Transportation Research Forum

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1 Transportaton Research Forum On the Impact of HOT Lane Tollng Strateges on Total Traffc Level Author(s): Sohel Sbdar and Mansoureh Jehan Source: Journal of the Transportaton Research Forum, Vol. 48, No. 3 (Fall 2009), pp Publshed by: Transportaton Research Forum Stable URL: The Transportaton Research Forum, founded n 1958, s an ndependent, nonproft organzaton of transportaton professonals who conduct, use, and beneft from research. Its purpose s to provde an mpartal meetng ground for carrers, shppers, government offcals, consultants, unversty researchers, supplers, and others seekng exchange of nformaton and deas related to both passenger and freght transportaton. More nformaton on the Transportaton Research Forum can be found on the Web at

2 On the Impact of HOT Lane Tollng Strateges on Total Traffc Level by Sohel Sbdar and Mansoureh Jehan Ths paper shows how tollng (or prcng) strateges can be used to control the congeston levels of both untolled and hgh occupancy toll (HOT) lanes. Usng a userequlbrum method, the paper calculates the number of travelers on each route durng the peak perod and provdes a numercal analyss that determnes the dstrbuton of travelers for dfferent tollng strateges. It shows that wth the rght tollng strategy some travelers who ntally plan to use the untolled lane durng the peak perod wll change both ther routes (.e., select the HOT lane) and departure tmes (.e., depart earler or later). Usng ths result, the paper compares statc and dynamc prcng strateges and shows that wth a dynamc strategy a larger proft can be earned and congeston reduced n the untolled lane. INTRODUCTION Traffc congeston s a serous problem n urban areas, challengng transportaton polcymakers and rasng the total cost of transportaton. Traffc demand management has become an ncreasngly popular tool for managng congeston. Ths approach assumes travel demand s a quantty requested by road users whose sze and dstrbuton can be controlled by dfferent polces. These polces nclude congeston prcng, flexble work hours, carpoolng, and publc transportaton. Many congeston prcng studes have developed polces that smply maxmze expected proft or mantan an acceptable level of congeston for a hgh occupancy toll (HOT) lane. However, many transportaton planners are more concerned wth reducng congeston levels than maxmzng profts, and would beneft from a prcng polcy that changes a traveler s behavor and sgnfcantly lowers overall congeston. Some researchers, ncludng Small and Yan (2001) and Feldng and Klen (1993), have addressed the concept of value prcng, whch lets travelers choose between a free but congested lane and a prced but freeflowng roadway. Also called HOT lanes, these prced roadways allow sngle drvers to pay for more hghly valued servces. HOT lanes currently n use n the Unted States nclude Interstate 15 n San Dego, Quck Rde System on the Kate Hghway and the Northwest Freeway n Houston, and the SR 91 Express Lanes n Orange County, Calforna (Evans et al. 1993). Value prcng can also solve some of the problems assocated wth hgh occupancy vehcle (HOV) lanes, such as when they become congested from vehcles wth two passengers (HOV2), whle those HOV lanes allowng vehcles wth at least three passengers (HOV3) are underutlzed. HOT lanes allow sngle occupant vehcles (SOVs) to enter a lane by payng a toll, whle carpools or buses can use the same lanes for free or at a dscounted rate. Allowng SOVs to use HOV lanes va an approprate toll can control congeston and generate proft for the tollroad operators. The exstng prcng lterature consders proft maxmzaton, secondbest prcng, socal welfare maxmzaton, and a mnmum level of servce requrement as obectves n ther model development. A secondbest prcng polcy maxmzes socal welfare subect to a zerotoll constrant on the alternatve roadways. The optmal soluton for ths polcy s a weghted average of the margnal external congeston costs between noncarpoolng and carpoolng vehcles. Proft (or revenue) maxmzaton allows a planner to maxmze proft subect to a zerotoll constrant on other roads. Small and Yan (2001) compared proft maxmzaton and revenue maxmzaton and found that travelers behavors under both polces are almost dentcal. They also compared the outcomes of proft maxmzaton and secondbest prcng, and showed that proft maxmzaton sets hgher 119

3 HOT Lane Tollng Strateges tolls and lowers total socal benefts. Another type of prcng regme, mantanng servce level, sets the toll hgh enough to keep the flow of the prced roadway at a mnmum specfed speed. Smlar to secondbest prcng, ths obectve acheves socal optmum and has an addtonal constrant that guarantees a mnmum level of servce n the express lane. A detaled revew of these prcng polces s provded by Yang and Huang (1999), Brad (1996) and Verhoef et al. (1996). Unlke the lterature above, ths paper examnes the effect of HOT lane prcng on the congeston level of a regular lane,.e., untolled lane. It nvestgates a stuaton where a regular lane s congested durng the peak (.e., 56 p m.) and the HOT lane s underutlzed durng the shoulders of the peak perod (.e., 34 p.m. or 78 p.m.). It shows that an approprate toll motvates some travelers who plan to use the untolled lane durng the peak perod to use the HOT lane durng the shoulders of the peak. To llustrate, consder a regular and a HOT lane of the same length and freeflow traffc tmes connectng two ponts. The toll for the HOT lane s $3 from 45 p m. and $6 from 56 p m. A traveler, who plans to depart at 5 p.m. and s not wllng to pay $6, mght change her travel tme to 4 p.m. to take advantage of the HOT lane s lght traffc and $3 rate. Ths behavor smoothes out the departure rate durng the peak perod and lowers the regular lane s traffc. LITERATURE REVIEW Most studes on value prcng use ether statc or dynamc models. Statc prcng assumes that travel demand and costs are not tme senstve. As a result, the ntertemporal mpact of tolls on longterm congeston levels s not consdered. Many studes n ths area use margnal cost prcng to estmate an optmal congeston prce on transportaton networks. Ths prcng polcy makes peak perod travel more expensve wth a toll that s calculated as the dfference between margnal socal and prvate costs. Dafermos and Sparrow (1971) used statc congeston prcng to determne the optmal toll for general transportaton networks. Yang and Meng (1998) calculated statc congeston prcng usng the margnal socal and prvate costs of travel, where roads are modeled as bottlenecks. Unlke statc congeston prcng, dynamc prcng methods affect travel demand, travel costs, and toll levels over tme. A semnal paper by Vckrey (1969) ntroduced a dynamc congestonprcng model for a sngle bottleneck wth a fxed capacty and number of travelers. He took nto account workrelated trps and assumed the same desred departure tmes for all travelers. However, some travelers could change ther departure tmes to avod traffc congeston. If a traveler left early, she faced no congeston, but could possbly encounter costs assocated wth early arrval. If she left to arrve on tme, she could also face congeston that would make her late. Vckrey (1969) showed that n these stuatons equlbrum congeston occurred when no drver could reduce her trp cost by changng her departure tme. Arnott et al. (1993) extended Vckrey s model by consderng heterogeneous travelers. They developed a determnstc mathematcal model to establsh the efffect of an optmal tmevaryng toll on socal welfare. They also compared the mpact of dfferent tollng schemes, such as unform and stepfunctons on system effcency. Wth a tmedependent congeston toll and optmal congeston prcng, they acheved a more unform departure rate and reduced both congeston cost and travelng tme for commuters. Arnott et al. (1990) also consdered fxed demand n a network wth parallel routes, and used dynamc traffc assgnment to examne the mpact of dfferent prcng regmes on network congeston and reproduced hypercrtcal flow condtons. Carey and Srnvasan (1993) used nonlnear programmng to address system margnal costs, user externalty costs, and optmal congeston tollng and developed optmal tolls under a tmedependent travel demand. They compared optmal dynamc and statc tolls and found that optmal dynamc tolls depended on the congeston level and whether the congeston was at the begnnng or endng of a peak perod. Yang and Huang (1999) developed a tmevaryng prcng model for a road bottleneck when demand s elastc and determnstc. They used a contnuoustme optmal control approach to maxmze socal beneft. 120

4 HOT Lane Tollng Strateges Lu and McDonald (1999) used economc and smulaton models to compare frstbest, secondbest, and notoll polces n a model wth two routes and perods (peak and prepeak). They found that secondbest prcng polces are effectve n reallocatng traffc volumes, but less effectve than frstbest tollng. The toll level n the secondbest prcng polcy s lower than n a frstbest tollng polcy, and the socal welfare beneft obtaned from the secondbest tollng polcy s smaller than the gans from a frstbest tollng polcy. All the papers revewed manage traffc on a route by prcng t to maxmze expected revenue or socal welfare and make route demand (.e., the number of passengers that use the route) solely a functon of the prce, thus neglectng the lnkage between prce and demand on alternatve routes. To fll ths gap, ths paper addresses ths lnkage and uses a traffc assgnment method that consders the prces charged on alternatve routes. The problem s modeled from the perspectve of a socal planner whose obectve s to control congeston. A prme factor mpactng our model s the presence of travelers who adust ther departure tmes wth respect to a route s prce level. Consderng ths factor, the planner can manage total congeston by chargng appropprate tolls. The secton below addresses the problem and presents the traffc assgnment method used. After that, a numercal analyss s conducted to determne the dstrbuton of travelers durng the peak perod under dfferent prcng polces followed by conclusons. MODEL Consder two roadways that connect ponts A and B. They have the same length, L, and freeflow travel tme, FFTT. One s tollfree (NT) and the other s tolled (T). The only ponts of access and egress for both roadways are A and B, and travelers can only choose between these two routes to travel from A to B. HOVs can use both roadways for free, but SOVs can only use NT for free and must pay to use the HOT lane. Next, consder a total of N travelers n SOVs wth homebasedwork (HBW) trps who plan to travel from A to B durng the peak perod. HBW trps, usually made by SOVs, are most mportant durng peak perods because they are less flexble n departure tmes snce each person must arrve at work at a certan tme. The length of the peak perod s n, and t s dvded nto n tme slots wth unt lengths. Wthout loss of generalty, consder an odd number of perods from 38 p.m., gvng fve tme slots, 34, 45, 56, 67, and 78 respectvely. Each traveler has a preferred departure tme, and snce only HBW trps are consdered, assume that all travelers prefer a specfc tme slot, say tme slot zero. Dependng on the congeston level and toll, travelers mght change ther departure tmes f those tmes ncrease ther dsutltes of travel. Next, assume D = [D 2, D 1, D 0, D 1, D 2 ] s the vector of dsutltes assocated wth each tme slot, where D 1 determnes the dsutlty of travelng n tme slot, for = 2,1,0,1,2. The sze of ths dsutlty s the same for all travelers, but vares by departure tme. For example, departng n tme slot zero mposes no dsutlty, whle travelng n the other tme slots mposes postve dsutlty, wth the dsutlty of travelng n later tme slots beng greater than travelng n earler tme slots. However, the dsutlty of travelng n tme slot 1 s not necessarly equal to the dsutlty of travelng n tme slot 1. Ths s also true for tme slots 1, 2, and 2. In addton to dsutlty, travel tme and the toll level affect a traveler s cost, and travelers choose a roadway and tme slot to mnmze ther costs. To analyze the problem descrbed, the paper reles on a traffc assgnment model to fnd an equlbrum soluton that specfes the number of travelers on each roadway and n each tme slot. A wde range of traffc assgnment models can be employed to assgn traffc flow between an orgn and a destnaton among dfferent routes, wth system optmal and user equlbrum beng the most popular. System optmal models fnd an assgnment that mnmzes total network travel tme based upon the assumpton that travelers cannot change ther route wthout ncreasng total system travel tme. In the user equlbrum model adopted n ths study, travelers cannot mprove ther travel tme by swtchng routes. Usng the Frank and Wolfe (1956) algorthm, all flows are assgned to the ntal shortest path and the lnks are teratvely updated usng a volume delay functon, whch shows the 121

5 HOT Lane Tollng Strateges relatonshp between the cost of traversng a lnk and the flow on t. The algorthm then fnds a new shortest path between each orgn and destnaton and assgns a convex combnaton of flow to the new and old shortest paths. The above traffc assgnment method s used to calculate the demand for each tme slot. For a gven prce menu, the equlbrum number of travelers on each route where no travelers have an ncentve to change ther departure tme or ther route choce s determned. By changng prce, the dstrbuton of passengers among these tme slots changes because some passengers change ther routes or departure tmes. Ths change allows the model to determne the mpact of each roadway s prce on travel on the other roadways. Several factors mpact a traveler s route choce, such as travel cost, costs of usng alternatve routes, dscomfort assocated wth a route, or tme of travel. The cost of travel ncludes outofpocket costs (e.g., gas, parkng fees, and tolls). Snce the study s on the effect of tolls and tme of travel on travel behavor, the effects of gas prce and parkng fees are not consdered. For dscomfort, t s assumed that tme slot 0 has no dsutlty, and the dsutltes of the other tme slots ncrease wth dstance from ths slot. To apply the algorthm, the lnks and ther generalzed cost functons are defned. Each tme slot (T and NT) s treated as a separate lnk, and the volume delay functon for each tme slot and roadway s defned as follows. V β (1) C ( V ) = δp + γd + ϕt [1+ α( ) ] Cap for = (n 1)/2,,0,, (n 1)/2, and = T, NT. where: C D : generalzed cost of travel on each lnk. : cost assocated wth the dsutlty of choosng tme slot rather than tme slot zero. Note that the values of D are the same for both roadways. : freeflow travel tme (n mnutes), whch s calculated as follows. t L = 60. FFS Cap : capacty durng tme slot n roadway. p : toll level durng tme slot n roadway T. : the flow durng tme slot n roadway. L : length of the lnks (n mles). : freeflow speed of the lnks (n mles per hour). α, β, δ, γ, and φ are the parameters of the model. If a road s so congested durng tme slot that ts travel tme s greater than the length of that tme slot, then all the vehcles cannot clear that slot. In ths stuaton, the overflowng vehcles wll enter the next tme slot. The flow of each lnk,, can then be calculated usng: (2) V = V + O 1 where V s the ntal flow n tme slot on roadway, and slot 1 to tme slot on roadway. It s assumed that for used to calculate : O O 1 O 3 0 s the overflow of traffc from tme = all and the followng equaton s 122

6 (3) O TT FFTT Cap ( ) αfftt = β HOT Lane Tollng Strateges The volume delay functon n Equaton 1 s the Bureau of Publc Roads (BPR) functon wth generalzed cost, whch uses travel tme as the cost of traversng a lnk. Ths lnk travel tme can be calculated usng the freeflow travel tme of a lnk and the rato of lnk flow to capacty. However, the BPR functon wth generalzed cost assumes that traversng a lnk conveys costs other than lnk travel tme. These costs are presumed to be the possble toll and dsutlty of selectng a tme slot over tme slot 0. The appled FrankWolfe algorthm can be explaned usng the followng procedure: 1. Intalzaton: Perform allornothng assgnment based on C (0). Ths gves the flow vector V (1). Set the teraton number m to Update Travel Cost: Update the lnk travel cost C ( m) = C ( V ( m)). 3. Drecton fndng: Perform allornothng assgnment wth the updated C. Ths gves the auxlary flow vector Y (m) (m). 4. Lne Search: Fnd the soluton to. 5. Move: Set the flows to V ( m + 1) = V ( m) + λ ( Y ( m) V ( m)). 6. Convergence Test: If λ < ε stop; otherwse go to Step 2, where ε s a very small number (e.g., ). The algorthm yelds an equlbrum soluton n whch travelers are dstrbuted among the 2n lnks wth mnmzed possble cost and all lnks have the same cost. Snce each lnk represents a tme slot and a roadway, the N travelers are dstrbuted among each tme slot on a tollfree or tolled lane so that they experence the same travel cost. NUMERICAL STUDY The smulaton model was valdated and calbrated usng the TransCAD platform. The calbrated model provded nformaton on total congeston on each road n each scenaro. Usng ths nformaton, total travel tme, total cost of travelers, and the revenue generated by the tolls were calculated. A total of N = 20,000 travelers are assumed to travel from an orgn to a destnaton by the regular and HOT lanes durng the fvehour perod noted. As n the prevous secton, there are 10 lanes of the same length ( L ( FFS = 10 mles), capacty (Cap = 2250 vehcles per hour per lane), and freeflow speed = 65 mles per hour) for all tme slots and roadway combnatons. Each lane s assgned to a tme slot and each traveler has 10 route choces. For example, the frst lane can be consdered as tme slot 34 p m. n the regular lane. Calbraton and Valdaton of Smulaton Model To accurately perform the smulaton, the model s parameters are estmated by calbraton. Ths technque develops travel demand by estmatng varous parameters and examnng the ablty of the model to replcate actual traffc patterns. 1 Ths s done by solvng the equatons for the parameters of nterest by supplyng the observed values of the varables from surveys and expermental analyss and by tral and error to fnd the values of the parameters wth the largest probablty of beng accurate wthn an acceptable margn of error (.e., wthn 0.5% of actual values). After satsfactory parameters are obtaned, the model s performance s checked by comparng the observed and smulated travel tmes and traffc counts. 123

7 HOT Lane Tollng Strateges The paper also uses the calbraton results n Jehan et al. (2008). These authors used a corrdor of Interstate 83 n Baltmore, Maryland, and calbrated demand parameters teratvely by comparng the outputs of ther models wth observed data. In each teraton, the parameters were adusted to replcate the observed condton. Then, followng Oketch and Carrck (2005), they used a modfed Chsquare test based upon the dfference between the smulaton model s output and observed traffc counts to test ther results. At the 5% level of sgnfcance, they could not reect the hypothess that ther model adequately smulated the traffc flow patterns. Smlarly, to estmate travel tme parameters and dsutlty levels, the results of Jehan et al. (2008) are used and we assume that α = 0.5, β = 4 and the dsutlty vector s D = [28,20,0,15,250]. The demand parameters, ncludng δ, φ, and γ are estmated followng the approaches n Ltman (2008) and Kumar et al. (2004), whch are based on arc elastctes of demand. Also, usng the same varables as n these authors works, the estmated generalzed cost functon s as follows: (4) C (V ) 4 p D + 0.8t [1+ 0.5( V Cap )4 ] Equaton 4 s used for the rest of the numercal study. Base Cases To determne the ntal dstrbuton of travelers among dfferent tme slots, the frst consderaton s what happens when the HOT lane does not exst (NoHOTLane). The next s to consder when a HOT lane s avalable for free, and all travelers, ncludng SOVs, can use the extra lane (FreeHOT Lane). Gven that durng the peak perod (38 p.m.), more passengers prefer to depart n the mddle tme slots (e.g., 56 p.m.), t seems reasonable to expect normal dstrbuton of travelers n these cases. Ths dstrbuton enables us to study the mpact of only departure tme on route choce. For example, n the case of NoHOTlane, the consumers generalzed cost functon conssts of departure tme dsutlty and travel tme snce travelers can choose among fve tme slots for the regular lane. The results are presented n Table 1. The frst set of columns shows road type and tme slots, and the second and thrd sets are the results for the NoHOTLane and FreeHOTLane scenaros. Further, the table s dvded nto two sets of rows, one for the regular lane and the other for the HOT lane. Four parameters are reported for each tme slot: number of vehcles (flow), volumecapacty rato (VOC), travel tme (TT), and average speed of vehcles (speed). As expected, the dstrbuton of travelers among tme slots follows normal dstrbuton, wth more travelers departng n tme slot 0 (56 p m.). In the NoHOTlane case, all tme slots n the regular lane are overutlzed, wth the volumecapacty rato rangng from 1.68 to 2.3. The length of each tme slot of 60 mnutes s exceeded n all the tme slots, whch means that some cars overflow nto the next tme slot. For example, n tme slot 2, where the flow s 3,772 vehcles per hour and travel tme s 91.4 mnutes, the road wll not be cleared for travelers who depart from 34 p m., and some of them wll contnue ther trps nto tme slot 1. Usng Equaton 2, the total overflow of tme slot 2 s 495 vehcles, and ths has been added to. Thus, 3, = 4,430. The overutlzed and overflow tme slots are hghlghted wth lght and dark shadows, respectvely. Based on the length of each tme slot and the estmated parameters, the free flow travel tme s mnutes. In Table 1, the travel tmes of all tme slots exceed 60 mnutes, wth the hghest beng tme slot zero (155 mnutes). The average speed of vehcles s also as low as 8 mles per hour whle the free flow speed s 65 mles per hour. Note that the dsutltes of usng dfferent tme slots are D = [28,20,0,15,25], whch means departng earler nduces hgher dsutltes than departng later. The values of the dsutltes gve a dstrbuton skewed toward the left (hgher flows n slots 1 and 2 compared wth slots 1 and 2 even wthout consderng the overflows). 124

8 Table 1: Flow, Travel Tme, and Speed of Tme Slot Speed Regular Lane Road Perod a 2 (34) 1 (45) 0 (56) 1 (67) 2 (78) Flow NoHOTLane VOC b TT c Speed Flow HOT Lane Tollng Strateges FreeHOTLane VOC TT Speed HOT Lane 2 (34) 1 (45) 0 (56) 1 (67) 2 (78) a There are fve tme slots for each road. The frst tme slot s 34 p m., and we label t as tme slot 2. The other tme slots are labeded accordngly. b If VOC 0.9 (or the flow of a tme slot exceeds 1835), the tme slot s consdered overutlzed (ndcated by lght shadow). c If TT 60, then the road has an overflow, (ndcated wth dark shadng). For the rest of our numercal analyss, HOT lane s added and the results compared n terms of the mpact of dfferent prcng scenaros on total traffc level. Frst, consder a benchmark scenaro where the HOT lane s avalable for free,.e., p = 0. When SOVs can use the HOT lane for free, t can be treated as a newly constructed regular roadway. The second set of columns n Table 1 shows the traffc flow, VOC, travel tme, and speed among fve tme slots of both lanes. A roadway s overutlzed f VOC 0.9. Snce the HOT lane s free, roadway use s almost even, wth three overutlzed tme slots (1, 0 and 1) and two underutlzed tme slots (2 and 2). The traffc flow s stll normally dstrbuted, and more travelers use tme slot zero than the others. Snce we assumed that the dsutlty of departng earler s hgher than later, the dstrbuton s skewed toward the left, whch means that and for = T, NT. Because of low demand for tme slots 2 and 2, the travelers can enoy the freeflow travel tme of 18 mnutes, whle the travel tme n other tme slots ncreases wth volume. The Hghway Capacty Manual (2000) grades the qualty or level of servce (LOS) of transportaton facltes on an AF scale, wth A beng the best and F the worst. Based on that manual, the LOS of tme slots mnus 2, 1, 0, 1, 2 are A, C, D, C, and A, respectvely. Statc Strategy In ths secton, a fxed prce s used for the HOT lane durng the peak perod (38 p.m.). Although a statc prcng plan does not gve the planner the ablty to control the traffc or maxmze revenue, ths type of prcng saves costs wth nfrequent prce changes. Wth dynamc tolls, the planner needs more equpment to control the toll and announce t to travelers. Statc tolls help travelers choose ther routes before approachng the entrance of HOT lane. However, there are some dsadvantages wth fxed tolls. The nablty to use prce as a tool to control traffc or to maxmze proft s sgnfcant. In addton, the constrant of havng a certan level of servce n the HOT lane makes the fxed toll polcy less attractve: a low toll makes the HOT lane overutlzed and ncreases the possblty that traffc n the mddle tme slots wll rse to unacceptable levels, and a hgh toll encourages travelers to avod the HOT lane n the shoulder tme slots (.e., 125

9 HOT Lane Tollng Strateges tme slots 2, 2), leadng to ther underutlzaton. To llustrate, Table 2 provdes a numercal study of two dfferent toll levels ($1 and $2) over the peak perod. Table 2: Statc Prcng Comparson Road Statc 1 Statc 2 Type Perod Flow VOC a TT Speed Flow VOC TT Speed 2 (34) Regular 1 (45) Lane 0 (56) (67) (78) (34) HOT 1 (45) Lane 0 (56) (67) (78) a If VOC 1, then the road s congested (ndcated by shaded cells). These two cases are Statc 1 and Statc 2, respectvely. Compared to the FreeHOTLane case, fewer travelers use the HOT lane because t s no longer free. In both cases, the regular lane s overutlzed, and there s one case of overflow n tme slot zero. The travel tme of tme slot zero s 60.9 mnutes and causes 18 travelers to overflow nto the next tme slot. The tme slots wth overflow are hghlghted wth dark shadows. Both cases result n the HOT lane not meetng ts mnmum level of servce (VOC 0.9). Because the tolls are not hgh enough to prevent a large number of travelers from usng tme slots 1, 0, 1 the HOT lane s underutlzed durng the shoulder tme slots and overutlzed n the mddle tme slots. Compared wth the FreeHOTLane case, the flows on the regular lane n Statc 1 and Statc 2 are large, wth a VOC of n the Free HOT lane (Table 1) and VOC from n the Statc 2 case (Table 2). On the other hand, the number of HOT lane travelers n tme slots 2 and 2 drops sgnfcantly, wth VOC near 0 n the shoulder tme slots of Statc 1 and Statc 2 versus a VOC of 0.75 n the shoulder tme slot of the FreeHOTlane case. A $2 toll results n some travelers leavng earler or later to pay no toll and enoy the better LOS n the regular lane durng tme slots 2 and 2. When Statc 1 and Statc 2 are compared, t s clear that those travelers who prefer to use the mddle tme slots n Statc1 change ther travel tmes n Statc 2 and use the shoulder tme slots. For nstance, the dfference between n Statc 2 and Statc 1 s 0.16 whle the dfference between n Statc 2 and Statc 1 s 0.1, suggestng that hgher tolls make travelers choose shoulder tmes. Ths fact mples that some travelers wll alter ther route choces and departure tmes f the rght toll s charged. Dynamc Strategy The prevous secton showed that a fxed toll durng the peak perod does not effcently allocate travelers among the tme slots. When the shoulder tme slots of the HOT lane are almost empty, a hgh fxed toll leads to the overutlzaton of the mddle tme slots and volates the mnmum LOS n a few cases. On the other hand, a low fxed toll causes the HOT lane to be overutlzed for most tme slots. To cope wth ths problem, a dynamc polcy s employed. 126

10 Table 3: Dynamc Prcng: Comparng Basc and Proposed Model Road Type Perod Flow Dynamc 1 p = [1, 2, 4, 2,1] VOC a TT Speed HOT Lane Tollng Strateges Dynamc 2 p = [0.8, 1.7, 3.5, 2,,0.6] Flow VOC TT Speed 2 (34) Regular 1 (45) Lane 0 (56) (67) (78) (34) HOT 1 (45) Lane 0 (56) (67) (78) a If VOC 1, then the road s congested (ndcated by shaded cells). To study the performance of the exstng toll strateges, the toll menu n effect on Interstate 15 n San Dego, Calforna, are used (I15 FasTrak Program 2005). In ths menu, the toll levels of tme slots 2, 1, 0, 1, and 2 were fxed at $1, $2, $4, $2, and $1, respectvely, or p = [1, 2, 4, 2, 1]. Ths s called Dynamc 1 n ths study. Table 3 llustrates the results for two dfferent toll menus (Dynamc 1 and Dynamc 2) where the frst set of columns llustrates the results for Dynamc 1. Due to the hgh prces charged n all the tme slots, many travelers use the regular lane, and the HOT lane s underutlzed. The VOC of the HOT lane ranges from 0.28 to 0.72, and, except for one tme slot, the average speed s above 60 mles per hour, whch s very close to the free flow speed. However, the regular lane s overutlzed, wth VOC as hgh as 1.53 n tme slot 0. Tme slot 0 also has an overflow of 170 travelers because the travel tme s greater than 60 mnutes. When p = [0.8, 1.7, 3.5, 2, 0.6], hereafter Dynamc 2, the utlzaton of the HOT lane ncreases sgnfcantly and congeston on the regular lane s reduced. The VOC of the shoulder tme slots of the HOT lane ncreases from 0.28 to 0.44, whle the overall VOC of the HOT lane remans n the accepted range. The traffc flow on the regular lane n the Dynamc 2 decreases by 1,056 travelers compared to Dynamc 1. More mddle tme slot travelers swtch routes snce deceases by 0.12, whle and decrease by 0.9. Many of these travelers also change ther travel tmes. The 1,056 travelers who swtch routes use tme slots 1, 2, 2 nstead of tme slots mnus 1, 0, 1. The VOC of slots 2, 2 ncreases by 0.11 and 0.16, respectvely, compared to ncreases of 0.02 and 0.08, respectvely, for tme slots 1, 1. Ths shows that assessng an approprate toll makes more travelers change ther travel tmes and depart when there s less traffc. Ths results n less congeston on the regular lane and ncreases HOT lane utlzaton whle mantanng the mnmum level of servce. Table 4 compares the performace of the dfferent prcng polces and summarzes other factors (.e., total proft from toll collecton, consumer generalzed cost, average travel tme of consumers, and total overflow). As expected, total profts n the NoHOTLane and FreeHOTLane cases are 0. Interestngly enough, although Dynamc 2 sgnfcantly reduces generalzed cost and average travel tme, the planner gans almost as much proft as usng Dynamc 1. The average generalzed cost for consumers s very hgh n the base model where one less lane s avalable for the travelers. But consumers enoy the lowest generalzed cost n the FreeHOTLane case because two free lanes are avalable. 127

11 HOT Lane Tollng Strateges Table 4: Comparng the Performances of Dfferent Tollng Polces Type Proft Consumer Cost Avg. TT E(TT FFTT) 2 Overflow NO HOT lane Free HOT lane Statc Statc Dynamc Dynamc Another ndex n the table s E(TT FFTT), 2 whch captures the expected squared dfference between travel tme and free flow travel tme n each tme slot. Ths ndex measures the overall performance of each tollng polcy n terms of consumer travel tme, and assgns a hgher penalty to those tme slots wth hgh travel tmes. Among the cases wth HOT lane, E(TT FFTT) 2 s the hghest n Statc 2 snce more travelers use the mddle tme slot on the regular lane. In comparng Dynamc 1 and Dynamc 2, E(TT FFTT) 2 sgnfcantly reduces n the latter because the dstrbuton of consumers among dfferent tme slots s flat and the tme slots are congested. CONCLUSIONS The fndngs n ths paper provde nsght nto congeston prcng and hghway management. Upon arrval at a multlane hghway where one lane s free and another one s subect to toll collecton, a traveler observes the current toll and decdes whether or not to enter the HOT lane. Some travelers are wllng to change ther departure tmes and or routes to face lower generalzed costs. A user equlbrum method was used to determne the dstrbuton of travelers between HOT and regular lanes durng the peak perod, and to compare the mpacts of dfferent prcng polces on traveler behavor. It s shown by numercal example that a planner can manage congeston levels wth tolls. Further, a prce menu currently n effect on Interstate 15 n San Dego, Calforna, was used to show that the congeston level on the regular untolled lane can be reduced wthout loss of proft from toll collecton. An extenson of ths study s to develop an optmzaton model to calculate a prcng menu n whch departure rates are unform durng the peak perod. A multobectve optmzaton model that both reduces the regular lane s congeston level and maxmzes the total expected proft can also be developed. Fnally, one can defne multple classes of travelers who have dfferent values of tme and dsutltes of changng ther departure tmes. Endnotes 1. For a comprehensve revew, the readers are referred to Kutz (1992). References Arnott, R., A. de Palma, and R. Lndsey. A Structural Model of Peakperod Congeston: A Traffc Bottleneck wth Elastc Demand. Amercan Economc Revew 83 (1), (1993): Arnott, R., A. depalma, and R. Lndsey. Departure Tme and Route Choce for the Mornng Commute. Transportaton Research B 24 (3), (1990): Brad, R. Peakload Prcng of a Transportaton Route wth an Unprced Substtute. Journal of Urban Economcs 40 (2), (1996):

12 HOT Lane Tollng Strateges Carey, M. and A. Srnvasan. Externaltes, Average and Margnal Costs, and Tolls on Congested Network wth Tmevaryng Flows. Operatons Research 41 (1), (1993): Dafermos, S. and F. Sparrow. Optmal Resource Allocaton and Toll Patterns n Useroptmzed Transportaton Network. Journal of Transportaton Economcs and Polcy 5, (1971): Feldng, G. and D. Klen. Hgh Occupancy Toll Lanes: Phasng n Congeston Prcng a Lane at a Tme. Reason Foundaton, Polcy Study 170, (1993). Frank, M. and P. Wolfe. An Algorthm for Quadratc Programmng. Naval Research Logstcs Quarterly 3 (1956): Hghway Capacty Manual, Transportaton Research Board, Washngton, D.C., I15 FasTrak Program, Wlbur Smth Assocates. I15 Express Lanes Toll Modfcaton Study. December Avalable n sandag.org. Jehan, M., P. James, and A. Saka. Measurng Nonrecurrng Postncdent Traffc Recovery Tme: Comparng Shock Wave Theory and Smulaton Modelng. Maryland State Hghway Admnstraton, Offce of Traffc Safety, Hanover, MD, Kumar, C. P., D. Basu, and B. Matra. Modelng Generalzed Cost of Travel for Rural Bus Users: A Case Study. Journal of Publc Transportaton 7 (2), (2004): Kutz, M. Handbook of Transportaton Engneerng. McGrawHll Handbooks, Ltman, T. Transportaton Elastctes, How Prces and Other Factors Affect Travel Behavor. Vctora Transportaton Polcy Insttute, Vctora, B.C. Canada, Lu, L. and J. McDonald. Economc Effcency of Secondbest Congeston Prcng Schemes n Urban Hghway Systems. Transportaton Research B 33 (3), (1999): Oketch, T. and M. Carrck. Calbraton and Valdaton of a Mcrosmulaton Model n Network Analyss. The Proceedngs of the 84rd TRB Annual Meetng, Washngton, D.C., Small, K. and J. Yan. The Value of Value Prcng of Roads: Secondbest Prcng and Product Dfferentaton. Journal of Urban Economcs 49 (2), (2001): Verhoef, E., P. Nkamp, and P. Retveld. Secondbest Congeston Prcng: the Case of an Untolled Alternatve. Journal of Urban Economcs 40 (3), (1996): Vckrey, W. Congeston Theory and Transportaton Investment. Amercan Economc Revew 59,s (1969): Yang, H. and H. Huang. Carpoolng and Congeston Prcng n a Multlane Hghway wth Hghoccupancyvehcle Lanes. Transportaton Research A 33 (2), (1999): Yang, H. and Q. Meng. Departure Tme, Route Choce and Congeston Toll n a Queung Network wth Elastc Demand. Transportaton Research Part B 32(4), (1998):

13 HOT Lane Tollng Strateges Sohel Sbdar holds the poston of assstant professor n the Charlton College of Busness, Unversty of Massachusetts at Dartmouth. He receved an M.S. n economcs and a Ph.D. n ndustral engneerng from Vrgna Tech. Hs research nterests nclude dynamc prcng, decson makng under uncertanty, transportaton scences, and game theory. Mansoureh Jehan s an assstant professor n the Department of Transportaton and Urban Infrastructure Studes at Morgan State Unversty. She receved a Ph.D. n cvl engneerng (transportaton systems) and an M.S. n economcs from Vrgna Tech, a master s degree n socoeconomcs systems engneerng from IRPD, and a B.Sc. n computer engneerng from Iran Natonal Unversty. Her research nterests are n transportaton plannng, transportaton economcs, ntellgent transportaton systems, and traveler behavor. 130