research article estimation of critical gap based on raff

Research ArticleEstimation of Critical Gap Based on Raffrsquos Definition

Rui-jun Guo12 Xiao-jing Wang1 and Wan-xiang Wang2

1 National ITS Center Research Institute of Highway Beijing 100088 China2 School of Traffic and Transportation Dalian Jiaotong University Dalian 116028 China

Correspondence should be addressed to Xiao-jing Wang xjwangriohcn

Received 26 July 2014 Accepted 5 October 2014 Published 9 November 2014

Academic Editor Xiaobei Jiang

Copyright copy 2014 Rui-jun 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

Critical gap is an important parameter used to calculate the capacity and delay of minor road in gap acceptance theory ofunsignalized intersections At an unsignalized intersection with two one-way traffic flows it is assumed that two events areindependent between vehiclesrsquo arrival of major stream and vehiclesrsquo arrival of minor streamThe headways of major stream followM3distribution Based onRaff rsquos definition of critical gap two calculationmodels are derived which are namedM3definitionmodeland revised Raff rsquos model Both models use total rejected coefficient Different calculation models are compared by simulation andnewmodels are found to be validThe conclusion reveals thatM3 definitionmodel is simple and valid Revised Raff rsquos model strictlyobeys the definition of Raff rsquos critical gap and its application field is more extensive than Raff rsquos model It can get a more accurateresult than the former Raff rsquos model The M3 definition model and revised Raff rsquos model can derive accordant result

1 Introduction

11 Literature Review of Critical Gap At unsignalized inter-sections with two traffic flows vehicles streaming at majorroad have the priority to pass intersections however vehiclesstreaming at minor road must wait for the enough gap ofmajor stream Critical gap is the minimum major-streamheadway during which a typical minor-stream vehicle canmake a maneuver as discussed by Luttinen [1] It is animportant parameter to determine the capacity and delay ofminor road Critical gap is a judgment threshold to whethera minor-stream vehicle can enter major stream That is tosay the vehicle can enter intersection when the headway ofmajor stream is larger than critical gap and the headwayis called accepted gap whereas the vehicle cannot enterintersection when the headway is smaller than critical gapand the headway is called rejected gap

It is obvious that different drivers or the same driverat different times have different critical gaps because ofdifferent driving operation This kind of difference is calledinconsistent and nonhomogeneous as discussed by Plankand Catchpole [2] So critical gap is a random variable andassumed following some distribution which can be describedby the average and variance

Many different methods for estimation of critical gap atunsignalized intersections have been presented Polus et al[3] found that critical gap decreased with increase of thewaiting time and the relation between them was ldquoSrdquo typecurve which could be expressed by the exponential modelHamed et al [4] thought that the distribution of critical gapwas related to driving years social and economic backgroundof drivers waiting time and travel destination The averagevalue of critical gap is related to conflicted flow the lanenumber of minor road proportion of turn-left lane andthe velocity of major stream They built multiple regressionmodel of critical gap Ashworth [5 6] analyzed the distri-bution character of accepted gap with different flow rate ofmajor road and modeled the average value and variance ofcritical gap on the assumption that the headways of majorroad follow negative exponential distribution and criticalgap and accepted gap follow normal distribution Based onAshworthrsquos model Miller [7] proposed the model of criticalgap provided that it follows 120574 distribution Raff and Hart [8]regarded the cross point of rejected gap number and acceptedgap number as critical gap The method is widely used inmany countries

Brilon et al [9] pointed that Sieglochrsquos method was validonly in bunched flow A majority of methods are valid

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2014 Article ID 236072 7 pageshttpdxdoiorg1011552014236072

2 Computational Intelligence and Neuroscience

Table 1 Critical gap and follow-up headway for roundabouts (HCM2000 [12]) (s)

119905119888

119905119891

Upper bound 41 26Lower bound 46 31

in free flow for example lag method can be regarded asthe theoretical reference value Logit method is similar tothe classical logit models of transportation planning Probitprocedure cannot consider critical gap as normal distributiondespite its obvious variance Hewittrsquos method [10] has afarther development than Probit procedure and even has cor-responding calculation software of GAPTIM and PROBITThe maximum likelihood method [11] estimates the averagevalue and variance on the assumption of normal distributionof accepted gap maximum rejected gap and critical gapRaff and Hart [8] found the maximum likelihood methodand Hewittrsquos method had an accurate result by analyzing andcalculating the field data

12 Experiential Value of Critical Gap Roundabout is atype of classic unsignalized intersection Many researchersobtained different values of critical gap by observation ofvarious types of roundabouts HCM (see Highway CapacityManual 2000 [12]) recommends the value range of critical gap119905119888and follow-up headway 119905

119891as in Table 1

Siegloch (see Wu [13]) thought that critical gap could beevaluated by the following equation

119905119888= 1199050+ 05119905

119891 (1)

where 1199050is minimum accepted gap

Akcelik [14] obtained parameter values at the field round-abouts 119905

0= 373 s 119905

119891= 231 s and 119905

119888= 489 s He

found that the fixed critical gap could not be applicable to allroundabouts contrarily it fluctuated at the interval of (2288 s) [15]

Australia model (see Akcelik [15]) used fixed follow-upheadway and critical gap which are respectively 2 s and 4 sThe follow-up headway changes in the interval of (12 4 s)usually are treated as 2 s when calculating capacity or ana-lyzing operational character at roundabouts The follow-upheadway fluctuates at the interval of (18 32 s) in Switzerland(see Lertworawanich and Elefteriadou [16]) nevertheless theusual values are from 18 s to 30 s

Tanyel [17] found that the acceptable gap changed atthe interval of (454 618 s) based on the survey of sixroundaboutsThe following parameter values (see Tanyel andYayla [18]) are adopted whenminor road vehicles always waitto enter single-lane roundabouts critical gap is 35 s follow-up headway is 2 s and minimum headway is 18 s

2 Calculation Model of Critical Gap

It is difficult to measure critical gap directly Usually it can beestimated by accepted gaps and rejected gaps As mentionedbefore there are many calculation methods of critical gapsuch as regression method maximum likelihood method

Sieglochrsquos method Ashworthrsquos method Raff rsquos method Hard-ersrsquo method Hewittrsquos method Logit procedure and Probitprocedure Ashworthrsquos method and Raff rsquos method are listedas follows

21 Raff rsquos Method The critical lag 119871 is the size lag which hasthe property that the number of accepted lags shorter than119871 is the same as the number of rejected lags longer than 119871(Raff and Hart [8]) A similar definition was proposed byDrew [19] but for gaps rather than lags So critical gap can bederived from the cross point between the number of curvesof accepted gaps and rejected gaps

Raff rsquos method can be expressed as (2) (see Brilon et al[9]) Consider

1 minus 119865119903(119905) = 119865

119886(119905) (2)

where 119905 is headway of major stream 119865119886(119905) is cumulative

probability of accepted gap 119865119903(119905) is cumulative probability of

rejected gapRaff rsquos method is also called threshold method The flow

rate of major road has a prominent influence on critical gapvalueThemethod is used widely in many countries owing toits simplicity and practicality

22 Ashworthrsquos Method Based on the assumption that theheadway of major stream follows negative exponential dis-tribution and critical gap and the accepted gap follow normaldistribution Ashworth gave the calculation formula of criti-cal gap as follows

119905119888= 119905119886minus 1199021205902

119886 (3)

where 119905119888is average critical gap (s) 119902 is flow rate of major

stream (vehs) 119905119886is average accepted gap (s) and 120590

2

119886is

variance of accepted gaps (s2)Similarly Miller [7] gave the calculation equation (4)

based on the hypothesis that critical gap followed 120574 distribu-tion Consider

119905119888= 119905119886minus 1198811199011205902

119888

120590119888= 120590119886

119905119888

119905119886

(4)

where 1205902119888is variance of critical gap (s2)

The computer iteration can be applied to calculate criticalgap using (4) We calculated critical gap by the use of (3)

3 Critical Gap Based on Raffrsquos Definition

31 Assumption For simplicity the following special condi-tion is based on the assumption of independence betweenarrival times of the minor-stream vehicles and the ones of themajor-stream vehicles

Based on the assumption the distribution form of allheadway samples in major stream should be the same as thedistribution form of part of stochastic samples when minor-stream vehicles arrive before the intersection Therefore wecan simulate the headway distribution in the major stream

Computational Intelligence and Neuroscience 3

by using the latter samples These headway samples canbe divided into accepted headway and rejected headwaybecause all surveyed headway samples are related to acceptedmaneuver or rejectedmaneuver of theminor-streamvehiclesThe distribution of accepted headway and rejected headwaycan be modeled Their proportions can be calculated as totalaccepted coefficient and total rejected coefficient

The following two circumstances are discussed (1) Ingeneral circumstance critical gap is a random variable Basedon Raff rsquos definition the estimation of critical gap is denotedas 119888 (2) In special circumstance critical gap follows normal

distribution and can be estimated by the average value 119905119888and

variance 1205902 The relation between 119905119888and 119888is discussed

32 Definition of Variables The distribution of headwayis very important for the calculation of capacity in gapacceptance theory Based on the negative exponential dis-tribution of headway (M1) and shifted negative exponentialdistribution (M2) a bunched exponential distribution (M3)was proposed (Cowan [20]) Generally an M3 distribution isas follows

119891 (119905) = 120572120582119890minus120582(119905minus119905

119898)

119905 ge 119905119898

0 119905 lt 119905119898

119865 (119905) = 1 minus 120572119890

minus120582(119905minus119905119898)

119905 ge 119905119898

0 119905 lt 119905119898

(5)

where 119891(119905) is probability density function of headway inmajor stream 119865(119905) is cumulative probability function ofheadway in major stream 120582 is decay constant (vehs) 120582 =

120572119902(1 minus 119902119905119898) 119905119898is minimum headway in major stream (s) 120572

is proportion of free vehiclesProvided that the headway of major stream follows M3

distribution headway samples are extracted from major-stream headways when minor-stream vehicles arrive beforean intersection Some variables are defined as follows

120573119886is total accepted coefficient or the proportion of

accepted gap number119873119886to total gap number119873

120573119886= 119873119886119873

120573119903is total rejected coefficient or the proportion of

rejected gap number119873119903to total gap number119873

120573119903= 119873119903119873 the relation between 120573

119886and 120573

119903is

120573119886+ 120573119903= 1 (6)

33 First Circumstance Raff considered that the number ofrejected gaps larger than critical gap was equal to the numberof accepted gaps smaller than critical gap Based on theRaff rsquos definition of critical gap the equation can be directlyexpressed as

119873119886119865119886(119888) = 119873

119903[1 minus 119865

119903(119888)] (7)

then

119865119886(119888) =

120573119903

120573119886

[1 minus 119865119903(119888)] (8)

Asmentioned earlier Raff rsquos definition can be expressed as(2) Equation (2) is only the special circumstance of (8) when120573119886= 120573119903 So (8) is the real equationwhich strictly accords with

Raff rsquos definition We name it revised Raff rsquos equationBased on Raff rsquos definition of critical gap the proportion

of rejected gaps larger than critical gap is equal to theproportion of accepted gap smaller than critical gap because119873 is fixed Two proportions can be counteracted so the totalaccepted coefficient is equal to the accumulative probabilityof headway 119905 larger than critical gap The equation can bederived as follows

120573119886= 119875 119879 ge

119888 + 120573119886119865119886(119888) minus 120573119903[1 minus 119865

119903(119888)]

= 119875 119879 ge 119888 = int

infin

119888

119891 (119905) 119889119905 = 1 minus 119865 (119888) = 120572119890

minus120582(119888minus119905119898)

(9)

then

119888= 119905119898minus1

120582ln(

120573119886

120572) (10)

where 119875sdot is probability of gap intervalEquation (10) is based on the M3 distribution of headway

in major stream We name it M3 definition method Bothmethods can be seen in the reference (see Guo and Lin [21])

34 Second Circumstance and Relation between 119905119888and

119888

Some assumptions are needed (1) Critical gap follows normaldistribution 119905

119888sim 119885(119906 120590

2

) (2) The headway in major streamfollows M3 distribution (3) The distribution of critical gap isindependent of the headway distribution

For discriminating from the first circumstance the prob-ability density function of headway in major stream isdenoted as 119891

119879(119905) and 119891

119879(119905) = 119891(119905) The accumulative

probability function of headway is denoted as 119865119879(119905) The

probability density function and accumulative probabilityfunction of critical gap are denoted as 119891

119879119862(119905119888) and 119865

119879119862(119905119888)

separatelyIt has been mentioned as before that

120573119886= 119875 119905 ge 119905

119888 = 119875 119905 minus 119905

119888ge 0 (11)

Provided that 119911 = 119905 minus 119905119888 its probability density function

is 119891119885(119911) and accumulative probability function is 119865

119885(119911) 119911 ge

119905119898minus 119905119888and 119911 is continuous at 0 for 119905 ge 119905

119898 Consider

120573119886= 119875 119911 ge 0 = 1 minus 119875 (119911 lt 0)

= 1 minus 119875 119911 le 0 = 1 minus 119865119885(0)

120573119903= 119865119885(0)

(12)

119891119885(119911) can be derived by use of 119891

119879(119905) and 119891

119879119862(119905119888) Accord-

ing to the assumption of independence the distribution


of 119905119888can be derived by convolution formula about two

independent variables

119891119885(119911) = 119891

119879(119905) 119891119879119862(119905119888) = int

+infin

minusinfin

119891119879(119905119888+ 119911) 119891

119879119862(119905119888) 119889119905119888

= int

+infin

minusinfin

120572120582119890minus120582(119905119888+119911minus119905119898)1

radic2120587120590119890minus(119905119888minus119906)2

21205902

119889119905119888

=120572120582

radic2120587120590int

+infin

minusinfin

119890minus((1199052

119888minus2119906119905119888+1199062

+21205902

120582119905119888)21205902

)minus120582119911+120582119905119898 119889119905119888

=120572120582

radic2120587120590119890120582(119905119898minus119911minus119906)+(120590

2

1205822

2)

int

+infin

minusinfin

119890minus(119905119888minus119906minus120590

2

120582)2

21205902

119889119905119888

=120572120582


2

1205822

2)radic2120587120590

= 120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

(13)

119911 follows M3 distribution When 119911 = minus119906 + (12059021205822) + 119905119898

119865119885(119911) = int

minus119906+(1205902

1205822)+119905119898

minusinfin

119891119885(119911) 119889119911

= 1 minus 120572int

+infin

minus119906+(12059021205822)+119905119898

119891119885(119911) 119889119911

= 1 minus int

+infin

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911

= 1 + 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)100381610038161003816100381610038161003816

infin

minus119906+(12059021205822)+119905119898

= 1 minus 120572

(14)

119905 follows M3 distribution and

119875 119905 = 119905119898 = 1 minus 120572 (15)

Similarly

119875119911 = minus119906 +1205902

120582

2+ 119905119898 = 1 minus 120572 (16)

When 119911 ge 119905119898+ (1205902

1205822) minus 119906

119865119885(119911) = 1 minus 120572 + int

119911

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911

= 1 minus 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

(17)

So the accumulative probability function of 119911 can beshown as follows

119865119885(119911) =

1 minus 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119911 ge 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(18)

The probability density function of 119911 is

119891119885(119911) =

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119911 gt 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(19)

When 119911 = 0 119905 = 119905119888 So

119865119885(0) = 1 minus 120572119890

minus120582(119906minus(1205902

1205822)minus119905119898)

= 120573119903

120572119890minus120582(119906minus(120590

2

1205822)minus119905119898)

= 120573119886

119906 = 119905119898+1205902

120582

2minus1

120582ln(

120573119886

120572)

(20)

The average critical gap can be derived as

119905119888= 119906 = 119905

119898+1205902

120582

2minus1

120582ln(

120573119886

120572) = 119888+1205902

120582

2 (21)

Namely

119905119888= 119888+1205902

120582

2 (22)

It is obvious that 119905119888= 119888when 120590 = 0 The average critical

gap is equal to Raff rsquos critical gap which is similar to the firstcircumstance 119905

119888= 119888when 120590 = 0 the relation of them can be

expressed as (22)

4 Simulation of Critical Gap

41 Generation of Rejected Gaps and Accepted Gaps Theheadway which follows M3 distribution is simulated in orderto analyze various models of critical gap On the assumptionof independence between arrival times of the minor-streamvehicles and the ones of the major-stream vehicles theheadways of major stream are divided into two sets includingrejected gap set and accepted gap set Critical gap can becalculated by various methods using the same traffic flow

An example is listed to introduce the generation ofheadways in major stream which followM3 distributionTheexponential rejected proportion function is assumed (Guoand Lin [21]) So

120582

119903=120572 minus 120573119886

120573119886

(23)

where 119903 is rejected coefficientFor an unsignalized intersection 119902 = 025 vehs 120572 = 06

119905119898= 2 s and 119903 = 0365 Other parameters can be calculated

from (4) (6) and (23) 120582 = 03 120573119886= 033 and 120573

119903= 067 So

the accumulative probability function of headway in majorstream is

119865 (119905) = 1 minus 06119890

minus03(119905minus2)

119905 gt 2

04 119905 le 2(24)


Table 2 Emulated values of rejected gaps and accepted gaps

Order U1 Headway(initial)

Headway(119905 = 2 if 119905 lt 119905

119898) Rejected proportion U2 State Rejected gap Accepted gap

1 061 196 200 100 064 0 2002 040 337 337 061 028 0 3373 089 068 200 100 037 0 2004 026 485 485 035 067 1 4855 050 262 262 080 029 0 2626 036 370 370 054 050 0 3707 031 421 421 045 014 0 4218 075 124 200 100 050 0 2009 070 148 200 100 063 0 20010 091 062 200 100 009 0 20011 099 033 200 100 024 0 20012 038 351 351 058 082 1 35113 098 036 200 100 056 0 20014 034 391 391 050 004 0 39115 005 1016 1016 005 053 1 1016

If 1198801 follows uniform distribution 1198801 sim 119880(0 1) 1198801 =1 minus 06119890

minus03(119905minus2) and the headway 119905 can be derived as follows

119905 = 2 minus10

3ln(53)sdot(1minus1198801) (25)

It is obvious that 1 minus 1198801 sim 119880(0 1) then

119905 = 2 minus10

3ln(53)1198801 (26)

119905 is the variable of headway in major stream The headwaycan be simulated by use of ldquoRnd()rdquo function in Visual Basicsoftware Headways smaller than 2 s need to be changed into2 s which means bunched stream

The rejected probability of 119905 can be calculated by useof exponential rejected proportion function (see Guo andLin [21]) The state of rejection or acceptation for 119905 can besimulated by comparing the rejected probability and anotherstochastic value 1198802 from ldquoRnd()rdquo function If the rejectedprobability is larger than 1198802 the headway will be rejectedotherwise it will be acceptedThe rejection state is denoted as0 and the acceptation state is denoted as 1 A part of simulatedaccepted gaps and rejected gaps are listed in Table 2

42 Calculation of Critical Gap with Different Stochastic SeedsThe above simulation process can be realized by use of VBsoftware The input parameters include 120572 = 06 119905

119891= 3 s 119902 =

025 vehs and 119903 = 0365 The different stochastic sequencesare produced with different stochastic seeds from minus1 to minus10One thousand headways are simulated and critical gap can becalculated by various methods in Table 3

FromTable 3 the results of Ashworthrsquos method and Raff rsquosmethod are adjacent the results of M3 definition methodand revised Raff rsquos method are adjacent and larger thanthe former calculations The calculations of M3 definitionmethod and revised Raff rsquos method have smaller fluctuation

Table 3 Critical gap comparison of various methods with differentseeds (s) (119905

119898= 2 s 120572 = 06 119905

119891= 3 s and 119902 = 025 vehs)

Seed M3 definition Ashworth Raff Revised RaffEquation (10) Equation (3) Equation (2) Equation (8)

1 390 389 361 4072 400 376 367 4133 386 337 345 3854 406 307 353 4095 399 263 349 3916 412 380 349 4057 399 417 361 4058 395 372 361 3979 398 302 347 39910 384 293 333 393119905119888

397 344 353 4001205902 0085 0503 0101 0089

Max minusmin 027 153 034 028

and the standard deviations are separately 0085 s and 0089 sAshworthrsquos method has largest fluctuation with differentstochastic seeds and the standard deviation is 0503 s Theassumption of Ashworthrsquos method is that accepted gap andcritical gap follow normal distribution whereas it is difficultto satisfy the assumption for the field or simulated data

The calculation values of Raff rsquos method are smaller thanones of revised Raff rsquos method because the proportion of freevehicles in major stream is 025 and the ratio of total rejectedcoefficient and total accepted coefficient is larger than 1

43 Calculation of Critical Gap with Different Flow Rates Therelation between flow rate and proportion of free vehicles canbe treated as 120572 = 1 minus 119902119905

119898(see Wu [13]) When seed = minus1


Table 4 Critical gap comparison of various methods with different flow rates (119905119898= 2 s 119905

119891= 3 s 119903 = 0365 and seed = minus1)

Order 119902 (vehs) 120572M3 definition Ashworth Raff Revised RaffEquation (10) Equation (3) Equation (2) Equation (8)

1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)

M3 definitionAshworth

RaffRevised Raff

Flow rate q (vehs)

Figure 1 Comparison of critical gap in different traffic rate

other parameters are similar to the ones in Table 3 Criticalgaps with different flow rates are calculated as in Table 4 andthe corresponding curve is as in Figure 1

FromTable 4 and Figure 1 critical gaps of all themethodshave a trend of decrease with increase of flow rate Thetendency is accordant with field traffic flow When the majorroad has a low flow rate drivers of theminor road are inclinedto reject some lager gaps since there are many available largegaps so critical gap increases When major road has a highflow rate drivers are inclined to accept some smaller gapsfor the lack of available large gaps in major stream which isan important reason why drivers will take a risk to enter theintersection with the increase of waiting time so critical gapdecreases

Ashworthrsquos method and Raff rsquos method have larger fluc-tuation M3 definition method and revised Raff rsquos methodhave smaller fluctuation and their values are adjacent to therecommended range [41 46 s] in HCM

All the methods use the same simulated data neverthe-less eachmethod has theoretically different calculation valuebecause of their different assumptions M3 definitionmethodand revised Raff rsquos method are based on the gap acceptancetheory Revised Raff rsquos method can be widely applied M3definition method is based on the M3 distribution of major-stream headway Ashworthrsquos method has a rigorous condition

of normal distribution for critical gap and accepted gapwhichrestricts its application scope So M3 definition method andrevised Raff rsquos method are worthy of recommendation

5 Conclusion

Based on gap acceptance theory two new methods areproposed on the assumption of independence between arrivaltimes of minor-stream vehicles and the ones of major-streamvehicles New models are verified by simulation of headwaydata and comparison of various critical gap methods

Both M3 definition method and revised Raff rsquos methoduse total rejected coefficient 120573

119903 M3 definition method is

simple and valid which can conveniently be substituted intothe equations of capacity and delay Revised Raff rsquos methodhas more universal application than Raff rsquos method thecalculation value is accurate Both methods have accordantresults whereas Raff rsquos method and Ashworthrsquos method havelarger fluctuation under different circumstance Ashworthrsquosmethod needs to satisfy a rigorous assumption conditionM3definition method and revised Raff rsquos method are worthy ofrecommendation

Notations

The following symbols are used in this paper

119905 Headway between successive vehicles inmajor stream

119891(119905) Probability density function119865(119905) Cumulative distribution function of

headway120572 Proportion of free vehicles120582 Decay constant (pcus)119905119898 Minimum headway in major stream (s)

119905119888 Critical gap (s)1199050 Minimum accepted headway (s)

119905119891 Minimum follow-up headway in minor

stream (s)119902 Flow rate of major stream (vehs)119905119888 Average critical gap (s)1205902 Variance of critical gap (s2)119905119886 Average accepted gap (s)


1205902

119886 Variance of accepted gaps (s2)

119888 Estimation of critical gap in accord with

Raff rsquos definition120573119886 Total accepted coefficient the proportion

of accepted gap number to total gap num-ber

120573119903 Total rejected coefficient the proportion of

rejected gap number to total gap number119865119886(119905) Cumulative distribution function of ac-

cepted gap119865119903(119905) Cumulative distribution function of reject-

ed gap119903 Rejected coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the Basic ResearchGrantProject (2014-9059) from the Research Institute of HighwayMOT of China

References

[1] R T Luttinen ldquoCapacity and level of service at Finnishunsignalized intersectionsrdquo Finnra Reports Finnish RoadAdministration Helsinki Finland 2004

[2] AW Plank andE A Catchpole ldquoA general capacity formula foran uncontrolled intersectionrdquo Traffic Engineering and Controlvol 25 no 6 pp 327ndash329 1984

[3] A Polus S S Lazar and M Livneh ldquoCritical gap as a functionof waiting time in determining roundabout capacityrdquo Journal ofTransportation Engineering vol 129 no 5 pp 504ndash509 2003

[4] M M Hamed S M Easa and R R Batayneh ldquoDisaggregategap-acceptance model for unsignalized T-intersectionsrdquo Jour-nal of Transportation Engineering vol 123 no 1 pp 36ndash42 1997

[5] R Ashworth ldquoA note on the selection of gap acceptance criteriafor traffic simulation studiesrdquo Transportation Research vol 2no 2 pp 171ndash175 1968

[6] R Ashworth ldquoThe analysis and interpretation of gap acceptancedatardquo Transportation Science vol 4 no 3 pp 270ndash280 1970

[7] A JMiller ldquoAnote on the analysis of gapmdashacceptance in trafficrdquoJournal of the Royal Statistical Society vol 23 no 1 pp 66ndash731974

[8] M S Raff and J W Hart A Volume Warrant for Urban StopSigns Eno Foundation for Highway Traffic Control SaugatuckConn USA 1950

[9] 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

[10] R H Hewitt ldquoMeasuring critical gaprdquo Transportation Sciencevol 17 no 1 pp 87ndash109 1983

[11] Z Tian M Vandehey BW Robinson et al ldquoImplementing themaximum likelihoodmethodology tomeasure a driverrsquos criticalgaprdquoTransportation Research Part A Policy and Practice vol 33no 3-4 pp 187ndash197 1999

[12] Transportation Research Board Highway Capacity ManualNational Research Council Washington DC USA 2000

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

[14] R AkcelikTheSimulation of Priority Intersection Systemand theFormulation of Vehicular Delay Istanbul Technical UniversityIstanbul Turkey 1987 (Turkish)

[15] R Akcelik ldquoA roundabout case study comparing capacityestimates from alternative analytical modelsrdquo in Proceedings ofthe 2nd Urban Street Symposium Anaheim Calif USA 2003

[16] P Lertworawanich and L Elefteriadou ldquoGeneralized capacityestimation model for weaving areasrdquo Journal of TransportationEngineering vol 133 no 3 pp 166ndash179 2007

[17] S Tanyel A capacity calculation method for roundabouts inTurkey [PhD thesis] Istanbul Technical University IstanbulTurkey 2001 (Turkish)

[18] S Tanyel and N Yayla ldquoA discussion on the parameters ofCowan M3 distribution for Turkeyrdquo Transportation ResearchPart A Policy and Practice vol 37 no 2 pp 129ndash143 2003

[19] R S Drew Traffic FlowTheory and Control McGraw-Hill NewYork NY USA 1968

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

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

Submit your manuscripts athttpwwwhindawicom

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Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

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Biomedical Imaging


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Advances in


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Human-ComputerInteraction

Advances in

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Table 1 Critical gap and follow-up headway for roundabouts (HCM2000 [12]) (s)

119905119888

119905119891

Upper bound 41 26Lower bound 46 31

in free flow for example lag method can be regarded asthe theoretical reference value Logit method is similar tothe classical logit models of transportation planning Probitprocedure cannot consider critical gap as normal distributiondespite its obvious variance Hewittrsquos method [10] has afarther development than Probit procedure and even has cor-responding calculation software of GAPTIM and PROBITThe maximum likelihood method [11] estimates the averagevalue and variance on the assumption of normal distributionof accepted gap maximum rejected gap and critical gapRaff and Hart [8] found the maximum likelihood methodand Hewittrsquos method had an accurate result by analyzing andcalculating the field data

12 Experiential Value of Critical Gap Roundabout is atype of classic unsignalized intersection Many researchersobtained different values of critical gap by observation ofvarious types of roundabouts HCM (see Highway CapacityManual 2000 [12]) recommends the value range of critical gap119905119888and follow-up headway 119905

119891as in Table 1

Siegloch (see Wu [13]) thought that critical gap could beevaluated by the following equation

119905119888= 1199050+ 05119905

119891 (1)

where 1199050is minimum accepted gap

Akcelik [14] obtained parameter values at the field round-abouts 119905

0= 373 s 119905

119891= 231 s and 119905

119888= 489 s He

found that the fixed critical gap could not be applicable to allroundabouts contrarily it fluctuated at the interval of (2288 s) [15]

Australia model (see Akcelik [15]) used fixed follow-upheadway and critical gap which are respectively 2 s and 4 sThe follow-up headway changes in the interval of (12 4 s)usually are treated as 2 s when calculating capacity or ana-lyzing operational character at roundabouts The follow-upheadway fluctuates at the interval of (18 32 s) in Switzerland(see Lertworawanich and Elefteriadou [16]) nevertheless theusual values are from 18 s to 30 s

Tanyel [17] found that the acceptable gap changed atthe interval of (454 618 s) based on the survey of sixroundaboutsThe following parameter values (see Tanyel andYayla [18]) are adopted whenminor road vehicles always waitto enter single-lane roundabouts critical gap is 35 s follow-up headway is 2 s and minimum headway is 18 s

2 Calculation Model of Critical Gap

It is difficult to measure critical gap directly Usually it can beestimated by accepted gaps and rejected gaps As mentionedbefore there are many calculation methods of critical gapsuch as regression method maximum likelihood method

Sieglochrsquos method Ashworthrsquos method Raff rsquos method Hard-ersrsquo method Hewittrsquos method Logit procedure and Probitprocedure Ashworthrsquos method and Raff rsquos method are listedas follows

21 Raff rsquos Method The critical lag 119871 is the size lag which hasthe property that the number of accepted lags shorter than119871 is the same as the number of rejected lags longer than 119871(Raff and Hart [8]) A similar definition was proposed byDrew [19] but for gaps rather than lags So critical gap can bederived from the cross point between the number of curvesof accepted gaps and rejected gaps

Raff rsquos method can be expressed as (2) (see Brilon et al[9]) Consider

1 minus 119865119903(119905) = 119865

119886(119905) (2)

where 119905 is headway of major stream 119865119886(119905) is cumulative

probability of accepted gap 119865119903(119905) is cumulative probability of

rejected gapRaff rsquos method is also called threshold method The flow

rate of major road has a prominent influence on critical gapvalueThemethod is used widely in many countries owing toits simplicity and practicality

22 Ashworthrsquos Method Based on the assumption that theheadway of major stream follows negative exponential dis-tribution and critical gap and the accepted gap follow normaldistribution Ashworth gave the calculation formula of criti-cal gap as follows

119905119888= 119905119886minus 1199021205902

119886 (3)

where 119905119888is average critical gap (s) 119902 is flow rate of major

stream (vehs) 119905119886is average accepted gap (s) and 120590

2

119886is

variance of accepted gaps (s2)Similarly Miller [7] gave the calculation equation (4)

based on the hypothesis that critical gap followed 120574 distribu-tion Consider

119905119888= 119905119886minus 1198811199011205902

119888

120590119888= 120590119886

119905119888

119905119886

(4)

where 1205902119888is variance of critical gap (s2)

The computer iteration can be applied to calculate criticalgap using (4) We calculated critical gap by the use of (3)

3 Critical Gap Based on Raffrsquos Definition

31 Assumption For simplicity the following special condi-tion is based on the assumption of independence betweenarrival times of the minor-stream vehicles and the ones of themajor-stream vehicles

Based on the assumption the distribution form of allheadway samples in major stream should be the same as thedistribution form of part of stochastic samples when minor-stream vehicles arrive before the intersection Therefore wecan simulate the headway distribution in the major stream







119891 (119905) = 120572120582119890minus120582(119905minus119905

119898)

119905 ge 119905119898

0 119905 lt 119905119898

119865 (119905) = 1 minus 120572119890

minus120582(119905minus119905119898)

119905 ge 119905119898

0 119905 lt 119905119898

(5)







120573119886= 119873119886119873




119886and 120573

119903is

120573119886+ 120573119903= 1 (6)


119873119886119865119886(119888) = 119873

119903[1 minus 119865

119903(119888)] (7)

then

119865119886(119888) =

120573119903

120573119886

[1 minus 119865119903(119888)] (8)




120573119886= 119875 119879 ge

119888 + 120573119886119865119886(119888) minus 120573119903[1 minus 119865

119903(119888)]

= 119875 119879 ge 119888 = int

infin

119888

119891 (119905) 119889119905 = 1 minus 119865 (119888) = 120572119890

minus120582(119888minus119905119898)

(9)

then

119888= 119905119898minus1

120582ln(

120573119886

120572) (10)




119888


119888sim 119885(119906 120590

2



119879(119905) and 119891




119879119862(119905119888) and 119865

119879119862(119905119888)


120573119886= 119875 119905 ge 119905

119888 = 119875 119905 minus 119905

119888ge 0 (11)



119885(119911) 119911 ge


119898 Consider

120573119886= 119875 119911 ge 0 = 1 minus 119875 (119911 lt 0)

= 1 minus 119875 119911 le 0 = 1 minus 119865119885(0)

120573119903= 119865119885(0)

(12)


119879(119905) and 119891

119879119862(119905119888) Accord-





119891119885(119911) = 119891

119879(119905) 119891119879119862(119905119888) = int

+infin

minusinfin

119891119879(119905119888+ 119911) 119891

119879119862(119905119888) 119889119905119888

= int

+infin

minusinfin

120572120582119890minus120582(119905119888+119911minus119905119898)1


21205902

119889119905119888

=120572120582

radic2120587120590int

+infin

minusinfin

119890minus((1199052

119888minus2119906119905119888+1199062

+21205902

120582119905119888)21205902

)minus120582119911+120582119905119898 119889119905119888

=120572120582


2

1205822

2)

int

+infin

minusinfin


2

120582)2

21205902

119889119905119888

=120572120582


2

1205822

2)radic2120587120590

= 120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

(13)


119865119885(119911) = int

minus119906+(1205902

1205822)+119905119898

minusinfin

119891119885(119911) 119889119911

= 1 minus 120572int

+infin

minus119906+(12059021205822)+119905119898

119891119885(119911) 119889119911

= 1 minus int

+infin

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911

= 1 + 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)100381610038161003816100381610038161003816

infin

minus119906+(12059021205822)+119905119898

= 1 minus 120572

(14)


119875 119905 = 119905119898 = 1 minus 120572 (15)

Similarly

119875119911 = minus119906 +1205902

120582

2+ 119905119898 = 1 minus 120572 (16)

When 119911 ge 119905119898+ (1205902

1205822) minus 119906

119865119885(119911) = 1 minus 120572 + int

119911

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911


2

1205822)minus119905119898)

(17)


119865119885(119911) =


2

1205822)minus119905119898)

119911 ge 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(18)


119891119885(119911) =

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119911 gt 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(19)

When 119911 = 0 119905 = 119905119888 So

119865119885(0) = 1 minus 120572119890

minus120582(119906minus(1205902

1205822)minus119905119898)

= 120573119903

120572119890minus120582(119906minus(120590

2

1205822)minus119905119898)

= 120573119886

119906 = 119905119898+1205902

120582

2minus1

120582ln(

120573119886

120572)

(20)


119905119888= 119906 = 119905

119898+1205902

120582

2minus1

120582ln(

120573119886

120572) = 119888+1205902

120582

2 (21)

Namely

119905119888= 119888+1205902

120582

2 (22)




expressed as (22)




120582

119903=120572 minus 120573119886

120573119886

(23)



from (4) (6) and (23) 120582 = 03 120573119886= 033 and 120573

119903= 067 So


119865 (119905) = 1 minus 06119890


119905 gt 2

04 119905 le 2(24)




Headway(119905 = 2 if 119905 lt 119905


1 061 196 200 100 064 0 2002 040 337 337 061 028 0 3373 089 068 200 100 037 0 2004 026 485 485 035 067 1 4855 050 262 262 080 029 0 2626 036 370 370 054 050 0 3707 031 421 421 045 014 0 4218 075 124 200 100 050 0 2009 070 148 200 100 063 0 20010 091 062 200 100 009 0 20011 099 033 200 100 024 0 20012 038 351 351 058 082 1 35113 098 036 200 100 056 0 20014 034 391 391 050 004 0 39115 005 1016 1016 005 053 1 1016



119905 = 2 minus10

3ln(53)sdot(1minus1198801) (25)


119905 = 2 minus10

3ln(53)1198801 (26)




119891= 3 s 119902 =




119898= 2 s 120572 = 06 119905

119891= 3 s and 119902 = 025 vehs)


1 390 389 361 4072 400 376 367 4133 386 337 345 3854 406 307 353 4095 399 263 349 3916 412 380 349 4057 399 417 361 4058 395 372 361 3979 398 302 347 39910 384 293 333 393119905119888

397 344 353 4001205902 0085 0503 0101 0089

Max minusmin 027 153 034 028







119891= 3 s 119903 = 0365 and seed = minus1)


1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)


RaffRevised Raff

Flow rate q (vehs)







5 Conclusion





Notations









1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









119891 (119905) = 120572120582119890minus120582(119905minus119905

119898)

119905 ge 119905119898

0 119905 lt 119905119898

119865 (119905) = 1 minus 120572119890

minus120582(119905minus119905119898)

119905 ge 119905119898

0 119905 lt 119905119898

(5)







120573119886= 119873119886119873




119886and 120573

119903is

120573119886+ 120573119903= 1 (6)


119873119886119865119886(119888) = 119873

119903[1 minus 119865

119903(119888)] (7)

then

119865119886(119888) =

120573119903

120573119886

[1 minus 119865119903(119888)] (8)




120573119886= 119875 119879 ge

119888 + 120573119886119865119886(119888) minus 120573119903[1 minus 119865

119903(119888)]

= 119875 119879 ge 119888 = int

infin

119888

119891 (119905) 119889119905 = 1 minus 119865 (119888) = 120572119890

minus120582(119888minus119905119898)

(9)

then

119888= 119905119898minus1

120582ln(

120573119886

120572) (10)




119888


119888sim 119885(119906 120590

2



119879(119905) and 119891




119879119862(119905119888) and 119865

119879119862(119905119888)


120573119886= 119875 119905 ge 119905

119888 = 119875 119905 minus 119905

119888ge 0 (11)



119885(119911) 119911 ge


119898 Consider

120573119886= 119875 119911 ge 0 = 1 minus 119875 (119911 lt 0)

= 1 minus 119875 119911 le 0 = 1 minus 119865119885(0)

120573119903= 119865119885(0)

(12)


119879(119905) and 119891

119879119862(119905119888) Accord-





119891119885(119911) = 119891

119879(119905) 119891119879119862(119905119888) = int

+infin

minusinfin

119891119879(119905119888+ 119911) 119891

119879119862(119905119888) 119889119905119888

= int

+infin

minusinfin

120572120582119890minus120582(119905119888+119911minus119905119898)1


21205902

119889119905119888

=120572120582

radic2120587120590int

+infin

minusinfin

119890minus((1199052

119888minus2119906119905119888+1199062

+21205902

120582119905119888)21205902

)minus120582119911+120582119905119898 119889119905119888

=120572120582


2

1205822

2)

int

+infin

minusinfin


2

120582)2

21205902

119889119905119888

=120572120582


2

1205822

2)radic2120587120590

= 120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

(13)


119865119885(119911) = int

minus119906+(1205902

1205822)+119905119898

minusinfin

119891119885(119911) 119889119911

= 1 minus 120572int

+infin

minus119906+(12059021205822)+119905119898

119891119885(119911) 119889119911

= 1 minus int

+infin

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911

= 1 + 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)100381610038161003816100381610038161003816

infin

minus119906+(12059021205822)+119905119898

= 1 minus 120572

(14)


119875 119905 = 119905119898 = 1 minus 120572 (15)

Similarly

119875119911 = minus119906 +1205902

120582

2+ 119905119898 = 1 minus 120572 (16)

When 119911 ge 119905119898+ (1205902

1205822) minus 119906

119865119885(119911) = 1 minus 120572 + int

119911

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911


2

1205822)minus119905119898)

(17)


119865119885(119911) =


2

1205822)minus119905119898)

119911 ge 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(18)


119891119885(119911) =

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119911 gt 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(19)

When 119911 = 0 119905 = 119905119888 So

119865119885(0) = 1 minus 120572119890

minus120582(119906minus(1205902

1205822)minus119905119898)

= 120573119903

120572119890minus120582(119906minus(120590

2

1205822)minus119905119898)

= 120573119886

119906 = 119905119898+1205902

120582

2minus1

120582ln(

120573119886

120572)

(20)


119905119888= 119906 = 119905

119898+1205902

120582

2minus1

120582ln(

120573119886

120572) = 119888+1205902

120582

2 (21)

Namely

119905119888= 119888+1205902

120582

2 (22)




expressed as (22)




120582

119903=120572 minus 120573119886

120573119886

(23)



from (4) (6) and (23) 120582 = 03 120573119886= 033 and 120573

119903= 067 So


119865 (119905) = 1 minus 06119890


119905 gt 2

04 119905 le 2(24)




Headway(119905 = 2 if 119905 lt 119905


1 061 196 200 100 064 0 2002 040 337 337 061 028 0 3373 089 068 200 100 037 0 2004 026 485 485 035 067 1 4855 050 262 262 080 029 0 2626 036 370 370 054 050 0 3707 031 421 421 045 014 0 4218 075 124 200 100 050 0 2009 070 148 200 100 063 0 20010 091 062 200 100 009 0 20011 099 033 200 100 024 0 20012 038 351 351 058 082 1 35113 098 036 200 100 056 0 20014 034 391 391 050 004 0 39115 005 1016 1016 005 053 1 1016



119905 = 2 minus10

3ln(53)sdot(1minus1198801) (25)


119905 = 2 minus10

3ln(53)1198801 (26)




119891= 3 s 119902 =




119898= 2 s 120572 = 06 119905

119891= 3 s and 119902 = 025 vehs)


1 390 389 361 4072 400 376 367 4133 386 337 345 3854 406 307 353 4095 399 263 349 3916 412 380 349 4057 399 417 361 4058 395 372 361 3979 398 302 347 39910 384 293 333 393119905119888

397 344 353 4001205902 0085 0503 0101 0089

Max minusmin 027 153 034 028







119891= 3 s 119903 = 0365 and seed = minus1)


1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)


RaffRevised Raff

Flow rate q (vehs)







5 Conclusion





Notations









1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119891119885(119911) = 119891

119879(119905) 119891119879119862(119905119888) = int

+infin

minusinfin

119891119879(119905119888+ 119911) 119891

119879119862(119905119888) 119889119905119888

= int

+infin

minusinfin

120572120582119890minus120582(119905119888+119911minus119905119898)1


21205902

119889119905119888

=120572120582

radic2120587120590int

+infin

minusinfin

119890minus((1199052

119888minus2119906119905119888+1199062

+21205902

120582119905119888)21205902

)minus120582119911+120582119905119898 119889119905119888

=120572120582


2

1205822

2)

int

+infin

minusinfin


2

120582)2

21205902

119889119905119888

=120572120582


2

1205822

2)radic2120587120590

= 120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

(13)


119865119885(119911) = int

minus119906+(1205902

1205822)+119905119898

minusinfin

119891119885(119911) 119889119911

= 1 minus 120572int

+infin

minus119906+(12059021205822)+119905119898

119891119885(119911) 119889119911

= 1 minus int

+infin

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911

= 1 + 120572119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)100381610038161003816100381610038161003816

infin

minus119906+(12059021205822)+119905119898

= 1 minus 120572

(14)


119875 119905 = 119905119898 = 1 minus 120572 (15)

Similarly

119875119911 = minus119906 +1205902

120582

2+ 119905119898 = 1 minus 120572 (16)

When 119911 ge 119905119898+ (1205902

1205822) minus 119906

119865119885(119911) = 1 minus 120572 + int

119911

minus119906+(12059021205822)+119905119898

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119889119911


2

1205822)minus119905119898)

(17)


119865119885(119911) =


2

1205822)minus119905119898)

119911 ge 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(18)


119891119885(119911) =

120572120582119890minus120582(119911+119906minus(120590

2

1205822)minus119905119898)

119911 gt 119905119898+1205902

120582

2minus 119906

0 119911 lt 119905119898+1205902

120582

2minus 119906

(19)

When 119911 = 0 119905 = 119905119888 So

119865119885(0) = 1 minus 120572119890

minus120582(119906minus(1205902

1205822)minus119905119898)

= 120573119903

120572119890minus120582(119906minus(120590

2

1205822)minus119905119898)

= 120573119886

119906 = 119905119898+1205902

120582

2minus1

120582ln(

120573119886

120572)

(20)


119905119888= 119906 = 119905

119898+1205902

120582

2minus1

120582ln(

120573119886

120572) = 119888+1205902

120582

2 (21)

Namely

119905119888= 119888+1205902

120582

2 (22)




expressed as (22)




120582

119903=120572 minus 120573119886

120573119886

(23)



from (4) (6) and (23) 120582 = 03 120573119886= 033 and 120573

119903= 067 So


119865 (119905) = 1 minus 06119890


119905 gt 2

04 119905 le 2(24)




Headway(119905 = 2 if 119905 lt 119905


1 061 196 200 100 064 0 2002 040 337 337 061 028 0 3373 089 068 200 100 037 0 2004 026 485 485 035 067 1 4855 050 262 262 080 029 0 2626 036 370 370 054 050 0 3707 031 421 421 045 014 0 4218 075 124 200 100 050 0 2009 070 148 200 100 063 0 20010 091 062 200 100 009 0 20011 099 033 200 100 024 0 20012 038 351 351 058 082 1 35113 098 036 200 100 056 0 20014 034 391 391 050 004 0 39115 005 1016 1016 005 053 1 1016



119905 = 2 minus10

3ln(53)sdot(1minus1198801) (25)


119905 = 2 minus10

3ln(53)1198801 (26)




119891= 3 s 119902 =




119898= 2 s 120572 = 06 119905

119891= 3 s and 119902 = 025 vehs)


1 390 389 361 4072 400 376 367 4133 386 337 345 3854 406 307 353 4095 399 263 349 3916 412 380 349 4057 399 417 361 4058 395 372 361 3979 398 302 347 39910 384 293 333 393119905119888

397 344 353 4001205902 0085 0503 0101 0089

Max minusmin 027 153 034 028







119891= 3 s 119903 = 0365 and seed = minus1)


1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)


RaffRevised Raff

Flow rate q (vehs)







5 Conclusion





Notations









1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Headway(119905 = 2 if 119905 lt 119905


1 061 196 200 100 064 0 2002 040 337 337 061 028 0 3373 089 068 200 100 037 0 2004 026 485 485 035 067 1 4855 050 262 262 080 029 0 2626 036 370 370 054 050 0 3707 031 421 421 045 014 0 4218 075 124 200 100 050 0 2009 070 148 200 100 063 0 20010 091 062 200 100 009 0 20011 099 033 200 100 024 0 20012 038 351 351 058 082 1 35113 098 036 200 100 056 0 20014 034 391 391 050 004 0 39115 005 1016 1016 005 053 1 1016



119905 = 2 minus10

3ln(53)sdot(1minus1198801) (25)


119905 = 2 minus10

3ln(53)1198801 (26)




119891= 3 s 119902 =




119898= 2 s 120572 = 06 119905

119891= 3 s and 119902 = 025 vehs)


1 390 389 361 4072 400 376 367 4133 386 337 345 3854 406 307 353 4095 399 263 349 3916 412 380 349 4057 399 417 361 4058 395 372 361 3979 398 302 347 39910 384 293 333 393119905119888

397 344 353 4001205902 0085 0503 0101 0089

Max minusmin 027 153 034 028







119891= 3 s 119903 = 0365 and seed = minus1)


1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)


RaffRevised Raff

Flow rate q (vehs)







5 Conclusion





Notations









1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119891= 3 s 119903 = 0365 and seed = minus1)


1 005 09 422 492 559 4212 01 08 458 469 487 4533 015 07 432 417 433 4474 02 06 404 381 391 4275 025 05 385 332 347 3996 03 04 395 314 305 4057 035 03 396 340 305 3918 04 02 373 310 235 4019 045 01 383 310 245 403

10

20

30

40

50

60

70

005 01 015 02 025 03 035 04 045

Criti

cal g

ap (s

)


RaffRevised Raff

Flow rate q (vehs)







5 Conclusion





Notations









1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1205902











Acknowledgment


References





























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article estimation of critical gap based on raff

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