LINK TRAVEL TIME PREDICTION FOR DECENTRALIZED ROUTE GUIDANCE ARCHITECTURES*

Transcription

1 LINK TRAVEL TIME PREDICTION FOR DECENTRALIZED ROUTE GUIDANCE ARCHITECTURES* Karl E. Wunderlch Mtretek Systems, Inc. 600 Maryland Avenue, SW, Sute 755 Washngton, DC Tel: (202) Fax: (202) Davd E. Kaufman AT&T Laboratores 379 Campus Drve, Room 2B19 Somerset, NJ Tel: (732) Fax: (732) Robert L. Smth Unversty of Mchgan Department of Industral and Operatons Engneerng Ann Arbor, MI Tel: (313) Fax: (313) Abstract A crtcal problem n decentralzed route gudance s to communcate antcpated congeston to ndvdual drvers n such a way that the routes chosen are lkely to be consstent wth the forecast. We propose a predcton technque for decentralzed route gudance archtectures to dentfy tme-dependent lnk travel tmes whch when communcated to drvers leads to tme-dependent fastest paths consstent wth ths forecast. The fxed pont property of the forecast s assured by an teratve process of traffc smulatons followed by dynamc route determnatons untl the routes and hence the resultng dynamc lnk tmes become stable. The resultng routes yeld an nherently accurate forecast of congeston as well as beng user-optmal by constructon. A novel back-datng process s utlzed to nsure the dscovery of a stable 1

2 routng after a fnte and usually small number of teratons. An emprcal case study based on the roadway network n Troy, Mchgan s ncluded. * Ths research was supported n part by the Unversty of Mchgan ITS Research Center of Excellen 2

3 1. Introducton. The archtecture of route gudance provson n ITS can be dvded nto two categores: a decentralzed archtecture, where the route selecton functon s located n-vehcle (or onpassenger, n the delvery of mult-modal route gudance); and a centralzed archtecture, where the route selecton functon s performed at some central ste by an Independent Servce Provder (ISP). These dstnctons and the lkely strengths and weaknesses of each archtecture have been addressed by researchers n precursor ITS archtecture studes [1,2,3] as well as documented n the Natonal ITS Archtecture [4,5]. Fgure 1, adapted from the Natonal ITS archtecture document: Theory of Operatons [6], llustrates sample concepts of route gudance archtecture, both decentralzed and centralzed. In the Autonomous form of decentralzed archtecture, all route gudance functonalty s n-vehcle. Here route gudance s relegated to prmarly statc route gudance applcatons, snce there s no mechansm to systematcally update the map database wth current lnk travel tmes. In the Decentralzed Dynamc archtecture, lnk-travel tmes are broadcast to vehcles to provde real-tme estmates of network congeston. Ths one-way communcatons lnk supports dynamc route selecton n-vehcle by performng map-matchng wth the data stream from an ISP. The ISP does not drectly track route requests or locaton/destnaton nformaton from ts clents. In the Centralzed archtecture, the ISP moves from the broadcast of lnk travel tmes to the provson of detaled, ndvdualzed routes to ts clents. In-vehcle functonalty s reduced to smple route dsplay, but the vehcle gans the capablty to communcate locaton and desred destnaton to the ISP. Each of these approaches are n complance wth the Natonal ITS Archtecture, and no sngle approach s favored overall by natonal polcy. However, the Natonal ITS Archtecture does project a gradual progresson over tme from Autonomous to Decentralzed to Centralzed route gudance provson [7]. Autonomous archtectures are replaced wth Decentralzed archtectures as dynamc route gudance develops as a vable servce. Centralzed archtectures are projected to replace Decentralzed archtectures n mature route gudance markets wth hgh market penetraton. Ths second step n the progresson over tme s projected because of nstablty n the allocaton of alternatve routes usng a dynamc decentralzed route gudance archtecture. The decentralzed feature 3

4 of generalzed lnk travel tme forecasts broadcast to an uncertan number of guded vehcles precludes detaled, ndvdualzed route assgnment. Therefore, route assgnment technques developed for decentralzed archtecture are typcally characterzed by all-or-nothng assgnments. At hgh market penetratons, ths may result n hghly neffcent routng too many vehcles followng the same path at the same tme. Allocaton of routes under hgh market penetraton s projected to be more stable under a centralzed archtecture because the route gudance servce provder can control more precsely the number of vehcles routed onto a specfc route. Research n the area of dynamc traffc assgnment suggests that predctve centralzed strateges can effectvely counter nstablty at hgh market penetratons. Lee [8], and others [9] have dealt wth nstablty at hgh market penetraton by employng mult-path routng strateges. These mult-path routng strateges are assocated wth centralzed route gudance archtectures because they rely on precse control of ndvdual vehcle routng. In some cases, arbtrary fractons of vehcle streams are routed, resultng n assgnments of fractonal vehcles to ndvdual paths. Ths paper argues that ths second evolutonary step, from Decentralzed to Centralzed, may not be necessary for effcent predctve route gudance at hgh market penetratons. A varant of the Decentralzed archtecture, Predctve Decentralzed Route Gudance, s proposed that ncludes a lnk travel tme predcton functon at the ISP (Fgure 2). Route requests from the vehcle to the ISP are optonal. Such an archtecture may be partcularly appealng to consumers who desre strct prvacy about travel habts. One example of a predctve approach consstent wth a decentralzed archtecture s the Smulaton of Antcpatory Vehcle Network Traffc (SAVaNT) developed at the Unversty of Mchgan [10, 11, 12]. Ths paper proposes a modfed verson of the SAVaNT travel tme predcton based on a heurstc search technque that mtgates a number of the pathologes assocated wth decentralzed, all-or-nothng route gudance provson at hgh market penetratons. Frst an overvew of the SAVaNT method s presented n Secton 2. Secton 3 provdes detals of the lnk travel tme predcton method n SAVaNT, whle Secton 4 dscusses the observed pathologes assocated wth the technque at hgh market penetratons (falure to converge and sub-optmal routng). Secton 5 demonstrates a technque, backdatng, to adjust predcted lnk travel tmes to make them nherently more accurate. However, ths 4

5 backdatng technque can also be demonstrated to exacerbate the problem of dvergence where the exact number of vehcles takng a partcular route cannot be determned (as s the case wth decentralzed route gudance). Secton 6 dscusses the relatonshp between lnk travel tme accuracy and the uncertanty surroundng the number of vehcles mpacted by route gudance wth respect to the ssue of dvergence n SAVaNT. Secton 7 proposes a heurstc search technque to fnd the most effcent, convergent amount of adjustment to predcted lnk travel tmes. Results from a smulated test network based on portons of the Troy, Mchgan arteral network demonstrate the effectveness of the mproved predcton technque when compared to both the SAVaNT method and non-predctve dynamc route gudance. Secton 8 presents some conclusons and extensons based on ths work. 2. Lnk Travel Tme Predcton and Antcpatory Route Gudance Antcpatory route gudance can be defned as paths calculated from a set of predcted future lnk travel tmes, whch when dssemnated to drvers cause the same set of predcted lnk travel tmes to be realzed by vehcles n the network. Thus the paths dstrbuted as route gudance were based on a correct assumpton about future congeston condtons on the network, correspondng to a user-optmal equlbrum condton. Chen and Underwood [13] provde a overvew of antcpatory route gudance and ts mplementaton. Intal applcatons of SAVaNT [10, 11, 12] demonstrated that antcpatory route gudance computed by the SAVaNT method can be employed effectvely n large-scale applcaton under low (<30%) market penetraton rates. Predctve route gudance at these levels was found to be superor to non-predctve route gudance methods. At market penetratons above 30%, however, SAVaNT sometmes produced less effcent routngs than nonpredctve methods and sometmes produced no soluton at all. Clearly, new ATIS consumers are unlkely to subscrbe to a SAVaNT-based decentralzed gudance servce as market penetratons approach 30%. Indeed, even f hgher market penetratons were realzed, drvers of guded vehcles would be unlkely to trust or comply wth the routes calculated n-vehcle usng SAVaNT-generated predctons of lnk travel tmes. The SAVaNT concept s llustrated n Fgure 3. Combnng a tme-dependent fastest-path calculaton, R, wth a traffc smulaton, A, n an teratve manner, we can dentfy a routng polcy, π, as a "fxed pont" f n consecutve teratons, the dynamc lnk travel tme profles, C, are dentcal,.e., a stable set of predcted dynamc lnk travel tmes has been dentfed. The lnk travel tme profle comprses lnks travel tmes for all lnks and tme perods over 5

6 the horzon of tme we are tryng to predct. Because we restrct our attenton n ths paper to all-or-nothng polces (.e., gvng the same route to all vehcles havng common locaton and destnaton at a gven tme), the set of possble polces s fnte. Thus, f executon of SAVaNT does not result n a fxed pont, the result must be the generaton of a repeatng sequence of polces, π 1, π 2, π 1, π 2, π 1, π 2,L, rather than convergence to a fxed pont. We call ths stuaton cyclng. The mechancs of the fastest path calculatons are dscussed n [14]. For an nvestgaton of ths ssue under multpath (as opposed to all-ornothng) routng, see [15]. SAVaNT uses a verson of the INTEGRATION traffc smulaton [16] for lnk-tme predcton, INTEGRATION-UM. The INTEGRATION model was orgnally developed by Mchel Van Aerde at Queen s Unversty (Canada) and modfed by researchers at the Unversty of Mchgan. INTEGRATION-UM retans the basc scope and modelng approach of the model: a strongly determnstc mesoscopc approach employng macroscopc travel-tme and flow relatonshps and mcroscopc ndvdual vehcle control and lnk queung. INTEGRATION allows the user to vary the fracton of vehcles on the network recevng route gudance. Non-equpped (background) vehcles are routed accordng to a statc fastest-path route. 3. Accuracy of Predcted Lnk Travel Tmes n SAVaNT. As observed n [10], there s an nherent naccuracy ntroduced nto SAVaNT by a dfference n the way the smulaton produces predcted tme-dependent travel tmes and nterpretaton of that data by the routng module. Note that we mpose a dscrete-tme lattce on the soluton horzon, H, over whch we predct lnk travel tmes. Let t be the number of seconds n each tme slce, and let t;t = 1,2KH correspond the tth tme slce n the horzon. Let c l () t be defned as the predcted lnk travel tme for lnk l durng tme-slce t. Consder the problem of constructng a lnk travel tme profle C = { c l (): t l,t}. As shown n Fgure 4, the smulaton reports experenced lnk travel tmes when vehcles fnsh traversng lnks. Let c l () t represent the travel tme reported by the vehcle makng the th departure from lnk l durng tme-slce t. Let τ l () t be the smulaton clock tme when c l () t s reported. In the verson of SAVaNT employed n [10], c l () t s updated each tme a 6

7 departure occurs accordng to an exponental smoothng functon, c l (t) = α c l (t) + ( 1 α)c l (), t where α = 0.4. The dynamc router nterprets c l () t as the expected average travel tme requred to traverse lnk l for vehcles whch begn travel on lnk l durng tme slce t. But the value of c l () t s computed based on vehcle reports made by vehcles fnshng travel on the lnk durng tme slce t. As llustrated n Fgure 5, when a lnk departure reports a travel tme for a lnk, t s more precsely gvng an estmate of vehcle travel tme whch begns earler, at τ l () t c l (). t Note that reducng the length of each tme slce t does not correct for ths partcular knd of error. Thus our estmatng procedure s nherently naccurate, but as we wll see, ths naccuracy s a crucal factor n the convergence of the SAVaNT teratve process. 4. Beneft Reducton Related to Inaccuracy n Lnk Tme Predcton. As stated above, routng polces are generated n SAVaNT by nterpretng c l () t as the expected average travel tme for vehcles begnnng travel on lnk l durng tme slce t. Ths naccuracy has the effect of causng a lag n the feedback control of vehcles under route gudance. For example, f travel tme s dynamcally rsng on a partcular path, there wll be a lag n accurate reportng of that change n that path tme equal to the maxmum ndvdual lnk travel tme n the path. Consder the stuaton where, based on perfect nformaton n the system, a path currently dentfed by the route as optmal becomes congested and an alternatve path becomes more attractve. In the next teraton, then, vehcles currently on ths path cannot be re-drected to the alternate path untl the router sees reports of travel tmes n the constructed travel tme profles. These profles contan the tme lag assocated wth lnkdeparture travel tme estmates, and thus the router msdrects vehcles for the duraton of that tme lag. Vehcles on our ntally optmal path wll be erroneously routed on what wll be experenced as a slower path but whch appears from the predcted travel tme profle as the fastest path. Ths naccuracy cannot be corrected n an future teraton of SAVaNT, snce the travel tme profle generated on all paths wll be constructed wth dentcal lag tme. Thus, the naccuracy s carred forward from teraton to teraton, and cannot be corrected wthn the current SAVaNT concept. 7

8 Hence, a fxed pont n SAVaNT corresponds to a routng polcy that s consstent wth the reproduced travel tme profle, a travel tme profle whch necessarly contans some naccurate nformaton. Thus the routngs dentfed by SAVaNT are the mnmum travel tme paths wth respect to a predcted travel tme profle, but not necessarly consstent wth experenced travel tme. Ths naccuracy can be compared to the nature of the naccuracy encountered n routng vehcles n a non-predctve manner. These methods provde a fastest-path routng polcy based on the assumpton that currently reported condtons persst ndefntely. The polces generated are consstent wth the expected (statc) travel tme profle, but not consstent wth experenced travel tme. The result of naccurate lnk travel tme predcton s a reducton n benefts to both the traffc system as a whole and to the ndvduals recevng route gudance. Fgure 6 llustrates the benefts seen n a 500-lnk, 200-node network of Troy, Mchgan. Note that at near 50 percent guded vehcles on the network, the performance of antcpatory vehcles drops to the same level as the background (unguded) vehcles. At 80 percent, the guded vehcles (assumed to follow fastest predcted paths) have longer trps on average than the background vehcles, whch would not be possble f ther antcpatory fastest paths were beng computed from perfectly accurate lnk travel tme forecasts. Ths dfference, although small, s statstcally sgnfcant. As noted n [10], when SAVaNT was confgured to correct for ths naccuracy, the method always termnated wth the constructon of a cycle, rather than a fxed pont. We wll dscuss the nature of ths phenomenon n more detal n secton 5. 8

9 5. Improved Predctve Fxed Pont Solutons SAVaNT can be constructed to provde accurate predctve lnk tmes. The lnk travel tme estmated by lnk departures whch occur n tme slce t can be "backdated" to the tme at whch travel on the lnk began, namely τ l () t c l (). t There may be several lnk travel tme estmates for each tme slce, so these values are averaged together to come up wth a value for c l ( t ) where τ l () t c l () t s contaned n tme slce t (the tme slce n whch travel on the lnk began). These values of lnk travel tme can be used n SAVaNT n the place of values dentfed above n Secton 2. However, such an mplementaton nvarably results wth SAVaNT termnatng n a cycle, even for market penetraton levels whch have fxed ponts for SAVaNT mplemented wthout backdatng. Ths result s consstent wth the observaton made Kaufman et al [15] that the exstence of a fxed pont n SAVaNT cannot be guaranteed gven the dscontnutes nherent n all-or-nothng routng polces and dscrete-tme traffc smulaton. The dlemma of SAVaNT can be summarzed by the followng: the naccurate lnk tme forecasts cause the method to dentfy sub-optmal solutons for antcpatory route gudance, yet only wth the naccuracy n lnk tme forecastng wll SAVaNT converge (albet nconsstently). However, we wll show that t s possble to mplement a heurstc based on ncrementally backdatng travel tme data that sgnfcantly mproves the performance of SAVaNT. Under ths heurstc approach, SAVaNT s demonstrated to be relably convergent and to result n mproved travel tmes for guded vehcles at all market penetratons (even above 30%). Whle the heurstc cannot guarantee the dentfcaton of an global optmal fxed pont, t does offer a practcal soluton to the dlemma of SAVaNT and can be vewed as a startng pont from whch further research on predctve gudance algorthms can be conducted for decentralzed route gudance archtectures. The heurstc tunng approach attempts to reduce the naccuracy to the smallest amount wthout causng a cycle to occur. Ths would allow for the maxmum amount of beneft to accrue to both the system and the guded vehcles wthout cyclng. We defne δ, 0 δ 1, as the fractonal amount of backdatng mplemented n SAVaNT. As llustrated n Fgure 7, as δ ncreases, the accuracy of the lnk tme predcton scheme ncreases. When δ = 0, SAVaNT s confgured as n [10]. When δ = 1, SAVaNT s confgured for accurate lnk travel tme predcton, but always dverges. By manpulatng ths fractonal amount of backdatng, we alter both the accuracy of the lnk travel tme forecast as well as the fracton 9

10 of vehcles mpacted teraton-to-teraton by forecasts whch nclude ther own experence n the prevous teraton. The remander of Secton 5 deals wth the mpact of mproved lnk travel tme accuracy. In Secton 6, the ssue of teraton-to-teraton forecasts are examned. To test the behavor of SAVaNT wth respect to δ, a corrdor subnetwork (TroyCor) of the Troy, Mchgan network was constructed (20 lnks, 10 nodes). Ths network corresponds to a par of parallel, sx-mle long arteral segments of John R and Dequndre Roads. The two facltes are comparable, although Dequndre Road has slghtly hgher capacty and hgher speed lmts. Whle both facltes have sgnals usng fxed tmngs coordnated for progresson based on free-flow lnk travel tmes, there are sgnfcant delays at each of the major sgnalzed ntersectons. Accuracy n lnk-tme predcton s especally mportant n sgnalzed corrdors snce the lnk travel tme experenced can be strongly nfluenced by the coordnaton of node arrval and the sgnal phasng. Lnks n ths arteral network are 1 mle or less n length. A network comprsed of longer lnks (e.g., 10 mles) lke those found n hghway networks would put an even larger premum on predcted travel tmes, because current lnk reports would be 15 mnutes or more out-of-date. An experment was performed on the TroyCor network usng 50 percent guded and 50 percent unguded vehcles. The unguded vehcles were assumed to take the fastest freeflow path, that s, the fastest expected path when the network s empty of vehcles (Dequndre). Over 2500 vehcles were ntroduced travelng southbound n the network over a 30 mnute perod. Ths travel demand exceeds the capacty of ether Dequndre or John R alone but does not exceed the capacty of the two facltes n combnaton. The network (Fgure 10) was ntally empty of vehcles. Therefore, the level of travel demand necesstates effcent allocaton of guded vehcles over tme between the two facltes takng nto account predcted delay at each of the downstream sgnalzed ntersectons. Travel tmes were measured for all the vehcles generated n the 30 mnute perod n ten second tme slces. An ntal travel tme forecast correspondng to a 100 percent unguded vehcle loadng was used to seed each run of the test. At the tested congeston level, vehcles experence between 4-6 mnutes of delay on a trp of between mnutes. Under no backdatng, δ = 0, a fxed pont was dentfed usng SAVaNT. The value of δ was then ncremented by 0.1 to 0.7 untl a cycle appeared at 0.8. A cycle also results when δ = The results are graphcally llustrated n Fgure 8. In comparson wth the ntal SAVaNT fxed pont, a 12 percent mprovement n benefts was obtaned for the system as a whole, and travel tme for guded vehcles mproved by 6 percent. Note that for smaller 10

11 values of δ, guded vehcles have hgher travel tmes than at larger values of δ but enjoy a mnute or more advantage over unguded vehcle travel tme. As delta ncreases, the mproved accuracy of the travel tme predcton allows guded vehcles to judcously avod the most serous ntermttent sgnal delays at ponts along both Dequndre and John R. The result s reduced travel tme for the guded vehcles as well reduced delay for unguded vehcles (both because the queues are smaller wthout the guded vehcles and because progresson n the corrdor s more lkely to be mantaned). When δ = 0.7, the travel tmes of the two vehcle classes are nearly the same. Although an nterestng result, as we wll see n Secton 7, dentcal travel tmes for guded and unguded vehcles are not always obtaned wth the largest possble values of δ. The number of teratons to convergence trended upwards wth an ncreasng level of accuracy n the lnk tme predcton method (Table 1). SAVaNT requres fewer teratons to dentfy routng polces that reman relatvely stable from tme-slce to tme-slce, and more teratons when dentfyng hghly tme-varant polces. The ncreased lnk tme accuracy provdes the router wth the ablty to mplement new routng polces wth a shorter tme-lag on control, and for the Troy Corrdor model, translates ths nto more frequent swtchng between dentfed fastest predcted paths. For comparson, alternatve routng schemes for guded vehcles were tested. A network of 100 percent unguded (fastest free-flow path) vehcles averaged mnutes to traverse the corrdor. Note that the use of fastest free-flow path n ths case results n all the travel demand attemptng to use Dequndre Road and no travel demand on John R, so ths value should be consdered a worst-case assgnment and not representatve of current condtons on the two facltes. A more complex test was also conducted usng a mx of 50 percent shortest free-flow path and 50 percent non-predctve guded vehcles (fastest path based on current travel tmes). Ths test resulted n an average travel tme of mnutes for the guded vehcles, for the unguded vehcles, and for the system as a whole. Note that the presence of sgnalzaton n TroyCor penalzes less accurate predctve methods compared to non-predctve methods. When δ = 0, SAVaNT returns a system travel tme of mnutes, compared wth for the test case usng non-predctve dynamc route gudance, an ncrease of 3.7 percent. However, when δ = 0.7 then system travel tme under SAVaNT s 7.3 percent smaller than the non-predctve method. 6. Resoluton of Cyclng Through Delayed Lnk-Tme Estmaton 11

12 The results descrbed n secton 4 suggest a method whch s ntalzed wth a convergent soluton n SAVaNT and then seeks the maxmum value of δ where a convergent soluton may stll be obtaned. From another perspectve, we mght consder the complement of that stuaton, namely when SAVaNT has termnated wth a cycle, rather than a fxed pont. We mght then ncrementally decrease the value of δ untl a convergent soluton s obtaned. As ndcated n Fgure 9, δ s not necessarly bounded below at δ = 0. The effects of decreasng δ are qualtatvely the same whether δ s postve or negatve. The frst effect s that lnk travel tmes generated are nherently less accurate. The second effect s that SAVaNT s made more lkely to fnd a convergent soluton. For example, SAVaNT does not fnd a fxed pont wth δ = 0 n the Troy corrdor model when 80 percent of the traffc on the network are guded vehcles. However, when δ = 0.2, a fxed pont s found resultng n a system travel tme of mnutes. In general, the hgher the fracton of guded vehcles traversng the network, the smaller the value of δ requred to force convergence n SAVaNT. If we consder the effect of travel tme profles n a sequence of teratons n SAVaNT, the effect of a changng the value of δ becomes more clear. Consder the smple case where δ = 1.0 and we have a sngle guded vehcle traversng the network. Let P 1 be the fastest volume-ndependent path n the smulaton, wth predcted freeflow travel tme CP ( 1 ). In the smulaton, a guded vehcle traverses the set of lnks L 1 :l P 1, and reports lnk travel tmes whch are hgher than freeflow tmes (because of the presence of traffc). Let the dfference between these hgher reported travel tmes and the predcted travel tme be C( P 1 ). Assume there exsts some alternatve path, P 2, whch contans at least one lnk not contaned n L 1. If CP ( 1 )< CP ( 2 ) and CP ( 2 )< CP ( 1 )+ CP ( 1 ), then a cycle must develop. When δ < 1.0, however, the sngle vehcle n the example above cannot mpact the same route gudance decson n followng teratons. But we may have many lnk departures wthn a tme slce, so the margnal cost mpact of our routng decson s also dependent on the number of vehcles gettng the same next-lnk nformaton. In ths case, for vrtually any δ, the margnal cost of a partcular assgnment can potentally yeld a cycle. When δ decreases, the margnal cost effects are pushed downstream n tme, causng some guded 12

13 vehcles to reman on a suboptmal path due to the tme lag n provdng nformaton about predcted condtons. The remanng vehcles, makng ther routng decsons later and hence unaffected by earler routngs despte the tme lag, wll be fewer n number and thus have lttle effect on observed lnk travel tmes. Therefore they wll be less lkely to change routes from one SAVaNT teraton to the next, makng t easer to obtan a fxed pont. Wthn SAVaNT, for example, f route gudance for all vehcles remans constant for two consecutve teratons through some ntermedate perod h, where h < H, our soluton horzon, then that partal routng polcy wll reman stable even f SAVaNT fals to converge. In the case of a fxed pont, stable routng forecasts are constructed wth monotoncally ncreasng values of h untl fnally h = H. 7. A Convergent Approach for SAVaNT Consder H, our soluton horzon. If we set δ to be strongly negatve ( δ large), then the effect n SAVaNT would be to push the delay downstream n tme past the soluton horzon. Thus, the smulated traffc would have no effect on the travel tme forecasts generated n SAVaNT. SAVaNT would only construct travel tme forecasts based on the default lnk tme travel tmes (free flow) and converge mmedately. A more precse proof of ths clam s stated below. Clam: Pf. There exsts some fnte value of δ such that SAVaNT must converge. Assume H fnte. Let L be the set of lnks n any fnte network. Under the non-restrctve assumpton that c l ()> t 0 l L, t = 1,2,LH, there exsts some fnte δ l () t δ l ()c t l ()< H t l L, t = 1,2,LH. Let δ < δ l () t l,t. Then δ s fnte snce every δ l () t s fnte. If we set δ l () t δ l L,t = 1, 2,LH, the smulaton module n SAVaNT produces lnk travel tme estmates c l ()= t c 0 l l L, t = 1,2,LH, where c 0 l s the free-flow lnk travel tme of lnk l. In any two consecutve teratons of SAVaNT, the travel tme profles 13

14 C n = C n 1 = c 0 l l L, t = 1,2,LH. Snce C n = C n 1, SAVaNT has converged to a fxed pont. We may thus force convergence n SAVaNT by choosng a suffcently small value of δ. However, we seek a value of δ as close to 1.0 as possble to realze the hghest lnk tme predcton accuracy and the resultng mprovement n travel tme savngs. The extreme case δ = δ corresponds to a predcton of free-flow travel tmes across the network for all tme slces, the least accurate depcton of a network whch has nonzero travel demand. A soluton procedure explotng a varable- δ accuracy method s outlned n Algorthm 1. Ths approach was tested on the Troy corrdor model wth σ = Note that we could have started from δ = 1.00and monotoncally decreased δ. These approaches are equvalent f there exsts δ * such that SAVaNT cycles for all δ > δ * and converges for all δ < δ *. Such a condton s dffcult to prove, but s emprcally borne out n TroyCor. Practcally, very small values of δ do not appear necessary for convergence. Convergence thresholds for SAVaNT-CNV are presented n Table 2. For TroyCor, δ s observed to be roughly However, SAVaNT obtaned stable results wth δ as hgh as 0.9, and no worse than -0.2 (mplyng the use of an nduced reportng delay as n Fgure 9), operatng at forecast accuracy far superor to the worst case δ. When δ δ, the percentage of lnks reported at free-flow over the entre horzon s 100%, whereas for δ n the stable range dentfed for TroyCor, there were no lnks reported at free-flow over the entre horzon. In general, how small δ must be for SAVaNT to fnd a stable forecast s lkely a functon of network geometry, the level of congeston, and market penetraton. Fxed ponts were dentfed at all market penetraton levels. Comparable evaluatons were made of both non-predctve route gudance and the verson of SAVaNT appled n [10]. As n the experment from Secton 4, lnk tmes are assumed to be dstrbuted every 10 seconds over a soluton horzon of 30 mnutes. In terms of both system and guded vehcle performance, SAVaNT-CNV dentfed better solutons than ether non-predctve route gudance or the verson of SAVaNT employed by Kaufman et al. Fgure 11 graphcally llustrates the beneft of mproved lnk tme predcton 14

15 accuracy n SAVaNT. Under gudance from SAVaNT-CNV, equpped vehcles experenced up to 9% percent faster travel tmes when non-predctve methods were employed. SAVaNT-CNV also sgnfcantly outperformed the older verson of SAVaNT. Travel tme savngs were most pronounced as the percentage of guded vehcles ranged between 40-90%. In all market penetraton levels, SAVaNT-CNV returned faster travel tmes for guded vehcles than ether of the methods tested, although dfferences for some lower market penetratons (<40%) were not statstcally sgnfcant. As ndcated from Fgure 12, SAVaNT-CNV also proved most effectve n reducng system travel tme. Solutons for SAVaNT-CNV provded 1-9% mprovements over the solutons obtaned by ether of the other two competng methods. Note that the travel tme reducton for SAVaNT-CNV s not monotoncally decreasng both because the forecasts are not perfectly accurate, and because ndvdual vehcles choose routes accordng to a useroptmal crteron rather than a system-optmal crteron. Table 3 summarzes the results obtaned for each approach. 8. Conclusons and Future Work Ths work demonstrates that an effectve predctve route gudance can be supported wthn the constrants of a decentralzed archtecture and all-or-nothng assgnment. The poor performance of prevous computatonal approaches under the constrants of decentralzed archtecture are shown to be mtgated by judcous adjustment n the accuracy of lnk travel tme profles. As an example, a verson of the SAVaNT antcpatory gudance method has been confgured to demonstrate ths effect. The revsed method, SAVaNT-CNV, confgured to produce the most accurate lnk travel tme predcton possble before the onset of cyclng, mproves over non-predctve route gudance, even n the scenaros most dffcult for predctve logc. Ths technque may prove useful to traveler nformaton servce provders utlzng current-day decentralzed archtectures who wsh to postpone or avod converson to a centralzed archtecture or reconfguraton to support mult-path assgnment. SAVaNT may requre addtonal teratons to fnd fxed ponts gven a more accurate method of lnk tme predcton, however. The SAVaNT-CNV method can requre the dentfcaton of several fxed ponts or cycles, snce we are applyng SAVaNT wth dfferent values of δ. In practce, ths can be done n parallel, so SAVaNT-CNV can be mplemented n the same real-tme applcaton as SAVaNT. 15

16 Several ssues reman to be addressed. A closer examnaton of antcpatory route gudance should be consdered n relaton to more complex characterzatons of drver behavor. In ths paper, we assume drvers of unguded vehcles follow fxed paths based on shortest free-flow routes and do not dvert or change behavor wth respect to experenced delay. Smlarly, we assume drvers of guded vehcles follow the fastest paths calculated from the predcted travel tmes. Other factors such as route complexty [17], drver famlarty, and route scenery all play nto traveler behavor. At ths pont, however, how one can predct traveler behavor s not well understood, partcularly n relaton to traveler nformaton. Another potental research area s the effect of ntal pont selecton. The selecton of a ntal pont whch s "close" to the best soluton wll drastcally speed the convergence of SAVaNT. However, ths must be weghed aganst the possblty that many fxed pont solutons may exst and that the selecton of a fxed pont may be crtcal n determnng whch of those solutons s dentfed. Fnally, the mpact of havng δ vary wth tme and locaton n the network has not yet been addressed. It may be possble to dentfy fxed ponts when lnk travel tme predcton s more accurate on some lnks and less accurate on others, yeldng addtonal mprovements n system average and guded vehcle travel tme. 16

17 REFERENCES [1] Cheslow, M., Hatcher, S., McGurrn, M., and Mertg, A. Alternatve Intellgent Vehcle Hghway Systems Archtectures, Techncal Report #92W48, The MITRE Corporaton, McLean, Vrgna (1992). [2] Cheslow, M., Hatcher, S., and Patel, V. An Intal Evaluaton of Alternatve Intellgent Vehcle Hghway Systems Archtectures, Techncal Report #92W63, The MITRE Corporaton, McLean, Vrgna (1992). [3] Cheslow, M., Hatcher, S., and Hsn, V. Communcaton, Storage, and Processng Load Requrements of Alternatve Intellgent Vehcle Hghway Systems Archtectures, Techncal Report, #93W18, The MITRE Corporaton, McLean, Vrgna (1993). [4] U.S. Department of Transportaton, The Natonal Intellgent Transportaton Systems Archtecture, Ver. 2.0, Washngton, DC (1998). [5] [6] U.S. Department of Transportaton, Theory of Operatons, Natonal Intellgent Transportaton Systems Archtecture, Washngton DC (1998). [7] bd, pp [8] Lee, C.K., A Multple-Path Routng Strategy for Vehcle Route Gudance Systems, Transportaton Research: C, Vol. 2, No. 3, pp (1994). [9] Jayakrshnan, R., H. Mahmassan, and Hu, T., An Evaluaton Tool for Advanced Traffc Informaton and Management Systems n Urban Networks, Transportaton Research: C, Vol., 2, No. 3, pp (1994). [10] Kaufman, D.E., Smth, R.L, and Wunderlch, K.E. "An Iteratve Routng/Assgnment Method for Antcpatory Real-Tme Route Gudance", SAE Vehcle Navgaton and Informaton Systems Conference Proceedngs, P-253, (1991). [11] Wunderlch, K.E and Smth, R.L. "Large Scale Traffc Modelng for Route- Gudance Evaluaton: A Case Study", Unversty of Mchgan IVHS Program Techncal Report #92-08, Ann Arbor, Mchgan (1992). [12] Wunderlch, K.E. and Smth, R.L. "Refnement and Calbraton n the Troy Case Study Model for Antcpatory Route Gudance Evaluaton", Unversty of Mchgan IVHS Program Techncal Report #93-04, Ann Arbor, Mchgan (1993). [13] Chen, K., and Underwood, S., "Research on Antcpatory Route Gudance", SAE Vehcle Navgaton and Informaton Systems Conference Proceedngs, P-253, pp (1991). [14] Kaufman, D.E. and Smth, R.L., "Fastest Paths n Tme-Dependent Networks For Intellgent Vehcle-Hghway Systems Applcaton", IVHS Journal, Vol. 1(1), pp.1-11 (1991). 17

18 [15] Kaufman, D.E., Smth, R.L., and Wunderlch, K.E. User-Equlbrum Propertes of Fxed Ponts n Iteratve Dynamc Routng/Assgnment Methods, Transportaton Research, Part C, Vol. 6, No. 1, pp (1998). [16] Van Aerde, M., Voss, J., and McKnnon, G. Integraton Smulaton Model User's Gude, Verson 1.1. Department of Cvl Engneerng, Queen's Unversty, Kngston, Ontaro, Canada (1989). [17] Adler, J., Blue, V., Wu, T., Assessng Network and Drver Benefts from B- Objectve In-Vehcle Route Gudance, Transportaton Research Board 78 th Annual Meetng, Preprnt CD-ROM, Washngton, DC (1999). 18

19 Decentralzed Autonomous Route Gudance Decentralzed Dynamc Route Gudance Centralzed Predctve Route Gudance In-Vehcle Route Gudance Map Database Vehcle Locaton Route Selecton Route Gudance Map Database Vehcle Locaton Route Selecton Route Gudance Accept/Reject Path Lnk Travel Tmes Route Request Suggested Route Informaton Servce Provder (ISP) Map Database Current Lnk Travel Tmes Map Database Predctve Route Selecton Fgure 1. Example Route Gudance Archtectures Decentralzed Predctve Route Gudance In-Vehcle Route Gudance Map Database Vehcle Locaton Route Selecton Route Request (optonal) Lnk Travel Tmes Informaton Servce Provder (ISP) Map Database Lnk Travel Tme Predcton SAVaNT Fgure 2. Decentralzed Predctve Route Gudance Archtecture 19

20 Dynamc Router R π Complete Routng Polcy Smulaton A Lnk Travel Tme Profle C NO Fxed Pont? Lnk Travel Tme Profle C Stop YES Fgure 3. The SAVaNT teratve method c 1 c 2 l (t) l (t) c 3 l (t) c 4 l (t) tme slce t t Smulaton Clock (seconds) Fgure 4. Lnk Departures Reportng Lnk Travel tmes durng Tme Slce t c l () t τ l () c t l () t τ l () t tme slce t Smulaton Clock (seconds) Fgure 5. Lnk Departures Accurately Predct Past Travel Tmes 20

21 Travel Tme (mn) %, 100% 8.6 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent Guded Vehcles Unguded Guded Fgure 6. SAVaNT Impact on Travel Tme Performance by Market Penetraton c l () t τ l () c t l () t τ l () δ t c l () t τ l () t tme slce t Smulaton Clock (seconds) Fgure 7. Fractonal Backdatng n SAVaNT 21

22 Travel Tme (mn) cycle for delta > Delta (Backdatng Parameter) Unguded Guded Fgure 8. Impact of Increasng Delta on Travel Tme Performance n SAVaNT c l () t τ l () c t l () t τ l () t tme slce t τ l () δ t c l () t Smulaton Clock (seconds) Fgure 9. Induced Reportng Delay, δ < 0 22

23 Fgure 10. The Troy Corrdor Network: Subset of the Troy Model 12 Guded Vehcle Average Travel Tme (mn) % 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent Guded Vehcles NP SAVaNT SAVaNT-CNV Fgure 11. Guded Vehcle Travel Tme Performance 23

24 19 18 Average System Travel Tme (mn) % 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent Guded Vehcles NP SAVaNT SAVaNT-CNV Fgure 12. System Travel Tme Performance 24

25 Table 1 Number of Iteratons to Convergence Under Varyng Predctve Accuracy δ Iteratons to Convergence CYCLE Algorthm 1 SAVaNT-CNV Method Step 1. Set δ = 0 and = 1, and choose some small ncrement value, σ. Step 2. Run SAVaNT, producng S, the result of the teratve process. Step 3a. If S s a cycle, set δ = δ σ. Goto Step 4. Step 3b. If S s a fxed pont, set δ = δ + σ. Goto Step 4. Step 4. If S,S 1 both cycles or both fxed ponts, set = + 1. Goto Step 2. Otherwse, STOP. Table 2 Convergence of SAVaNT-CNV Maxmum Convergent Pct. Guded Vehcles Value of δ 10% % % % % % % % % %

26 Table 3 Summary of Travel Tmes Obtaned n Troy Corrdor Model Average System Travel Tme Improvement vs. NP (mn) Mkt. Pen. NP SAVaNT SAVaNT-CNV SAVaNT SAVaNT-CNV 0% % 0.00% 10% % 2.67% 20% % 4.00% 30% % 3.67% 40% % 5.43% 50% % 7.29% 60% % 1.08% 70% CYCLE NA 3.76% 80% CYCLE NA 8.34% 90% CYCLE NA 4.08% 100% % 1.03% Average Guded Vehcle Travel Improvement vs. NP Tme Mkt. Pen. NP SAVaNT SAVaNT-CNV SAVaNT SAVaNT-CNV 10% % 1.04% 20% % 1.17% 30% % 0.00% 40% % 3.94% 50% % 6.28% 60% % 2.49% 70% CYCLE NA 5.85% 80% CYCLE NA 8.08% 90% CYCLE NA 3.63% 100% % 1.03% 26