theory Many methods can be used to calculatethe critical gap such as Raff rsquos method Maximum likelihoodmethod Sieglochrsquos method and Ashworthrsquos method [30ndash35]

The maximum likelihood method used log normal dis-tribution of critical gap It was used in measuring valuesfor the Highway Capacity Manual Brilon et al [30] reacheda conclusion that the maximum likelihood and Hewittsrsquosmethods provided the more consistent estimates with theleast bias The probability equilibrium method had a signif-icant bias that was dependent on the flow in the prioritystream The modified Raff technique (Troutbeck 2011) isan acceptable alternative Guo [36] proposed the theoreticalmethod which can be substituted into the capacity equationscontaining critical gap variable The equation had basicallyidentical critical gap with the maximum likelihood method

000010020030040050060070080090100

005 01 015 02 025 03 035 04 045q (vehs)

Eq (7)Eq (8)

Eq (9)Eq (10)

Figure 1The ratio of free flow using different equation (119905119898 = 2119904 119887 =25 119896119889 = 22)Table 3 Critical gap and following gap estimated in HCM 2010 (s)

Single lane Left lane in Multi lanes Right lane in Multi-lanes119905119888 48 47 44119905119891 25 22 22

Brilon [37] found the traditional log normal distribution isnot an essential part of themaximum likelihoodmethodThelogistic distribution is easier to handle

The constant critical gap in HCM 2010 is listed in Table 3based on the field data of roundabouts in California

4 Entrance Capacity of a Roundabout

41 The Theoretical Model of Entrance Capacity in Priority-Controlled Intersections Based on gap acceptance theorythe equation of theoretical capacity 119862 of the minor road inunsignalized intersections is as follows (Siegloch 1973 referto [2])

119862 = 119902intinfin0

119891 (119905) 119892 (119905) 119889119905 (11)

where 119892(119905) is the number of vehicles entering the intersectionwhen headways of vehicles on the major road are equal to119905 It can be divided into continuous function and piecewisefunction because the number of vehicles can be expressed asan integer or decimal number

Siegloch (1973 refer to [2]) proposed the continuouslinear function of 119892(119905) as follows

119892 (119905) = 119905 minus 1199050119905119891 119905 gt 11990500 119905 le 1199050

1199050 = 119905119888 minus 1199051198912(12)

where 119905119891 is the following headway 1199050 is the minimumacceptable headway the intercept in horizontal coordinate ofheadway The integration interval is [119905119888infin) in (11)


0

1

2

3

4

5

6

0 tc tc+tf

Eq (12)Eq (13)

tc+2tf tc+3tf tc+4tf tc+5tf

g(t) (veh)

Headway (s)

Figure 2 The g(t) of continuous function and piecewise function

When 119892(119905) is a piecewise function it has an equation asfollows (Harders 1976 refer to [2])

119892 (119905) = 119899119875119899 (119905) 119899 = 1 2 sdot sdot sdot infin119875119899 (119905) =

1 119905119888 + (119899 minus 1) 119905119891 le 119905 lt 119905119888 + 1198991199051198910 119905 lt 119905119888

(13)

where 119905119888 is the critical gap The integration interval is [119905119888infin)in (11)

From Figure 2 the interval of circulating headway whichcan be accepted is [1199050 +infin) in continuous 119892(119905) function Theinterval of circulating headway which can be accepted is[119905119888 +infin) in piecewise 119892(119905) function

Equation (11) shows that the capacity is related to thetraffic volume on the major road headway distribution and119892(119905) function The lower limit of integral and 119892(119905) function isdependent on the critical gap

Capacity model can be divided into two basic typesaccording to critical gap which can be a constant or stochasticdistribution function On the other side the capacity modelcan be divided into two basic types including the continuousfunction and the piecewise function of 119892(119905) when the criticalgap is constant The basic equations are all based on a single-lane roundabout

42 Entrance Capacity Model When Critical Gap Is Constant

421 When g(t) Is Piecewise Function When the critical gapis constant the model varies with the headway distributionAs mentioned before M1 M2 M3 Erlang log-normaldistribution etc usually were used to describe the arrivalof circulating vehicles M3 model was mainly recommendedbecause it can be regarded as a general equation of M1 M2and M3T

Troutbeck [38] proposed (14) of entrance capacity wherethe headway on the major road followed M3 distribution

and g(t) is piecewise function Equation (14) is used inAUSTROADS [29]

119862 =

120572119902119890minus120582(119905119888minus119905119898)1 minus 119890minus120582119905119891 119902 gt 01119905119891 119902 = 0 (14)

The following are the capacity equations in which thedistribution parameters are specially determined in M3distributionWhen headway followsM1 distribution vehiclesare under the free flow status and 119905119898 = 0 where multiplecirculating lanes were usually regarded as single lane Whenheadway follows M2 distribution vehicles are under the freeflow status and 119905119898 = 0 in one lane When headway followsM3T distribution a dichotomized headway model assumesthat a proportion 120572 of all vehicles are free and the 1-120572bunched vehicles have the same 119905119898(1) M1 Distribution 120582 = 119902 when 119905119898 = 0 and 120572 = 1 Buckley etal ([39] refer to [40]) proposed (15)

119862 =

119902119890minus1199021199051198881 minus 119890minus119902119905119891 119902 gt 01119905119891 119902 = 0 (15)

(2) M2 Distribution 120582 = 119902(1 minus 119902119905119898) when 119905119898 = 0 119905119898 is aconstant and 120572 = 1 Equation (16) can be referred to Luttinen[14] as follows

119862 =


(3) M3T Distribution 120582 = 119902 when 119905119898 = 0 119905119898 is a constantand 120572 = 1 minus 119902119905119898 The equation can be referred to Tanner [28]and Luttein [14]

119862 =

(1 minus 119902119905119898) 119902119890minus120582(119905119888minus119905119898)1 minus 119890minus120582119905119891 119902 gt 01119905119891 119902 = 0 (17)

(4) M3 Distribution and Critical Gap Model When theheadway on the major road follows M3 distribution (18)can be deduced when the function of critical gap [34 36] issubstituted into (14)

119862 = 1205731198861199021 minus 119890minus120582119905119891 (18)

where 120573119886 is total acceptance coefficient the proportion of thenumber of accepted gaps to the number of total gaps


422When g(t) Is Continuous Function Theentrance capac-ity can be expressed as the product of the saturated flow rate119904 of entrance and the probability larger than the minimumacceptable gap 1199050 119875119905 gt 1199050 (Akcelik 1998) That is to say119862 = 119904119875119905 gt 1199050 where 119875119905 gt 1199050 is the probability larger thanthe minimum acceptable gap 1199050(1) M1Model Siegloch (1973 refer toWu 1999) proposed (19)in which the headway on the major road followed negativeexponent distribution as in (15)

119862 = 119890minus1199021199050119905119891 (19)

Equation (19) was used in HCM 2010 and parameterswere set as 119905119888 = 48119904 and 119905119891 = 25119904(2) M2 Model Jacobs (1980 refer to [2]) proposed (20)in which the headway on the major road followed shiftednegative exponent distribution as (16)

119862 = 119902120582119905119891 119890minus120582(1199050minus119905119898) = 1 minus 119902119905119898119905119891 119890minus120582(1199050minus119905119898) 1199050 gt 119905119898 (20)

The equation also can be expressed in similar form as

119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)

McDonald and Armitage [41] used the above equationsas the capacity of roundabouts They can be comparedto signalized intersections where 119905119891minus1 is regarded as thesaturated flow rate and 1199050 is regarded as lost time

(3) M3 Model In Cowanrsquos M3 distribution the minimumacceptable gap is larger thanminimum headway 1199050 gt 119905119898Thedifference betweenM3distribution andM2distribution is theratio of free flow 120572 The potential capacity is as

119862119901 = 119902119905119891 [120572120582intinfin

1199050

119905119890minus120582(119905minus119905119898)119889119905 minus 1199050119877 (1199050)]= 120572119902120582119905119891 119890

minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)

The M3 model is used in AUSTROADS [29] and SIDRA(Akcelik 1998) Multilane traffic flow can be regarded as asingle-lane traffic flow and the traffic volume is equal to thesum of all circulating traffic volume

43 Entrance Capacity Model When Critical Gap Followsa Distribution Different drivers have various critical gapsbecause the driversrsquo behaviors or the types of vehiclesare different Ideally driver behaviors are supposed to behomogeneous and consistent [21 42] When a driver isconsistent every driving behavior is identical making thecritical gap of the driver a constant When different driversare homogeneous their drive behaviors are similar so theyhave the same critical gap whether constant or a distribution

Catchpole and Plank [21] proposed the classification ofentrance capacity models including the constant critical gapand critical gap distribution The models in Section 42 arethe first part of Section 431 Heidemann and Wegmann[4] proposed the models based on a certain distributionof 119905119888 119905119898 119905119891 and M3 distribution of the circulating streamWursquos equations [2] were based on Erlang distribution of thecirculating headway whereas Guorsquos equations [34 36] werebased on M3 distribution of the circulating headway and theexponent function of rejected proportion

431 Equations of Catchpole and Plank Catchpole and Plank[21] proposed the capacity model based on different driverbehaviors When g(t) is piecewise function the capacity canbe expressed as follows

(1) When the driver behaviors are homogeneous orconsistent that is to say 119905119888 is a constant the capacity modelsare as (14) to (22) mentioned before

(2) When the driver behaviors are consistent but nonho-mogeneous the capacity is as follows

119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)

where L(119905119888(119902)) is the Laplace transformation of 119905119888 distributionand L(119905119891(119902)) is the Laplace transformation of 119905119891

(3) It can be regarded as a special style of the followingnumber (4) situation when the driver behaviors are homoge-neous but inconsistent

(4) When the driver behaviors are inconsistent andnonhomogeneous the capacity is as follows

119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)

where 120579119895 is the proportion of the 119895th kind of entrance vehicleL(120579119895(119902)) is the Laplace transformation of 120579119895432 Equations of Heidemann and Wegmann Heidemannand Wegmann [4] proposed that the capacity equation ofunsignalized intersections follow a certain distribution for119905119888 119905119898 119905119891 Based onM3 distribution of the headway on amajorroad the capacity is (25) when 119892(t) is the piecewise functionand it is (26) when 119892(t) is the continuous function

119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)

where 119861 = 119905119898120572 1199051015840119888 = 119905119888 minus 119905119898 L(1199051015840119888(120582)) is the Laplacetransformation of 1199051015840119888 at 120582 L(1199051015840119888(120582)) = L(119905119888(120582))L(119905119898(minus120582))L(119905119891(120582)) is the Laplace transformation of 119905119891 at 120582 L(119905119898(minus120582))is the Laplace transformation of 119905119898 at minus120582

433 Wu Equation Wu [2] proposed the probability densityfunction when the headway on a major road follows Erlangdistribution as follows

119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)

where 120574 = 120572119905119909119905119909 119905119909 is the average value of 119905119909 120572119905119909 is theconstant in Erlang distribution of 119905119909(1) When g(t) Is a Piecewise Function It is supposed that theminimum value of 119905119888 is 120591119905119888 and 120591119905119888 gt 120591119905119898 the minimum valueof 119905119891 is 120591119905119891 gt 0 and 120591119905119891 gt 0 the minimum value of 119905119898 is120591119905119888 and 120591119905119898 gt 0 and 119905119888 119905119898 119905119891 all follow negative exponentdistribution The capacity is as follows when the drivers arenonhomogeneous

119862119887119906119899119888ℎ = 120572119902(120582 (119905119888 minus 120591119905119888) 119896119905119888 + 1)minus120572119905c 119890minus120582120591119905119888 (minus120582 (119905119898 minus 120591119905119898) 119896119905119898 + 1)minus120572119905119898 1198901205821205911199051198981 minus (120582 (119905119891 minus 120591119905119891) 119896119905119891 + 1)minus120572119905119891 119890minus120582120591119905119891 (28)

The capacity is as follows when the drivers are homoge-neous

119862119887119906119899119888ℎ = 120572119902(minus120582 (119905119888 minus 120591119905119888) 119896119905119888 + 1)120572119905c 119890minus120582120591119905119888 (120582 (119905119898 minus 120591119905119898) 119896119905119898 + 1)120572119905119898 1198901205821205911199051198981 minus (120582 (119905119891 minus 120591119905119891) 119896119905119891 + 1)minus120572119905119891 119890minus120582120591119905119891 (29)

(2) When g(t) Is Continuous Function Some assumptionshave been given that the headway on a major road followsshifted Erlang distribution and 119905119891 follows random distri-bution The capacity is as follows when the drivers arenonhomogeneous

119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050

sdot 119890minus1205821205911199050 (minus120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)minus120572119905119898 119890120582120591119905119898

(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)

434 Guo Equation The equation of Guo [34 36] is basedon M3 distribution All the headways of the free flow on themajor road have rejected proportions that is to say every

headway in the interval of (119905minfin) on the major road isprobably to be accepted The rejected proportion function isthe exponent function and the accepted proportion increaseswhen the headway on the major road increases

The rejected proportion function 119891119903119900(119905) is the exponentfunction as follows

1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)

where 119903 is the rejected proportion coefficientThe capacity can be deduced when g(t) is continuous

function as follows

119862 = 120572119902[ 119903(120582 + 119903) + 119890minus120582119905119891119905119891120582 minus 120582119890minus(120582+119903)119905119891119905119891 (120582 + 119903)2] (33)

The capacity can be deduced when g(t) is piecewisefunction as follows

119862 = 120572119902(120582 + 119903) [ 119903

(1 minus 119890minus120582119905119891)minus 120582119890minus(120582+119903)119905119891[1 minus 119890minus(120582+119903)119905119891] (

119903120582 + 119903)(120582+119903)120582]

(34)


44 The Capacity of Limited Priority Merge Under somespecial situations gap acceptance theory can be based ondifferent assumptions such as limited priority merge orpriority conversion The driving behavior under limitedpriority merge [43ndash46] means that vehicles on the circulatinglane need to adjust the gaps for the vehicles entering theintersection

Another priority style priority conversion means thatvehicles on the entrance are forced to enter into the intersec-tion and the circulating vehicles have to yield for the entrancevehiclesThese driving behaviors aremore feasible in the fieldand their equations are more complicated to be applied forcalculation

Troutbeck [45] proposed the following equation of lim-ited priority merge

119862 = 120572119891119871119902119890minus120582(119905119888minus119905119898)1 minus 119890minus120582119905119891 119905119898 lt 119905119888 lt 119905119891 + 119905119898119891119871 = 1 minus 119890minus120582119905119891

1 minus 119890minus120582(119905119888minus119905119898) minus 120582 (119905119888 minus 119905119891 minus 119905119898) 119890minus120582(119905119888minus119905119898)(35)

When 119905119888 ge 119905119891 + 119905119898119891119871 = 1 the vehicles on the major roadare not delayed after merging and the system can be modeledas an absolute priority system It can be referred to Bunkerand Troutbeck [46] Troutbeck [44 45] Troutbeck and Kako[43]

If both 119905119888 and 119905119891 are set up as 119905119898 all the gaps are 119905119898 aftermerging The capacity is as follows

119862 = 1119905119898 minus 119902 (36)

where 119891119871 = (1 minus 119890minus120582119905119898)120582119905119898 and 119862 = 1205721199021205821199051198985 Comparison of Different Models

When critical gap follows a stochastic distribution that is tosay a driver might reject a gap which he accepted before ordrivers have different critical gaps this effect results in anincrease of capacity comparingwith the condition of constantcritical gap

The limited priority merge can have a significant effecton the entry capacity at roundabouts The limited prioritycapacity is very close to the empirical results from the UKlinear regression method The capacity prediction based onthe limited priority and the UK method are reasonable ingeneral

It is difficult to compare different models in the fieldbecause each model adapts in different road geometrical andtraffic conditionsThe headway distribution of the circulatingvehicles the ratio of free flow the type of g(t) function thecritical gap 119905119888 and other parameters such as 119905119891 119905119898 vary withdifferent geometrical and traffic conditions and they affectthe calculation value of entrance capacity in roundabouts

Suppose that the headway of circulating vehicles followthe M3 distribution as (1) the decay constant 120582 is (3) andthe ratio of free flow is (7) Equations (14) (18) (22) (34)and (35) can be compared using the same parameter values

0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)

Capacity of entrance

Eq 14(fixed tc)Eq 18Eq 22

Eq 34Eq 35(fixed tc)

Figure 3 Calculation of entrance capacity under different flow rateson circulating vehicles

where 119905119898 = 2119904 and 119905119891 = 3119904 Due to different g(t) functions andcalculationmodels of 119905119888 some different parameters need to beset in some Equations In (14) and (35) 119905119888 = 435119904 In (22) and(34) 1199050 = 119905119898 = 2119904 In (18) and (35) 119903 = 0365 The entrancecapacity 119862 can be calculated using the five equations under adifferent flow rate 119902 The results can be shown in Table 4 andFigure 2

From Figure 3 it can be concluded that the values of(22) are larger than values of other equations The values of(14) (18) (34) and (35) are so similar that different curvescannot be clearly recognized Because the number of vehiclesentering the intersection can be considered as a decimalif continuous g(t) (22) with continuous g(t) function willresult in a larger value of entrance capacity than the otherequations based on the piecewise g(t) functionThe influenceof different critical gap models on capacity value is limitedThe capacity values of (18) and (34) are slightly larger than(14) and (35) when the circulating vehicles are close to thesaturated flow rate where q=04 and q=045 The capacityvalues of the four equations are almost equal with each other

The above capacity models have very small differencesof each other regardless of the critical gap type and the g(t)type Therefore the simple calculating models based on theconstant critical gap and piecewise g(t) function should berecommended in the field

6 Application Conclusions ofDifferent Models

The capacity models based on gap acceptance theory areanalysis methods in which traffic parameters have a certainphysical meaning When the critical gap is constant thedeviation of the capacitymodel is conveniently used to obtainthe equations and calculate the accurate capacity valuesDriversrsquo behavior characteristics are different in differentcountries For example the critical gap of the single-laneroundabout is 48s in America however it is about 35s inChina Driversrsquo nonhomogeneous and inconsistent character


Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020


is more realistic than the fixed critical gap and following gapAlthough the calculation results of capacity are similar asthe field capacity under the assumption of homogeneity andcontinuance there is only a little percent deviation

Several widely used models and the parameters can becompared as follows

(1) The headway was usually regarded as negative expo-nent distribution M3 distribution was also paid enoughattention [4]

(2) Based on a field traffic survey in Australia [22] thecapacity Equation (22) ofM3distribution and continuous g(t)was applied in aaSIDRA (Akcelik et al 1994 1997 1998) inwhich the following gap and critical gap are varied from thegeometry of the roundabout the flow rate of entrance andcirculating lanesThe capacity is related to the circulating flowrate lane use ODmetric of flow rate queue on entrance lanesand the ratio of free flow All the lanes on entrance can bemodeled at the same time

(3) Based on the survey of a limited number of round-abouts in America and in comparison with the experiencevalues of other countries the capacity equation (15) ofM1 distribution and piecewise function g(t) was used inHCM (2000) In the model the constant parameters of gapacceptance were used which did not vary from the geometryof the roundabout and traffic volume The calculation resultshave limitations when the circulating volume increased up to1200pcuh for a single-lane roundabout

(4) The capacity equation (19) of M1 distribution andcontinuous function g(t) was used in HCM (2010) in whichboth 119905119888 and 119905119891 were determined and some equations werebuilt based on a different number of entrance lanes anddifferent number of circulating lanesThe calculating processof traffic characteristics was proposed including capacitydelay level of service and queue length

(5) The capacity equation (18) was deduced by directlyplugging the critical gap model into the capacity equation(14) Equation (18) has reasonable results in accordance withsome classicalmethods and it hasmore accurate results undersome conditions for example the low capacity is closer to thereality than linear regression under low flow rate conditions

(6) The method based on stochastic critical gap is morerealistic with operation in roundabouts butmost of the equa-tions are too complicated to calculate quickly Simplificationis needed to apply in the field

With the increase of traffic volume the application oftraditional roundabouts becomes more and more difficultMany roundabouts especially in China were dismantled andfew roundabouts were newly builtThefield capacity becomesvery small becausemany drivers do not obey the priority ruleTraffic jams or even traffic accidents occurred at roundaboutsIn reality signal timing was applied in some roundabouts inwhich the inner lanes of the weaving area were designed asleft-turn waiting areas Left-turn vehicles need to watch thesignal light two times when passing through the roundaboutThe validity needs to be verified further

Drivers are sensitive to some problems of roundaboutsat peak hours but ignore the advantages at off-peak timesuch as safe environment friendly suitable to large-volumeleft turns To the researchers and engineers in the traffic field

traffic organization and optimization should be the key focusto increase capacity and reliability at peak hours

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] W Grabe Leistungsermittlung von nicht lichtsignalsteuertenKnotenpunkten des Stranverkehrs [Capacity calculation ofunsignalized intersections in road traffic] Forschungsarbeitenaus dem Strassenwesen Neue Folge 11 Kirschbaum VerlagBielefeld Germany 1954

[2] N Wu ldquoA universal procedure for capacity determination atunsignalized (priority-controlled) intersectionsrdquo Transporta-tion Research Part B Methodological vol 35 no 6 pp 593ndash6232001

[3] C F Daganzo ldquoTraffic delay at unsignalized intersectionsClarification of some issuesrdquo Transportation Science vol 11 no2 pp 180ndash189 1977

[4] D Heidemann and H Wegmann ldquoQueueing at unsignalizedintersectionsrdquo Transportation Research Part B Methodologicalvol 31 no 3 pp 239ndash263 1997

[5] W Brilon ldquoStudies on Roundabouts in Germany LessonsLearnedrdquo in Proceedings of theThe 3rd International Transporta-tion Research Board 2010

[6] W Wang H Gao W Li et al Capacity Analysis Method ofHighway Intersection Science Press 2001

[7] R J Guo B L Lin and W X Wang ldquoThe Iterative Caculationof Full Capacity of Roundaboutsrdquo in Proceedings of the SecondInternational Symposium on Information Science and Engineer-ing pp 570ndash573 IEEE Shanghai China 2009

[8] P K Jeffrey N Gautami and L Elefteriadou ldquoQueuing delaymodels for single-lane roundaboutstrdquo Civil Engineering andEnvironmental Systems vol 22 no 3 pp 133ndash150 2001

[9] E Chevallier and L Leclercq ldquoA macroscopic theory forunsignalized intersectionsrdquo Transportation Research Part BMethodological vol 41 no 10 pp 1139ndash1150 2007

[10] A J Miller An Empirical Model for Multilane Road TrafficNorthwestern University for the US Bureau of Public Roads1968

[11] R M Kimber ldquoGap-acceptance and empiricism in capacitypredictionrdquo Transportation Science vol 23 no 2 pp 100ndash1111989

[12] D R McNeil and J T Smith A Comparison of Motorist Delaysfor Different Merging Strategies United States Department ofTransportation 1968

[13] E M Hollis M C Semmens and S L Denniss ldquoARCADY AComputer Program to Model Capacities Queues and Delays atRoundaboutsrdquo Transport and Road Research Laboratory 940TRRL Laboratory Report Crowthorne Berkshire UK 1980

[14] R T LuttinenCapacity and Level of Service at Finnish Unsignal-ized Intersections Finnish Road Administration 2004

[15] H M N Al-Madani ldquoCapacity of large dual and triple-lanes roundabouts during heavy demand conditionsrdquo ArabianJournal for Science and Engineering vol 38 no 3 pp 491ndash5052013


[16] H L Khoo and C Y Tang ldquoRoundabout system capacity esti-mation and control strategy with origin-destination patternrdquoJournal of Transportation Engineering vol 142 no 5 Article ID04016017 2016

[17] J Bie H K Lo and S C Wong ldquoCapacity evaluation of multi-lane traffic roundaboutrdquo Journal of Advanced Transportationvol 44 no 4 pp 245ndash255 2010

[18] X Qu L Ren S Wang and E Oh ldquoEstimation of entrycapacity for single-lane modern roundabouts Case study inQueensland Australiardquo Journal of Transportation Engineeringvol 140 no 7 Article ID 05014002 2014

[19] Y H Yap H M Gibson and B J Waterson ldquoModels of round-about lane capacityrdquo Journal of Transportation Engineering vol141 no 7 Article ID 04015007 2015

[20] R Mauro and F Branco ldquoComparative analysis of compactmultilane roundabouts and turbo-roundaboutsrdquo Journal ofTransportation Engineering vol 136 no 4 pp 316ndash322 2010

[21] E A Catchpole and A W Plank ldquoThe capacity of a priorityintersectionrdquo Transportation Research Part B Methodologicalvol 20 no 6 pp 441ndash456 1986

[22] O Hagring ldquoEffects of OD Flows on Roundabout Entry Capac-ityrdquo Transportation Research Circular E-C018 4th InternationalSymposium on Highway Capacity pp 434ndash445 2000

[23] A Khattak and P Jovanis ldquoCapacity and delay estimation forpriority unsignalized intersections Conceptual and empiricalissuesrdquo Transportation Research Board pp 129ndash137 1990

[24] D Branston ldquoModels of single lane timeheadway distributionsrdquoTransportation Science vol 10 no 2 pp 125ndash148 1976

[25] China Highway SocietyManual of Transportation EngineeringChina communication press Beijing China 2001

[26] R C Cowan ldquoUseful headway modelsrdquo TransportationResearch vol 9 no 6 pp 371ndash375 1975

[27] O C Yigiter and S Tanyel ldquoLane by lane analysis of vehicle timeheadways mdash Case study of Izmir ring roads in Turkeyrdquo KSCEJournal of Civil Engineering vol 19 no 5 pp 1498ndash1508 2015

[28] J C Tanner ldquoThe capacity of an uncontrolled intersectionrdquoBiometrika vol 54 no 3-4 pp 163ndash170 1967

[29] AUSTROADS Roundabouts Guide to Traffic Engineering Prac-tice Association of Australian State Road and TransportAuthorities Sydney Australia 1993

[30] W Brilon R Koenig and R J Troutbeck ldquoUseful estimationprocedures for critical gapsrdquo Transportation Research Part APolicy and Practice vol 33 no 3-4 pp 161ndash186 1999

[31] Z Z Tian M Vandehey B W Robinson et al ldquoImplementingthe maximum likelihood methodology to measure a driverrsquoscritical gaprdquoTransportation Research Part A vol 33 pp 187ndash1971999

[32] Z Tian R M Troutbeck W Kyte et al ldquoFurther investi-gation on critical gap and follow-up timerdquo in Proceedings ofthe Transportation Research Circular E-C018 4th InternationalSymposium on Highway Capacity 2002

[33] R Guo ldquoEstimating critical gap of roundabouts by differentmethodsrdquo Transportation of China pp 84ndash89 2010

[34] R J Guo X J Wang and W X Wang ldquoEstimation of criticalgap based on Raff rsquos definitionrdquo Computational Intelligence andNeuroscience vol 2014 Article ID 236072 7 pages 2014

[35] R Guo and Y Zhao ldquoCritical gap of a roundabout based on alogit modelrdquo in Proceedings of the Fifth International Conferenceon Transportation Engineering pp 2597ndash2603 Dailan China2015

[36] R J Guo and B L Lin ldquoGap acceptance at priority-controlledintersectionsrdquo Journal of Transportation Engineering vol 137no 4 pp 269ndash276 2011

[37] W Brilon ldquoSome remarks regarding the estimation of criticalgapsrdquo Transportation Research Record vol 2553 no 1 pp 10ndash19 2016

[38] R J Troutbeck ldquoAverage delay at an unsignalized intersectionwith two major streams each having a dichotomized headwaydistributionrdquoTransportation Science vol 20 no 4 pp 272ndash2861986

[39] D J Buckley ldquoA semi-poissonmodel of traffic flowrdquoTransporta-tion Science vol 2 no 2 pp 107ndash133 1968

[40] M Kyte W Kittelson Z Z Tian et al ldquoAnalysis of traffic oper-ations at all-way stop-controlled intersections by simulationrdquoin Proceedings of the Annual Meeting of Transportation ResearchBoard Washington DC USA 1996

[41] M McDonald and D J Armitage ldquoThe capacity of round-aboutsrdquo Traffic Engineering and Control vol 19 pp 447ndash4501978

[42] R J Troutbeck and W Brilon ldquoTraffic Flow Theoryrdquo in Part 8Unsignalized Intersection Theory Gartner and Messer Eds pp81ndash847 1997

[43] R J Troutbeck and S Kako ldquoLimited priority merge atunsignalized intersectionsrdquo Transportation Research Part APolicy and Practice vol 33 no 3-4 pp 291ndash304 1999

[44] R J Troutbeck ldquoTheperformance of uncontrolledmerges usinga limited priority processrdquo in Transportation and TrafficTheoryin the 21st Century M A P Taylor Ed pp 463ndash482 TaylorAustralia 2002

[45] R J Troutbeck ldquoThe capacities of a limited priority mergerdquoPhysical Infrastructure Centre Research Report 98-4 Queens-land University of Technology Australia 1998

[46] J Bunker and R Troutbeck ldquoPrediction of minor stream delaysat a limited priority freeway mergerdquo Transportation ResearchPart B Methodological vol 37 no 8 pp 719ndash735 2003

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method and gap acceptance method were since developed(Ashworth 1989) [8 9]

Philbrick proposed the linear regression method (referto [10]) which is mainly used in England The exponentialregression methods (Brilon and Stuwe 1993) were based onnumerous survey data among the saturated entrance flow rateand conflicting flow rate geometry etc

The linear regression models in England were developedfrom traditional roundabouts which had the large-diameterisland design and entrance priority The capacity such as inthe models proposed by Kimber [11] McNell and Smith [12]and Hollis and Semmens et al [13] was regressed withgeometry parameters of roundabouts and the total circulatingtraffic volume [14] regardless of traffic status of every lane

In gap acceptance theory some parameters should bedetermined including headway distribution of circulatingvehicles the critical gap and following gap although theyare variable with different geometry and traffic conditionsof roundabouts Vehicles enter the roundabout by use ofacceptable gaps of circulating flow and the capacity is mainlydetermined by circulating flow rate and headway distribu-tion

Some countries such as USA Germany Australia UKJapan France and Russia have built complete capacitymethods which are suitable for their own traffic conditionsincluding Highway Capacity Manual in USA aaSIDRAAUSTROADS andNAASRA in Australia Swedish CAPCALSETRA method and CETUR method in Germany

The gap acceptance can be regarded as the signal timingprocess which was proposed by Achelik and Chung (1994)Both the gap acceptance theory and regression method canbe used to calculate the delay and queue length The gapacceptance models are identical among roundabouts two-way stop and all-way stop intersections

The capacity models and traffic control methods ofroundabouts were studied in many countries Al-Madani[15] collected the data of 13 large roundabouts includingcirculating and exiting flows the number of lanes andlateral position of vehicles in Bahrain The developed modelmatched the field data reasonably well and fell well in othermethods Khoo and Tang [16] proposed a control strategywhich was effective in reducing system travel time andincreasing volume especially duringmedium to high levels ofdemand A case study of a two-lane roundabout in Malaysiawas developed in a microscopic simulation environmentto study the roundabout system properties and to test theeffectiveness of the proposed control strategy Macioszek(2016) applied the capacity method of HCM 2010 in the fieldroundabouts in PolandThe calculated values were similar asthe experience values

Biel andWong [17] proposed the entrance capacitymodelof circulating multilanes roundabout based on the regressionmodel and HCM model Under the limited priority condi-tion Qu et al [18] calculated the entrance capacity basedon Raff rsquos critical gap model and the maximum likelihoodmethod Yap and Gibson et al [19] proposed two regressionmethods and compared variousmethods based on the surveydata of roundabouts in UK The capacity model of negativeexponential distribution showed a coincidence with the

reality than the regression model under both the high flowrate and low flow rate By using of the Germany capacityprogramsMauro andBranco [20] compared the capacity anddelay model between circulating multilane roundabout andTurbo roundabout

Weaving theory was used to describe the traffic charac-teristics including weaving capacity in weaving sections It isapplicable in the traditional roundabouts where the centralisland is large and entry flow stream has the priority But theregression method and gap acceptance theory can be used tocalculate entrance capacity in modern roundabouts

Linear regression method is mainly used in EnglandThe entrance capacity is regressed with conflicting flowrate geometry etc regardless of traffic status of every laneRegression method usually can obtain satisfied results aftera large number of data survey of roundabouts under thecondition of saturated traffic flow

Gap acceptance characters are related to traffic conditionsof roundabouts including the headway distribution of circu-lating vehicles circulating volume critical gap and followinggap Queuing theory was used to describe the gap acceptanceprocess and to deduce the capacity equation Gap acceptancetheory and regression method can be used to calculate thedelay and queue length too By choosing the reasonable trafficparameters such as critical gap value or its distribution gapacceptance theory can obtain the consistent capacity resultsas the field data Gap acceptance theory was usually appliedin the science research of traffic flow whereas the regressionmethod was more appropriate in the traffic engineering field

Based on the gap acceptance theory the basic assump-tions were induced the headway distributions were intro-duced and the relevant parameters were summed up Mostentrance capacity models were classified according to thedifferent critical gap types and the g(t) functions Theirapplicable conditions and current application fields wereconcluded

2 The Assumptions of Gap Acceptance

The headway is the time interval that elapses between thearrival of the leading vehicle and the following vehicle at thedesignated test point Gap is a little different from headway inthat gap is the measure of the time between the rear bumperof the first vehicle and the front bumper of the second vehiclerather than front-to-front time

Gap acceptance usually happens in unsignalized intersec-tions that are controlled by priority The entrance vehicleswill enter the intersection when there are no vehicles on themajor road or wait for the acceptable gap at the stop lineTheentrance vehicle will enter the intersection if the gap on themajor road is larger than or equal to the critical gap otherwisethe entrance vehicle will decelerate or stop so as to wait forthe next gap when it is larger than the critical gapThe criticalgap can be regarded as the driverrsquos judgment threshold whichdetermines whether the entrance vehicle has enough timeto enter the intersection safely or not Driversrsquo behaviorsare different from each other in reality so the critical gapsof various drivers are different The critical gap is usually














119891 (119905) = 120572120582119890minus120582(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (1)



and120582= 1205721199021 minus 119902119905119898 (3)




120582 = 1(1119899)sum119899119894=1 119905119894 minus 120577 (5)




120572 = 119890minus119896119902 (6)


120572 = 119890minus119887119902119905119898 (7)



120572 = 1 minus 119902119905119898 (8)


120572 = 075 (1 minus 119902119905119898) (9)




119905119898 18 09 063600119905119898 2000 4000 6000119887 05 03 07119896119889 02 03 03


119905119898 20 10 083600119905119898 1800 3600 4500119887 25 25 25119896119889 22 22 22


q(vehs) 005 01 015 02 025 03 035 04 045Eq (7) 078 061 047 037 029 022 017 014 011Eq (8) 090 080 070 060 050 040 030 020 010Eq (9) 068 060 053 045 038 030 023 015 008Eq (10) 080 065 051 041 031 023 016 010 005









000010020030040050060070080090100

005 01 015 02 025 03 035 04 045q (vehs)

Eq (7)Eq (8)

Eq (9)Eq (10)







119862 = 119902intinfin0

119891 (119905) 119892 (119905) 119889119905 (11)



119892 (119905) = 119905 minus 1199050119905119891 119905 gt 11990500 119905 le 1199050

1199050 = 119905119888 minus 1199051198912(12)



0

1

2

3

4

5

6

0 tc tc+tf

Eq (12)Eq (13)


g(t) (veh)

Headway (s)




1 119905119888 + (119899 minus 1) 119905119891 le 119905 lt 119905119888 + 1198991199051198910 119905 lt 119905119888

(13)









119862 =



119862 =



119862 =



119862 =



119862 = 1205731198861199021 minus 119890minus120582119905119891 (18)




119862 = 119890minus1199021199050119905119891 (19)




119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)



119862119901 = 119902119905119891 [120572120582intinfin

1199050


minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)







119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)




119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)


119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















119891 (119905) = 120572120582119890minus120582(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (1)



and120582= 1205721199021 minus 119902119905119898 (3)




120582 = 1(1119899)sum119899119894=1 119905119894 minus 120577 (5)




120572 = 119890minus119896119902 (6)


120572 = 119890minus119887119902119905119898 (7)



120572 = 1 minus 119902119905119898 (8)


120572 = 075 (1 minus 119902119905119898) (9)




119905119898 18 09 063600119905119898 2000 4000 6000119887 05 03 07119896119889 02 03 03


119905119898 20 10 083600119905119898 1800 3600 4500119887 25 25 25119896119889 22 22 22


q(vehs) 005 01 015 02 025 03 035 04 045Eq (7) 078 061 047 037 029 022 017 014 011Eq (8) 090 080 070 060 050 040 030 020 010Eq (9) 068 060 053 045 038 030 023 015 008Eq (10) 080 065 051 041 031 023 016 010 005









000010020030040050060070080090100

005 01 015 02 025 03 035 04 045q (vehs)

Eq (7)Eq (8)

Eq (9)Eq (10)







119862 = 119902intinfin0

119891 (119905) 119892 (119905) 119889119905 (11)



119892 (119905) = 119905 minus 1199050119905119891 119905 gt 11990500 119905 le 1199050

1199050 = 119905119888 minus 1199051198912(12)



0

1

2

3

4

5

6

0 tc tc+tf

Eq (12)Eq (13)


g(t) (veh)

Headway (s)




1 119905119888 + (119899 minus 1) 119905119891 le 119905 lt 119905119888 + 1198991199051198910 119905 lt 119905119888

(13)









119862 =



119862 =



119862 =



119862 =



119862 = 1205731198861199021 minus 119890minus120582119905119891 (18)




119862 = 119890minus1199021199050119905119891 (19)




119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)



119862119901 = 119902119905119891 [120572120582intinfin

1199050


minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)







119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)




119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)


119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119905119898 18 09 063600119905119898 2000 4000 6000119887 05 03 07119896119889 02 03 03


119905119898 20 10 083600119905119898 1800 3600 4500119887 25 25 25119896119889 22 22 22


q(vehs) 005 01 015 02 025 03 035 04 045Eq (7) 078 061 047 037 029 022 017 014 011Eq (8) 090 080 070 060 050 040 030 020 010Eq (9) 068 060 053 045 038 030 023 015 008Eq (10) 080 065 051 041 031 023 016 010 005









000010020030040050060070080090100

005 01 015 02 025 03 035 04 045q (vehs)

Eq (7)Eq (8)

Eq (9)Eq (10)







119862 = 119902intinfin0

119891 (119905) 119892 (119905) 119889119905 (11)



119892 (119905) = 119905 minus 1199050119905119891 119905 gt 11990500 119905 le 1199050

1199050 = 119905119888 minus 1199051198912(12)



0

1

2

3

4

5

6

0 tc tc+tf

Eq (12)Eq (13)


g(t) (veh)

Headway (s)




1 119905119888 + (119899 minus 1) 119905119891 le 119905 lt 119905119888 + 1198991199051198910 119905 lt 119905119888

(13)









119862 =



119862 =



119862 =



119862 =



119862 = 1205731198861199021 minus 119890minus120582119905119891 (18)




119862 = 119890minus1199021199050119905119891 (19)




119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)



119862119901 = 119902119905119891 [120572120582intinfin

1199050


minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)







119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)




119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)


119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

1

2

3

4

5

6

0 tc tc+tf

Eq (12)Eq (13)


g(t) (veh)

Headway (s)




1 119905119888 + (119899 minus 1) 119905119891 le 119905 lt 119905119888 + 1198991199051198910 119905 lt 119905119888

(13)









119862 =



119862 =



119862 =



119862 =



119862 = 1205731198861199021 minus 119890minus120582119905119891 (18)




119862 = 119890minus1199021199050119905119891 (19)




119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)



119862119901 = 119902119905119891 [120572120582intinfin

1199050


minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)







119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)




119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)


119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119862 = 119890minus1199021199050119905119891 (19)




119862119901 = 119890minus120582(1199050minus119905119898)119905119891 (1 + 120582119905119898) 1199050 gt 119905119898 (21)



119862119901 = 119902119905119891 [120572120582intinfin

1199050


minus120582(1199050minus119905119898) 1199050 gt 119905119898(22)







119862 = 119902 (1 minus 119902119905119898) 119890119902119905119898L (119905119888 (119902))1 minus L (119905119891 (119902)) (23)




119862 = 1sum119895 (120579119895119902119895) = 120572119902119890120582119905119898

1 minus 119890minus120582119905119891 sdot1

sum119895 120579119895L (120579119895 (119902)) (24)


119862 = 1205821 + 120582119861

L (1199051015840119888 (120582))1 minus L (119905119891 (120582))

= 1205821 + 120582119861

L (119905119888 (120582))L (119905119898 (minus120582))1 minus L (119905119891 (120582))(25)


119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



119862 = 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (1199051015840119888 (minus120582))

= 1205821 + 120582119861

1(1 minus L (119905119891 (120582)))L (119905119888 (minus120582))L (119905119898 (120582))

(26)



119891 (119905119909) = 120574(120572119905119909 minus 1) (120574119905119909)

120572119905119909minus1 119890minus120574119905119909 (27)






119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

120582 (1199050 minus 1205911199050)1198961199050 + 1)minus1205721199050


(30)


119862119887119906119899119888ℎ = (1 minus 119902119905119898) 1119905119891 (

minus120582 (1199050 minus 1205911199050)1198961199050 + 1)1205721199050

sdot 119890minus1205821205911199050 (120582 (119905119898 minus 120591119905119898)119896119905119898 + 1)120572119905119898 119890120582120591119905119898

(31)




1198911199030 (119905) = 119890minus119903(119905minus119905119898) 119905 ge 1199051198980 119905 lt 119905119898 (32)


function as follows



119862 = 120572119902(120582 + 119903) [ 119903


119903120582 + 119903)(120582+119903)120582]

(34)









119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










119862 = 1119905119898 minus 119902 (36)






0

005

01

015

02

025

03

035

005 01 015 02 025 03 035 04 045

C (v

ehs

)

q (vehs)











Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table4Ca

lculationof

entrance

capacityun

derd

ifferentfl

owrateso

ncirculatingvehicle

s(119905 119898=

2119904119905119891=3119904

)

No

q(vehs)

alph

a120582(1

s)120573a

tc(s)

Eq(14)(fixed

tc)(vehs)

Eq(18)(vehs)

Eq(22)(vehs)

Eq(34)(vehs)

FLin

Equ(35)

Eq(35)(fixed

tc)(vehs)

1005

0779

0043

0696

4589

0289

0286

0300

0285

0997

0288

201

0607

0076

0502

4489

0249

0247

0267

0246

0995

0248

3015

0472

0101

0370

4418

0213

0212

0233

0210

0994

0212

402

0368

0123

0275

4362

0179

0179

0200

0177

0993

0178

5025

0287

0143

0206

4311

0146

0147

0167

0146

0991

0145

603

0223

0167

0153

4255

0114

0116

0133

0115

0990

0113

7035

0174

0203

0112

4179

0083

0086

0100

0084

0989

0082

804

0135

0271

0078

4050

0052

0056

0067

0055

0986

0051

9045

0105

0474

004

63756

0021

0027

0033

0027

0982

0020















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
















References


















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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