Modelling urban travel time variability with the Burr regression technique

Transcription

1 Ausralasian Transpor Research Forum 2011 Proceedings Sepember 2011, Adelaide, Ausralia Publicaion websie: hp:// Modelling urban ravel ime variabiliy wih he Burr regression echnique Susilawai 1, Michael A P Taylor 1, Sekhar Somenahalli 1 1 Barbara Hardy Insiue, Universiy of Souh Ausralia for correspondence: sussy002@mymail.unisa.edu.au Absrac The need for more reliable ravel ime in urban areas in order o provide beer ranspor service o he communiy has araced many sudies o model ravel ime reliabiliy and variabiliy. The ravel ime disribuion is basic knowledge for his modelling, and sudies o fi coninuous parameric disribuions o ravel ime disribuion have been conduced since he early 1950s. Two ses of empirical ravel ime daa colleced by GPS equipped vehicles in Adelaide indicae ha ravel ime disribuions are posiively skewed and have long upper ails. The Burr disribuion has been found o provide a good fi o he daa. Uilising he Burr disribuion properies and he Burr regression echnique, his paper models he Adelaide urban arerial ravel ime variabiliy by considering raffic variables such as link lengh, congesion index and degree of sauraion. This sudy suggess how o fill some curren gaps in ravel ime variabiliy modelling, especially for urban arerial roads. I should also be useful for furher ravel ime variabiliy sudies such as he valuaion of ravel ime variabiliy effecs. 1. Inroducion Travel ime is one of he mos imporan variables in assessing raffic sysems performance. Boh ravellers and raffic engineers would like o have high qualiy raffic performance including shor ravel imes, as a resul, here are many raffic demand managemen iniiaives ha have been implemened o achieve his goal. However, raffic sysems are complex and have a sochasic naure, for example shor and long incidens can disrup performance and lead o high ravel ime. On a daily basis on urban arerial roads, he flucuaion of demand and supply will also grealy affec he raffic performance in general and he ravel ime in paricular. Given he need for more reliable ravel ime in urban areas in order o provide beer ranspor service o he communiy, he variabiliy of ravel ime by ime of day, day of week or even due o he seasonal facors has araced many sudies. Those sudies have modelled and assessed he ravel ime variabiliy boh in urban arerial roads and freeway by fiing coninuous parameric disribuions such as Normal and Log Normal disribuion o ravel ime disribuion (Faouzi and Maurin, 2007). Having he bes fi of disribuion, he variabiliy of ravel ime can be measured by using is properies such as he sandard deviaion and he coefficien of variaion. Given he rapid developmen of raffic daa collecion echniques, he recen sudies no only model he ravel ime variabiliy bu also he relaionship beween he ravel ime variabiliy merics, he raffic parameers such as link lengh, raffic flow, capaciy, congesion index and also he occurrence of incidens using he regression echnique (Li e al., 2006, Eliasson, 2006, 2007). Those sudies have significanly increased he insigh abou he naure of he ravel ime and he possible facors ha migh influence is variabiliy. Travel ime variabiliy merics ha measure he dispersion of he acual ravel ime from he preferred ravel ime daa have been inroduced. However, here is a subjecive judgmen in selecing he preferred ravel ime. Some sudies ha assume ravel ime variabiliy daa are symmeric have used he mean ravel ime as he preferred ravel ime and he sandard deviaion as he ravel ime variabiliy meric. Oher sudies have suggesed ha ravel ime 1

2 ATRF 2011 Proceedings daa are skewed and have a long upper ail, have ended o use he median (50 h percenile) ravel ime and difference beween he 90 h percenile and he 50 h percenile (quanile range) as preferred ravel ime and is variabiliy respecively (Lin and Zuylen, 2006). Mos previous sudies assumed ha he Normal and Log Normal disribuions are he bes fi disribuion o he ravel ime daa, even hough recen ravel ime sudies have suggesed ha he ravel ime variabiliy disribuions are more likely o have long ails and very posiive skew. Susilawai e al. (2010) confirmed ha he Burr disribuion has been found o provide a good fi o he empirical ravel ime daa colleced a wo urban arerial roads in Adelaide. The burr disribuion has wo parameer, c and k respecively. I has been widely used in reliabiliy engineering and acuarial science. I was found he c parameer of he Burr disribuion give shape of he disribuion (he lower value of c he sharper fall of cdf curve). Previous sudies also found ha he Burr disribuion is very sensiive o he c parameer, consequenly his parameer plays a very imporan role in he disribuion properies. The closed form and algebraic racabiliy of his disribuion provide flexibiliy in shape of he pdf. The closed form makes he calculaion of is perceniles easy o do. Once he c and k parameers have been esimaed, he percenile can be esimaed easily as well. Thus, his sudy invesigaes he ravel ime variabiliy of wo urban arerial roads in Adelaide by uilising he properies of Burr disribuion. In corresponding o he facors ha migh affec he ravel ime variabiliy, i has been shown ha he raffic variables play an imporan role in influencing he ravel ime variabiliy. Recen sudies by Eliasson (2006), Peer e al. (2009) and Black and Chin (2009) developed eiher linear or non linear ravel ime variabiliy formulas which were formed by ravel ime variabiliy merics including he sandard deviaion and coefficien of variaion and he raffic variables. Those sudies suggesed ha he link lengh, free flow speed and capaciy are he mos imporan facors in ravel ime variabiliy. Given he consideraion ha he c parameer shapes he Burr disribuion densiy funcion and deermine he Burr disribuion s properies including he mean, median and perceniles his sudy herefore observe he influence of he raffic variables on he c parameer using he Burr regression echnique. The Burr regression echnique suggesed ha he c parameer allows o vary wih he explanaory variables (e.g speed, volume, capaciy, degree of sauraion). This sudy herefore replaces he c parameer wih he exponenial funcion of degree of sauraion daa and used his funcion in parameer esimaion. I was found ha he proposed echnique allows o gain insigh he influence of degree of sauraion associaed wih he shape of he burr disribuion and can be also useful for furher daa analysis in corresponding o ravel ime variabiliy and reliabiliy valuing. The nex secion reviews he pas sudies abou he ravel ime variabiliy modelling and how hose sudies esimaed he influence of he raffic variables o he ravel ime variabiliy. The hird secion discusses he Burr disribuion and is properies and how o maximise hese properies o develop he ravel ime variabiliy merics. This secion also reviews he Burr regression echnique and how o incorporae his echnique o he ravel ime variabiliy modelling. The fourh secion is abou he sudy area, ravel ime daa collecion and briefly discuss abou he Sydney Coordinaed Adapive Traffic Sysems (SCATS) daa exracion. The fifh secion is daa analysis ha covers he uilisaion of SCATS daa in his modelling and he discussion of he daa analysis. The las secion provides conclusions and direcions for furher research. 2. Travel ime variabiliy modelling Travel ime is he produc of he complex raffic sysems varies across ime of day and day of week (Oh and Chung, 2006, Emam and Al-Deek, 2006, Zhang e al., 2007). Day by day ravel ime sudies have confirmed ha daily ravel ime generally has wo disinc peaks 2

3 Modelling urban ravel ime variabiliy wih he Burr regression echnique which are morning peak (e.g. beween 7 and 9AM) and afernoon peak, perhaps 4PM o 7PM (Zhang e al., 2007). Those sudies also proven ha Monday and Friday ravel ime paern differ o oher weekdays ravel ime paern. Given he same period of ime, he weekend ravel ime is much differen o he weekdays ravel ime (Emam and Al-Deek, 2006). More ineresingly, he holiday season ravel ime commonly also differs from he non holiday season ravel ime, suggesing ha seasonal facors also have influence. In recen imes here have been several sudies abou ravel ime variabiliy, which have quanified no only he variabiliy bu also considered he facors ha migh conribue o ravel ime variabiliy. The impac of variabiliy on raveller behaviour is also of ineres. How variabiliy affecs raveller rip managemen such as deparure ime, roue and mode choices, and how he variaions may influence he qualiy of raffic performance can be explored using sae preference (SP) and reveal preference (RP) surveys. Iniial research used he normal disribuion properies such as mean and sandard deviaion o measure ravel ime variabiliy. To invesigae he role of raffic variables on ravel ime variabiliy, previous sudies develop models of he mean and sandard deviaion as funcions of hose raffic variables. For his modelling mehod, muliple linear regressions is he mos commonly used echnique because i is easy o apply ye can deliver good resuls. Back in lae 1970s, he sudies by Herman and Lam (1977) in Deroi, Richardson and Taylor (1978) in Melbourne Ausralia and Polus (1979) in Michigan can be considered as he pioneer sudies on ravel ime variabiliy. Those sudies used similar echniques o collec ravel ime daa and buil simple linear regression models of he variabiliy. Herman and Lam (1974) presened saisical analysis for o and from work rip imes on 25 roues in Deroi, colleced over 20 monhs. Each rip ime was he sum of sop ime and cruise ime and i was found ha he rip ime was linear in boh sop ime and cruise ime. This sudy found ha on average from work rip ime daa are longer han o work rip ime daa. The o work rip ime hisogram ended o follow a normal disribuion while from work rip ime hisogram was more like a uniform disribuion. However, as his sudy did no conduc he goodness of fi es, for simpliciy purposes, hey assumed all he ravel ime disribuions followed he normal disribuion. Moreover, a model of he mean, sandard deviaion and coefficien of variaion was been developed using he power and linear regression funcion. Ineresingly, i was found ha he linear funcion gave beer resuls han power funcion. Wih a similar objecive o he Herman and Lam s work, Polus (1979) also colleced he ravel ime daa in Michigan. The main purpose was o indicae he degree of ravel ime reliabiliy by observing he variabiliies in ravel imes. Insead of direcly using he sandard deviaion as he ravel ime variabiliy merics, he suggesed ha he ravel ime reliabiliy funcion is he inverse of he sandard deviaion. In his way, he more reliable ravel ime roues would have a larger value of he reliabiliy index, Richardson and Taylor (1978) colleced and analysed longiudinal ravel ime daa in Melbourne. They assessed he correlaions beween ravel imes on successive secions of he sudy roue, and developed relaionships beween he ravel ime variabiliy and he level of congesion. They concluded ha ravel imes on a link were independen of hose on oher links along he roue, and ha he observed ravel ime variabiliy migh be represened by a lognormal disribuion In parallel o rapid developmen of daa collecion echniques and he ongoing growh in compuer capabiliy o deal wih huge daa ses during he las decade, here are number of sudies ha have looked in more deail a ravel ime daa. The auomaic vehicle idenificaion (AVI) echnology allows he raffic engineer o record he ime when a vehicle passes a sar poin and he ime when he vehicle leaves he sudy area. The ime differences beween sar and end imes can be used as he ravel ime beween hese poins. In-vehicle GPS sysems have ransformed ravel ime collecion echniques. GPS equipped axi and bus flees have become major sources of ravel ime daa. The ravel ime daa allow he raffic engineer o obain no only vehicle o vehicle ravel ime variabiliy bu 3

4 ATRF 2011 Proceedings also differen roues and differen ime period ravel ime variabiliies, due o he abiliy of his echnique o cover wide areas in urban road neworks. Vehicle o vehicle ravel imes vary according o he speed limi and driving syle. In he urban arerial conex, he vehicle o vehicle ravel ime differ grealy due o he acual imes when he sudied vehicles pass hrough he inersecion. Vehicles ha encouner delay (e.g. a raffic signals) will have higher ravel imes han hose ha do no encouner delay. These ypes of ravel ime measuremens have been used in recen sudies. Eliasson (2007) forecased he ime of day and day o day ravel ime variabiliy on a Sockholm bypass road by uilising ravel ime daa colleced horough an auomaic camera sysem. The ravel ime daa colleced from 6AM o 10 PM, and were hen analysed in 15min inerval. This sudy used a nonlinear regression model o seek he relaionship beween he ravel ime, free flow ravel ime, speed and Sandard Deviaion of vehicle o vehicle ravel ime daa across he day. This sudy used he following equaion based on a volume delay funcion: Equaion 1 where =acual ravel ime 0 = free flow ravel ime L = link lengh λ TOD and λ SPD are dummy variables represening ime of day and he speed limi, λ is a consan, and α, γ and ω are esimaed parameers. Eliasson (2007) also discussed he ravel ime paern in he sudy area. He found ha ravel ime is less skewed under higher congesion condiions and he ravel ime covariance is relaively small. I is hen possible o direcly sum link ravel ime variances in order o ge he roue ravel ime variance. A similar echnique has been used for analysing he ravel ime variabiliy in Peer e al. (2009). This sudy aimed o forecas ravel ime variabiliy for use in cos benefi analysis of a ransporaion projec. Peer e al. (2009) also assessed 15 minue inerval ravel ime ha was derived from speed daa colleced by loop deecor. The speed daa hen can be convered o ravel ime daa used piecewise-linear-speed based (PLSB) mehod. This sudy used 255 working days ravel ime daa in is analysis. Similar o Eliasson s model, Peer a al also used a regression echnique. However, insead of modelling he relaionship beween sandard deviaion and speed and ravel ime, his sudy focused on he link beween he sandard deviaion, delay and he V/C raio. The fied model was: Equaion 2 Equaion 3 where VCR = flow capaciy raio L=link lengh β=esimaed parameer In he Unied Kingdom, a model of link and corridor ravel ime variabiliy, using GPS daa colleced by individual vehicles for 34 roues in en of he larges urban areas in England, was developed by Black and Chin (2007). The ravel ime daa from each vehicle were hen 4

5 Modelling urban ravel ime variabiliy wih he Burr regression echnique grouped o min inerval daa. Similar o previous mehods ha look a he relaionship beween he ravel ime variabiliy and oher raffic parameers, his sudy relaed he coefficien of variaion of ravel ime variabiliy (CV ) o he congesion level in he sudy area CV = CI Equaion 4 where β α CI is a congesion index, defined as / 0 CI =, where is acual ravel ime 0 is free flow ravel ime and α and β are esimaed parameers. They firs considered ravel ime variabiliy a he link level, and hen used sandardised link ravel imes o develop a corridor ravel ime reliabiliy model: CV = 0.16CI L Equaion 5 where L (km) is he link lengh and is an esimaed parameer (he elasiciy of CV wih respec o disance). A similar model was developed by Richardson and Taylor (1978), who showed ha under cerain resricive condiions he heoreical value of β would be 0.5. Given he assumpion ha he day by day ravel ime variabiliy can be reaed as he ime series daa, Sohn and Kim (2009) used he auoregressive moving average-generalized auoregressive condiional heeroscedasiciy (ARMA-GARCH) mehod o invesigae he ravel ime paern for road neworks in Souh Korea. This sudy also esed he impac of he differen combinaion of raffic and geomeric condiion ha migh influence he ravel ime reliabiliy. I was found ha he link widh and congesion level play an imporan role in ravel ime variabiliy. The quesion of serial correlaion beween ravel imes on individual links along a roue is of ineres, and here are a number of differen resuls in he repored sudies. Richardson and Taylor (1978) found no significan saisical eveidence of such correlaion, and as indicaed above, neiher did Eliasson (2007). Faouzi and Maurin (2007) suggesed how o rea serial correlaion beween link ravel imes if hose imes were log normally disribued. Nicholson and Nicholson and Munakaa (2009) found evidence of srong serial correlaion in a sudy of ravel imes on he Tokyo expressway nework. 2.1 Travel ime variabiliy merics Given he definiion of he ravel ime variabiliy he dispersion of he acual ravel ime o he preferred ravel ime he mean is he commonly used preferred ravel ime while sandard deviaion, coefficien of variaion and a quanile range (difference beween he 95 h or 90 h and 50 h percenile) are suggesed variabiliy merics. The use of he 90 h percenile is noionally jusified by he implicaion ha i allows an employee o possibly arrive lae in he office wice in 20 working days. Some alernaive ravel ime variabiliy merics have been proposed. Federal Highway Adminisraion (FHWA) (2006) inroduced a buffer ime (BT ) o represen he addiional ime above he average ravel ime ( ) required for on-ime arrival. The buffer ime is he difference beween he 95h percenile ravel ime ( 95 ) and he mean ravel ime: BT = 95 Equaion 6 FHWA (2006) also esablished a ravel ime reliabiliy index (Planning Index, PI ), which is he raio of he 95h percenile ravel ime o he ideal ravel ime, aken o be he free flow ravel ime ( 0 ): 5

6 ATRF 2011 Proceedings PI = 95 / 0 Equaion 7 Addiionally, van Lin and van Zuylen (2005) proposed he so-called skew-widh mehods. skew The skewness of he ravel ime λ is defined as he raio of difference of he 90 h percenile ( 90 ) and 50 h percenile ( 50 ) ravel imes and he difference beween 50 h percenile and 10 h var percenile ( 10 ) ravel imes. The widh of ravel ime λ is defined as he raio of he differences of 90 h percenile and 50 h percenile ravel ime and he 10 h percenile ravel ime. The equaion of he skewness and he widh ravel ime merics are as follows; skew λ = Equaion = var λ Equaion 9 50 From he review, i can be seen ha he exising ravel ime variabiliy model use differen approaches as well as differen shapes of a parameric disribuion. There is commonaliy in he research in he use of he sandard deviaion, congesion index, link lengh and he mean of ravel ime o measure variabiliy. Despie he fac ha previous sudies have aemped o model ravel ime variabiliy and examine he conribuion of each raffic variables o he ravel ime variabiliy, mos of his research has simply used he normal and log normal disribuions and heir properies in modelling ravel ime variabiliy. There is sill lef room o sudy he bes fi ravel ime disribuions for ravel ime variabiliy sudies. Therefore, he nex secion discusses he use of he Burr disribuion as a proposed ravel ime variabiliy model. 3. The Burr disribuion and Burr regression echnique Recen colleced empirical ravel ime daa exhibi posiive skew and a long ail, and he normal disribuion herefore does no really fi hese daa. Through exhausive goodness of fi esing, using he eigh years of coninuous ravel ime daa from he Adelaide daabase, Susilawai e al (2010) found ha neiher he Normal nor Log Normal disribuions could fi he observed ravel ime disribuions. Susilawai e al (2010) found ha he Burr disribuion fied he empirical ravel ime daa from Adelaide urban arerial roads and concluded ha he Burr disribuion could be used o represen he observed daa. The nex secion reviews he Burr disribuion as a proposed ravel ime disribuion and he applicaion of his disribuion in reliabiliy engineering and oher reliabiliy sudies. The Burr disribuion was developed by Burr (1942) for he express purpose of fiing a cumulaive disribuion funcion (cdf) o a diversiy of frequency daa forms covering a wide range of disribuion shapes. I can also be used o model an unspecified disribuion. In is basic form i has wo parameers, c and k. The probabiliy densiy funcion (pdf) f(x c,k) of he Burr disribuion is c 1 c ( k+ 1) f ( x, c, k) = ckx (1+ x ) Equaion 10 where x > 0, c > 0 and k > 0. The cdf F(x, c, k) is given by F c k ( x, c, k) = 1 (1+ x ) Equaion 11 6

7 Modelling urban ravel ime variabiliy wih he Burr regression echnique The disribuion has some ineresing saisical properies (Tadikamalla, 1980). In he firs insance he rh momen of he disribuion (E(x r )) will only exis if ck > r, in which case E( x r r r kγ( k ) Γ( +1) ) = µ = c c r Γ( k+ 1) Equaion 12 where Γ ( y) is he mahemaical Gamma funcion. In addiion, he modal value x m is given by x m c 1 = ck 1 + 1/ c Equaion 13 bu x m will only exis if c > 1. [If c 1, hen he disribuion is L-shaped.] Based on previous research, he value of c plays a very imporan role in deermining he shape of he cdf curve -he lower he value of c, he sharper he fall of cdf curve (c = 1 he sharper fall of cdf curve, c = 2 shallower fall of curve) while he lower he k parameer, he sharper he iniial rise of he curve (k=2, sharper he iniial rise of cdf curve, k = 3, shallower iniial rise of curve) as shown in figure 1. Ineresingly, he k parameer can also define he skewness, for k more han one means ha he disribuion are righ skew and less han one is lef skewed. See Figure 1 for some alernaive Burr disribuion shapes. Figure 1: Burr disribuion cdf for differen c and k parameer 7

8 ATRF 2011 Proceedings Ineresingly, he wide range of possible shapes of he Burr disribuion and is well behaved algebraic properies make i is useful for fiing many ypes of daa and for approximaing many differen disribuions (Zimmer e al, 1998). For insance, he Weibull disribuion is specific ype of Burr disribuion where one of he parameers is gamma disribued. When he k parameer is one hen he Burr disribuion is a special case of log logisic disribuion. The versailiy and flexibiliy of he Burn disribuion makes i quie aracive as a enaive model for daa whose underlying disribuion is unknown The Burr disribuion is very popular disribuion and has gained a srong ineres in reliabiliy engineering for modelling lifeime daa and monoone failure raes. A number of reliabiliy engineering applicaions have uilised i o model he produc life process (Abdel-Ghaly e al., 1997). Also, Shao e al. (2004) sudied he models for exended hree parameer of Burr disribuion and used his disribuion o model exreme even wih applicaion o flood frequency. The hree parameer Burr disribuion or Singh Maddala disribuion is he exended version of he Burr disribuion which has a scale parameer for he independen variable x. The scale parameer defines he poin where he daa is mos concenraed. For purposes of simpliciy, he scale parameer can be used o esimae he median of he daa. The disribuion has an algebraic ail ha is useful in modelling less frequen failures (Soliman, 2005). As is cdf can be wrien in closed form, is perceniles are easily compued. To compue perceniles for he Burr disribuion, use he following approach Given he CDF defined by equaion (10), for a given value of F(x, c, k), he percenile P, i.e. P c k = 1 (1+ x ) Equaion 14 c ( 1+ x ) x c k = (1 P) 1 = 1 P 1/ k 1 x = (1 P) 1 c 1/ k P he median is hen c k x 2 1/ 50 1 = Equaion h percenile is c k x 10 1/ 90 1 = Equaion h percenile is k 10 1/ x c 10 1 = Equaion 17 9 On he basis of equaions 15, 16 and 17, he nex secion indicaes how o compue curren ravel ime variabiliy merics direcly from he Burr disribuion parameers. 8

9 Modelling urban ravel ime variabiliy wih he Burr regression echnique 4. Proposed ravel ime variabiliy merics Preceding secion reviewed he ravel ime variabiliy modelling and meric developmen and also he properies of he Burr disribuion. Equaion 4 and 5 show ha he 50 h percenile and he 90 percenile are easy o obained once he parameer c and k are known. Since he ravel ime variabiliy can also be measured as he dispersion of he preference ravel ime and he acual ravel ime, his sudy used he similar mehod used by Lin and Zuylen (2005) o develop he ravel ime variabiliy merics. According o Lin and Zuylen (2005), is he raio beween he difference of he 90 h percenile and he 50 h percenile and he 50 percenile. This meric is useful o illusrae he ravel ime dispersion. Replacing he 90 h percenile and he 50 h percenile in he Lin and Zuylen (2005) merics, skewness of ravel ime (λ skew ) and widh of ravel ime (λ var ) in erms of Burr parameers is possible, given ha hese wo merics are boh defined in erms of perceniles. Therefore we obain refined ravel ime variabiliy merics as below. λ λ skew var = = Given x 50 and x 90 as defined by equaion 15, 16 and 17, i follows ha and c 1/ k c 1/ k skew λ = Equaion 18 1/ k c 1/ k 10 c c 1/ k c 1/ k var λ = Equaion 19 c 1/ k 5. The ravel ime daa daabase This sudy used wo of he ses of longiudinal ravel ime daa in he daabase. The daa was colleced by using GPS during regular journey o work rips for differen commuers, each of which sared a abou he same ime of day and wih similar driving behaviour. The wo ses of ravel ime daa are (1) he Glen Osmond Road (GOR) daa se, including 16 successive links from Glen Osmond o he Adelaide CBD. The lengh of links vary beween 150 m o 600 m wih speed limis of eiher 50km/h or 60km/h, and (2) he Souh Road (SR) daa se, which is he collecion of ravel ime from a major freigh corridor in Adelaide. This roue serves he souhern suburbs of Adelaide meropolian area. The links lengh of his roue also varies from 165 m o more han 4000 m and he speed limi varies beween 40 km/h (because of road works) and 80km/h. 9

10 ATRF 2011 Proceedings Figure 2: shows each of hese roues on a map of he Adelaide meropolian area. 10

11 Modelling urban ravel ime variabiliy wih he Burr regression echnique Using maximum likelihood esimaion, he hree parameers (c, k and scale) of he Burr disribuion were obained. These parameers herefore can be used for calculaing h 90 h and 50 h percenile and he λ var. The hree parameers and he variabiliy of each links are shown in Table 1 (Souh Road) and Table 2 (Glen Osmond Road). Table 1: The Burr parameers and he link variabiliy of Souh Road daa se Link No Lengh (m) Parameer Percenile k c scale 50h 90h N/A : no available daa due o he burr disribuion did no fi he ravel ime daa λvar free flow 0(s) N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A CI 11

12 ATRF 2011 Proceedings Table 2 : The Burr parameers and he link variabiliy of Glen Osmond Road daa se Link No link Lengh (m) Parameer Percenile c k scale λ var CI N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A : no available daa due o he burr disribuion did no fi he ravel ime daa I can be seen ha he ravel ime variabiliy merics for boh daa ses do vary. The highes ravel ime variabiliy is for SR daa se is link 14 (2.36) while he lowes is for link 1 (0.03). On average he ravel ime variabiliy for SR daa se is abou 0.6 which means ha he dispersion of he ravel ime from he median 0.6 imes median. On he oher hand, he ravel ime variabiliy for GOR daa se is much more diverse han he SR daa se. The highes is for link 15 wih λ var equalling This means ha he ravel ime variaion is almos riple he median ravel ime. Link 1 has he lowes ravel ime variabiliy which is only 0.45 of he median. Figure 3 he scaer plo of Scale parameer and he median of he SR and GOR daa Scale Parameer y = x R² = Series1 Linear (Series1) h percenile 12

13 Modelling urban ravel ime variabiliy wih he Burr regression echnique Using he resuls in Tables 1 and 2, Figure 3 shows he scaer plo of he scale and he median of he SR and GOR daa ses. I clearly shows ha here is a linear relaionship beween hese wo variables (wih R 2 =0.93). I can be suggesed here ha for purposes of simpliciy he scale parameer of Burr disribuion can be esimaed he median of ravel ime. These ables also show ha he ravel imes do vary link by link. Since he ravel ime daa used in his sudy is he link ravel ime where he link lenghs also vary, furher daa analysis o assess he role of raffic variables on ravel ime variabiliy is conduced. There are wo quesion arose. The firs quesion is how he link ravel ime will likely affec he ravel ime variabiliy. In order o answer he firs quesion, a similar formula o ha used by Black and Chin (2009) Coefficien of Variaion (CV) as a funcion of link lengh and congesion Index (CI) is adoped. Given ha formula (see equaion 5), he λ var is a funcion of he link lengh and he congesion index as a refined ravel ime variabiliy formula is developed. To measure he congesion index, insead of using he mean as he denominaor, he median (50 h percenile) was using. Adoping he new proposed ravel ime variabiliy meric, he link by link ravel ime variabiliy and he congesion index modelling is abulaed in able 3. Table 3: Esimaed Parameer for ravel ime variabiliy modelling Parameer Descripion Value α consan β 1 esimaed parameer for Congesion Index β 2 esimaed parameer for link lengh I shows ha boh link lenghs give he negaive value of he power funcion, while congesion index give he posiive value o he λ var. This resul is quie differen o Black and Chin s research, bu his may reflec he differen naures of he wo daa ses The proposed mehod can be exended o invesigae he influence of oher raffic variables for furher daa analysis. However, for his model, he daa analysis limis for jus esing he affec of he link lengh on he λ var. A second objecive of his research is o seek he role of he degree of sauraion in influencing he c parameer. Before conducing furher daa analysis, here is a judgemen required in selecing he degree of sauraion as he main facor in his sudy. Based on he lieraure, i is found o be an imporan variable in deermining urban raffic sysem performance. The degree of sauraion gives he raio of he capaciy and flow. These variables hen can be inpu ino he ravel ime variabiliy modelling. This paper does no discuss he mehod o obain he degree of sauraion and how o relae he SCATS degree of sauraion daa o he heoreical degree of sauraion. The discussion abou ha issue can be found in Suksri e e al (2010). The degree of sauraion daa obained from SCATS daa of each link has been used as he covariaes facors. Nex secion reviews he Burr regression echnique and how o uilise his echnique in he ravel ime variabiliy modelling. 6. Burr regression The Burr regression echnique was firsly inroduced by Beirlan e al. (1998) in relaed o he special case of he Burr disribuion. I has been known for some ime ha he log logisic disribuion is a special case of he Burr. Adoping he similar parameerisaion as used in he log logisic regression, Beirlan e al. (1998) proposed Burr disribuion parameerisaions. Paramerizaion: 13

14 ATRF 2011 Proceedings The shape parameer (c) is allowed o vary wih y where y is covariae (dimensional vecor)(beirlan e al., 1998) Given he cdf of he Burr disribuion c 1 c ( k+ 1) f ( x, c, k) = ckx (1+ x ) Equaion 20 hen based on he firs parameerizaion, he new cdf of Burr disribuion is as follows ~,. Since exp can be ermed as an exponenial funcion of and,hen replacing he c wih he c(y) he cdf becomes ~,,,,, Equaion 21 Where, k=shape parameer c=shape parameer x= ravel ime y = Degree of sauraion Maximum likelihood esimaion can be used for Burr regression parameer esimaion by replacing he c wih he exp expression. The maximum likelihood analysis can be done as follows: The likelihood funcion is, Equaion 22, Equaion 23 Then,, Equaion 24, Equaion 25 For illusraive purposes, his sudy merely applied he firs parameerisaion of Burr disribuion for seleced GOR and SR links. The SCATS degree of sauraion daa has been used as he covariae marix (y) - represening he raffic variables. The Burr regression modelling resul is given in Table 3 14

15 Modelling urban ravel ime variabiliy wih he Burr regression echnique Table 4: Burr regression modelling for Glen Osmond daa se Link No Parameer Burr Disribuion Parameer Burr regression echnique Log Likelihood es raio cv*= c k scale k θ scale 6 (GOR) (GOR) (SR) (SR) (SR) (SR) cv*=criical value a he 95 h confidence level Acceped ( ) Acceped ( ) Acceped ( ) Acceped ( ) acceped ( ) acceped ( ) The resul of he general form Burr disribuion and Burr regression echnique parameer esimaion are shown in Table 4. To es he similariy of hese wo echniques he log likelihood raio es was conduced. This resul is promising as he value of k and scale parameers from boh echniques are similar. From he log likelihood es, i was found ha he Burr regression echnique for seleced links of GOR and SR perform well, as indicaed by he p value of he log likelihood es. The p values are less han he criical value a 95 h confidence level. The θ as he esimaed parameer for degree of sauraion daa se is showing ha for differen link wih differen raffic condiion gave differen esimaed variables. By assuming ha he link which have higher variabiliy index, higher congesion index migh have higher degree of sauraion, hen his able can describe ha behaviour. For example link 12 has higher variabiliy and also has higher θ han link 6. According o he variabiliy and congesion index values, he GOR daa se may well represen oversauraed condiions. For undersauraed condiion, he similar echnique was applied for seleced SR links. 7. Conclusion The need for more reliable ravel ime has gained srong ineres in inernaional research. There are numerous sudies ha have suggesed merics o quanify he variabiliy of ravel ime and have looked ino he facors ha migh conribue o he higher ravel ime variabiliy. Those sudies suggesed ha he basic raffic variables were playing an imporan role in modelling he ravel ime variabiliy and used he normal disribuion properies as he ravel ime variabiliy merics. However, he recen empirical ravel ime daa exhibi posiive skew and long ails, and so are no well represened by sandard pdfs such as he Normal disribuion. This sudy herefore suggesed refined ravel ime variabiliy merics by uilising he Burr disribuion properies. From wo ses of link ravel ime daa colleced in Adelaide, i was found ha he link ravel ime variabiliy does vary. The firs hypohesis needed o es is 15

16 ATRF 2011 Proceedings wheher he link lengh is he main conribuor o he ravel ime variabiliy, he second is wha congesion index role in modelling he ravel ime variabiliy. The resuls confirmed he posiive effec of he link lengh and he power effec congesion index, respecively, on he link ravel ime variabiliy. This resul is similar o he earlier research conduced by Black and Chin (2007). This paper gives an insigh of he use of he Burr disribuion properies o refine he curren ravel ime variabiliy merics. This echnique has been esed for seleced links of he GOR and SR daa ses. Under he 95 h confidence level, he resul of his proposed modelling echnique is similar o he resul of he general form of Burr regression echnique. Ineresingly, his proposed echnique can be exended o invesigae more raffic variables in relaed o he Burr disribuion parameer. This new approach can be also used for furher ravel ime variabiliy modelling. Appendix Noaion σ λ λ TOD λ speed L 0 α γ ω β β ij VCR CI CV BT PI λ skew λ var f F c k G(y) x m P x 90 x 50 x 10 y θ Descripion sandard deviaion consan dummy variables represening ime of day dummy variables represening he speed limi link lengh acual ravel ime free flow ravel ime esimaed parameer esimaed parameer esimaed parameer esimaed parameer esimaed parameer flow capaciy raio congesion Index coefficien of Variaion buffer ime planning Index 95 h percenile ravel ime 90 h percenile ravel ime 50 h percenile ravel ime 10 h percenile ravel ime mean ravel ime he skewness of he ravel ime he widh of ravel ime pdf cdf shape parameer Burr disribuion shape parameer Burr disribuion Gamma funcion mode percenile 90 h percenile Burr disribuion 50 h percenile Burr disribuion 10 h percenile Burr disribuion degree of sauraion esimaed parameer 16

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