Predicting Corridor-level Travel Time Distributions Based on Stochastic Flow and Capacity Variations DRAFT. Hao Lei

Transcription

1 Abstract Predicting Corridor-level Travel Tie Distributions Based on Stochastic Flow and Capacity Variations Hao Lei Departent of Civil and Environental Engineering University of Utah Salt Lake City, Utah, 84 Eail: Xuesong Zhou Assistant Professor Departent of Civil and Environental Engineering University of Utah Salt Lake City, Utah, 84 Tel: Fax: Eail: George List Professor Departent of Civil, Construction, and Environental Engineering North Carolina State University Raleigh, NC Eail: Jeffrey Taylor Departent of Civil and Environental Engineering University of Utah Salt Lake City, Utah, 84 Eail: Trip travel tie variability is an iportant easure of transportation syste perforance and a key factor affecting travelers choices. This paper ais to establish a point-queue based end-to-end travel tie prediction ethod on a corridor with ultiple erges and diverges. A set of analytical equations is developed to calculate the nuber of queued vehicles ahead of the probe vehicle and further capture any iportant factors affecting end-to-end travel ties: the prevailing congestion level, queue discharge rates at bottlenecks, and flow rates associated with erges and diverges. Based on ultiple rando scenarios and a vector of arrival ties, the experienced delay at each bottleneck along a corridor is recursively estiated to produce end-toend travel tie distributions. The proposed odel can incorporate stochastic variations of bottleneck capacity and deand, and explain the travel tie correlation between sequential links.

2 A high-fidelity vehicle trajectory data set available fro the NGSIM project is used to verify the proposed ethod, and the sources of prediction error are systeatically exained. Key words: travel tie variability, stochastic capacity and deand, point-queue, NGSIM. Introduction Travel tie has long been regarded as one of the ost iportant perforance easures in transportation systes. Recently, significant attention has been devoted to evaluating and quantifying travel tie variability due to its influences on travelers ode, route and departure tie choices. Operating and anageent agencies have also increased efforts for onitoring and iproving the reliability of transportation systes through probe-based data collection, integrated corridor anageent and advanced traveler inforation provision. In particular, vigorous data collection efforts have been ade to iprove easureent quality and uncover the root sources of travel tie unreliability. For instance, a wide range of corridor anageent strategies have been designed to balance traffic between freeway and arterial corridors in response to various non-recurring traffic congestion sources. In addition, advanced traveler inforation provision systes have been enhanced to provide reliability-related inforation to enable travelers to eet their liited travel tie budget constraints. There also appears to be a growing trend toward incorporating end-to-end trip travel tie variability easures, and their related traveler behavior coponents, into traffic network analysis and anageent odels. Although noteworthy progress has been ade in travel tie variability quantification, there are still a nuber of theoretical challenges to be addressed in the above practically iportant applications. The first challenge is how to establish a systeatic estiation and prediction fraework to capture the stochasticity of the traffic flow under both recurrent and nonrecurrent congestion conditions. Such a fraework is vital for both travelers and operating agencies (e.g., traffic anageent tea) to obtain accurate tie-dependent travel tie variability inforation to ake infored decisions. Past research on travel tie variability usually assues either stochastic deand or stochastic capacity. Few studies have considered the travel tie variability under both stochastic deand and capacity situations. Additionally, erging and diverging cause significant disturbances on both ainline lanes and raps. Hence, all of these sources of uncertainty ust be carefully exained and addressed for corridor-level travel tie variability quantification. The objective of this paper is to develop an integrated travel tie variability prediction fraework which can characterie the travel tie dynaics through tie-dependent deand and capacity fluctuations under both recurrent and non-recurrent congestion conditions. Potential applications for our travel tie variability prediction ethods are investigated, and an illustrated exaple application is presented. Particularly, we use high-fidelity NGSIM vehicle trajectory data to validate the predicted travel tie distributions under different traffic conditions, and investigate lane-by-lane travel tie variability.. Literature review and otivations This section first reviews deterinistic link-based travel tie odels and exaines the underlying distributions of capacity and deand eleents. This is followed by a discussion on the various odeling approaches for connecting travel tie variability with its root sources... Travel tie odels and capacity/deand eleent distributions

3 Within the subject of analytical dynaic traffic network analysis, the whole-link odel is widely adopted to describe link travel tie evolution due to its siple description of traffic flow propagation through an analytical for. The link travel tie function introduced by Fries et τ t on a single link at a tie t as a linear function of the al. (993) defines the travel tie ( ) nuber of vehicles ( ) xt on the link at tie t: ( t) a bx( t) τ = + () where a and b are constants in the above general linear for. A nondecreasing and continuous function is defined to calculate the nuber of vehicles on the link based on the inflow and vt, at tie t: outflow rates, u( t ) and ( ) t ( ) = ( ) + ( ( ) ( )) x t x u s v s ds () Meanwhile, soe ore general non-linear travel tie functions have been proposed as: ( ) ( t) f xt ( ), ut ( ), vt ( ) τ =. (3) A special case of this for, introduced by Ran et al. (993), decoposes the link travel tie as two different functions: g accounts for flow-independent travel tie and g accounts for the queuing delay. A detailed atheatical representation is shown below. ( t) g xt ( ), ut ( ) g xt ( ), vt ( ) They later showed that, by assuing g and and g g xt ( ) g vt ( ) a b τ = + (4) = +, and Eq. 4 can be rewritten as τ g are separable, i.e., g g xt ( ) g ut ( ) ( t) α f( ut ( )) gvt ( ( )) hxt ( ( )) = + a b = (5) where α is the free flow travel tie, and f (), g() and h() correspond to the functions of link inflow rate, link outflow rate and the nuber of vehicles on the link, respectively. Dagano (995) draws attention to probles with the general for in Eq. (3), indicating vt would lead to that either a rapid decline in the inflows u( t ) or a rapid increase in outflow ( ) unrealistic travel tie. Thus, he recoended oitting u( t ) and ( ) link travel tie to a function of the nuber of vehicles on the link, that is, ( t) f xt ( ) vt fro Eq. 3, reducing the ( ) τ =. Although the link travel tie function odels provide soe degree of siplification on travel tie analysis, there is one significant drawback. Traffic congestion usually occurs at soe bottleneck, and queues are produced and often grow beyond the bottleneck, which is difficult for any travel tie function to capture (Zhang and Nie, 5). In dynaic traffic assignent and other applications, the vertical queue or point-queue odel (Dagano, 995) was widely adopted to describe bottleneck traffic dynaics (Zhang and Nie, 5). In a queuing-based travel tie odel, it is iportant to capture the variations of queue discharge flow rates and incoing deand to a bottleneck. 3

4 Conventionally, freeway capacity is viewed as a constant value the axiu discharge flow rate before failure (HCM, ). However, the capacities vary according to different external factors in real life situations. Conceptually, capacity or discharge flow rate can be represented as the reciprocal of the average of vehicle headways. Over the past decades, any researchers have developed a nuber of headway odels to describe its distribution. Representatives of these odels include the exponential-distribution by Cowan (975), noral distribution, gaa-distribution, and lognoral-distribution odels by Greenberg (966). Incidents are one of the ajor contributing factors in capacity reductions, and the agnitude and duration of capacity reductions are directly related to the severity and duration of incidents (Kripalari and Scherer, 7; Guiliano, 989). In quantifying capacity reduction, the HCM provides guidance for estiating the reaining freeway capacity during incident conditions. Using over two years of data collected on freeways in the greater Los Angeles area, Golob et al. (987) found that accident duration fit a lognoral distribution. By extending the research of Golob et al., Guiliano (989) applied a lognoral distribution when analying incident duration for 5 incidents in Los Angeles. It is coonly observed that travel deand fluctuates significantly within a day. During the orning and evening peak hours, surging deand ay overwhel a roadway s physical capacity and results in delays (FHWA, 9). Waller and Ziliaskopoulos (), Chen et al. (3) and La et al. (8) have used the noral distribution for odeling travel deand variation. Other researchers have odeled travel deand using the Poisson distribution (Haelton ; Clark and Watling, 5) and the unifor distribution (Ukkusuri et al. 5)... Methods for estiating travel tie variability Substantial efforts have been devoted to travel tie variability estiation over the last decade, producing several different approaches for estiating travel tie variability. Statistical approaches (Richardson 3; Oh and Chung 6) have been widely adopted to quantify travel tie variability fro archived sensor data. In recent studies investigating the different sources of travel tie variability, Kwon et al. () proposed a quantile regression odel to quantify the 95th percentile travel tie based on the congestion source variables, such as incidents, weather. In their ulti-state travel tie reliability odeling fraework, Guo et al. () provided connections between the travel tie distributions and the uncertainty associated with the traffic states, e.g., with incidents vs. without incidents. In addition, they (Park et al. ) show that a ulti-ode odel could lead to better representations of real-world observations copared to single-ode odels (represented by ean and variance paraeters). A second approach uses nuerical approxiation ethods to characterie travel tie variability distributions as a result of stochastic capacity and stochastic deand. Given a stochastic capacity probability distribution function (PDF), a Mellin transfors-based ethod was adopted by Lo and Tung (3) to estiate the ean and variance of travel tie distributions. Using a sensitivity analysis fraework, Clark and Watling (5) developed a coputational procedure to construct a link travel tie PDF under stochastic deand conditions. Given various sets of traffic flow assignent results, Ng and Waller () introduced a fast Fourier transforation approach to approxiate the travel tie PDF fro underlying stochastic capacity distributions. Although it can quantify the ipacts of deand and capacity variation on the travel ties, the steady-state travel tie function-based approach is still unable to address the underlying tie-dependent traffic dynaics. In order to account for the inherent tie-dependent traffic dynaics, soe researchers have incorporated point-queue odels into travel tie variability estiation techniques. 4

5 Assuing lognoral distributions on capacity and deand, Zhou et al. () adopted a pointqueue odel and a cuulative count curve approach to quantify the day-to-day travel tie variability. For single bottlenecks, the travel tie variability is analytically derived fro the variation paraeters in deand and capacity. The challenging issue in extending their odel on a corridor-level analysis is how to quantify end-to-end travel tie along several corridors where downstrea and upstrea traffic states are correlated. Using a dynaic traffic assignent siulator, Alibabai () developed an algorithic fraework to investigate the properties of the path travel tie function with respect to various path flow variables. While realistic siulation results require significant efforts in siulation/assignent odel calibration, this approach is particularly suited for studying the effects of various uncertainty sources and assessing the benefits of traffic anageent strategies and traffic inforation systes. The rest of the paper is organied as follows. Section describes the point-queue based end-to-end travel tie estiation fraework with deterinistic inputs. Monte Carlo siulation is applied in Section 3 to copute the travel tie distribution with stochastic inflow, outflow and discharge rates. High-fidelity vehicle trajectory data fro the NGSIM project is utilied to validate our ethods in Section 4.. Coputing end-to-end tie-dependent delay. Proble stateent Consider a corridor with M bottlenecks, where each node in a node-link structure represents a bottleneck, and the road segents between consecutive bottlenecks are odeled as a link with hoogeneous capacity. That is, node is the starting point of the corridor, node corresponds to bottleneck, and each link between bottlenecks is denoted as link (-, ), for M. Link (-, ) can also be described as link interchangeably. Figure illustrates a node-link representation for a corridor with M bottlenecks. Possible erge or diverge nodes are connected to bottleneck and are denoted as ' or '', respectively, so that the on-rap before node is denoted as (', ), and the off-rap before node is denoted as (, ''). In other words, we assue erge and diverge links are directly connected to the bottleneck. Link (,) Link (',) ' '' 3 M- M Link (-,) Figure : A node-link representation of a corridor with M bottlenecks In this study, we are interested in how to predict the travel tie fro bottleneck to bottleneck for a probe vehicle, departing at tie t =. The proposed odel ais to find the end-to-end travel tie, p, based on the following given conditions: () the nuber of vehicles x ( t ) on each link along the path at tie t, () the discharge flow rate for each bottleneck c, net and (3) the on-rap or off-rap flow rates f. For this study, the end-to-end travel tie is defined as the difference between the departure tie at bottleneck and the departure tie at bottleneck for probe vehicle. The departure tie at bottleneck is defined as the tie the probe vehicle leaves the queue at bottleneck, which is the tie when the nuber of vehicles in the queue before probe vehicle at bottleneck is. 5

6 The nuber of vehicles on a link is assued to be observable fro sensors, such as loop detectors, and the discharge flow rates and net flow rates fro on-raps and off-raps are assued to be estiated fro historical flow patterns or predicted based on prevailing traffic conditions (e.g., capacity reduction due to incidents). Indices: The notation for end-to-end travel tie is described below. : index for identifying a probe vehicle; k: index of a siulation instance used in Monte Carlo siulation; : index of bottlenecks and links along the corridor. Input: t : starting tie, t =. M: nuber of bottlenecks along the corridor of interest; FFTT : free-flow travel tie over link (-, ); c : queue discharge rate of bottleneck ; net f : net flow rate at a erge or diverge corresponding to bottleneck, that is, fro an on-rap to the ainline segent or fro the ainline to the off-rap; ( ) x t : nuber of vehicles on link ( -, ) at tie ( t ) µ : arrival rate of link (-, ) at tie t; v t : departure rate of link (-, ) at tie t; ( ) Variables to be calculated: ( t ) t ; τ : travel tie on link for probe vehicle entering the link at tie t; ( t ) λ : nuber of vehicles waiting at the bottleneck at tie t, that is, the nuber of queued vehicles behind bottleneck ; w : waiting tie in the vertical queue of bottleneck for probe vehicle ; t : arrival tie for probe vehicle at bottleneck ; p : end-to-end path travel tie fro node to bottleneck. 6

7 . Travel tie calculation In a point-queue odel, a link can be considered as two segents: the free-flow segent and the queuing segent. A vehicle can always travel at the free-flow speed on the free-flow segent until reaching the beginning of the queuing segent, where this vehicle joins the end of the queue waiting to be discharged. A queue is only fored if the link deand exceeds the bottleneck capacity, or the link arrival rate exceeds the link departure rate. To construct a deterinistic, nuerically tractable for for calculating the end-to-end travel delay along a corridor with ultiple bottlenecks, several iportant assuptions are ade in this study. ) A point-queue odel is adopted to calculate the delay on each link. On each link, a FIFO (First-In, First-Out) property is assued to assure that any vehicles that enter the link before tie t will exit the link before those entering after tie t. ) The link traversal tie is coposed of free-flow travel tie and queuing delay. The free-flow travel tie is constant and flow-independent. The queuing delay is deterined by the nuber of vehicles in the queue when the probe vehicle arrives at the bottleneck λ ( t + FFTT ) and the bottleneck queue discharge rate c. Thus, the link traversal tie is λ( t + FFTT) τ ( t) = FFTT + w( t + FFTT) = FFTT + (6) c where w( t FFTT ) + is the queuing delay when vehicle reaches the vertical queue of bottleneck at tie t + FFTT. 3) The erge and diverge links are connected to the beginning of the queuing segent for each bottleneck. 4) The bottleneck reains congested in the prediction horion, which extends fro the current tie t to the arrival tie of the probe vehicle at the bottleneck, t. The corresponding queue discharge rates c and net flow rates horion are assued to be constant. f net in the prediction The first two assuptions are widely used in a typical queuing odel. The third assuption akes it easy to incorporate the flow rate fro a erge/diverge without iplicitly considering the driving distance and free-flow travel tie fro the erge/diverge point to the bottleneck. Eq. (6) considers the arrival tie at the beginning of a link. By considering the arrival tie at the queue of bottleneck for vehicle, t, the link traversal tie can be rewritten as ( t) λ τ ( t FFTT) = FFTT + (7) c For a general queue with tie-dependent arrival and departure rates, a continuous transition odel can be used in Eq. (6) to update the nuber of vehicles in the queue at any given tie t. 7

8 dλ dt ( t) The nuber of queued vehicles λ ( t ) = µ ( t FFTT ) v () t (8) the queue length updating Eq. (8), as shown in Eq. (9). ( ) ( ) ( ) ( t) at tie t on bottleneck can be derived fro dλ λ t λ t dt λ t µ t FFTT v t dt t = + = + t dt t t t = λ t + µ ( t FFTT ) dt v () t dt t t t ( ) [ ( ) ()] Since the fourth assuption states that the bottleneck reains congested in the prediction horion, the departure rate is equal to the bottleneck capacity reaining challenge is to estiate the unknown queue length ( ) coplex integral of t µ ( t FFTT) dt. t t t t (9) v () t dt = c ( t t ). The λ at tie t, and calculate the For illustrative purposes, the following discussions start with the first bottleneck fro net x t and the net flow rate f Figure, where =. In this case, the nuber of vehicles ( ) associated with bottleneck are given. For the specific starting tie t, a probe vehicle enters the vertical queue of bottleneck at tie t = t + FFTT, and the nuber of vehicles in the queue at tie t is ( ) ( ) t t = t + t t FFTT dt c t t λ λ µ ( ) ( ) () Let us consider a sipler case without erge and diverge points, i.e., f net =. Thanks to t the first-in and first-out property, we can now show that λ ( t ) + µ ( t FFTT ) dt = x ( t ) left-hand side ( ) t t t. The λ t + µ ( t FFTT ) dt is the total nuber of vehicles stored in both the freeflow segent and the queuing segent before the probe vehicle. The right-hand side is the actual nuber of vehicles observed on the physical link. One can use Figure to ap or rotate soe of the vehicles fro the physical link (shaded) to the vertical stack queue, and the other vehicles on the physical link (not shaded) correspond to the vehicles that will arrive at the vertical queue between tie t and tie FFTT (that is, right before the probe vehicle). Notice that the length of the queue segent in the point-queue odel is equal to ero and has unliited storage capacity. Interested readers are referred to the paper by Hurdle and Son () to exaine the connection between physical queues and vertical stack queues. The individual coponents in Equation () are described visually using the cuulative vehicle count curves shown in Figure 3. The cuulative arrival curve A is equivalent to the A() t = µ t dt, and the cuulative arrival curve at the vertical integral over the arrival rate, ( ) stack queue V is the cuulative arrival curve shifted by the free-flow travel tie, V () t = A( t FFTT ), and thus V () t = µ ( t FFTT) dt. The cuulative departure curve D is 8

9 equivalent to the integral over the departure rate, D() t = v() t dt = c ( t t). Substituting t with values of t and t for Vt, () Figure 3 shows that V ( t) V ( t) = µ ( t FFTT) dt and thus ( ) ( ) and ( ) ( ) t λ t = λ t + µ ( t FFTT ) dt c ( t t ) Cuulative Vehicle Count Direction of traffic flow x(t) Free-flow segent Physical link Figure : A vertical stack queue λ(t) FFTT t V(t ) V(t) t t t x t = λ t + µ ( t FFTT) dt. A λ(t ) t w FFTT c ( t t) Figure 3: Visual representation for Equation () V Queue D Tie By further considering the net flow rate fro the erge or diverge point connected to the bottleneck, we now have t ( ) ( ) [ ( ) ] ( ) λ t = λ t + µ t FFTT c dt = x t + f t c t () net t 9

10 Continuing to link in Figure, the probe vehicle will arrive at the queue of bottleneck at t. Again, considering the FIFO assuption, the nuber of vehicles transferring fro the first link to the second link before the probe vehicle includes two ters, ( ) are the nuber of queued vehicles λ( t ) λ t + c t, which when the probe vehicle arrives at the first bottleneck at tie t, and those vehicles c t already entering the second link before tie t. Following the derivation logic for Eq. (), the nuber of vehicles waiting in the queue ahead of vehicle when it arrives at the second bottleneck at tie t is By substituting ( t ) ( ) λ ( ) ( ) λ t = t + c t + x t + f t c t () net λ fro Eq. (), Eq. () reduces to ( ) ( ) ( ) λ t = x t + f t c t + c t + x t + f t c t net net net net x( t) f t x( t) f t c t = More generally, for bottleneck : () The nuber of vehicles waiting at the vertical queue of bottleneck at tie t can be expressed as λ (3) net ( t ) = x ( t ) + ( f t ) c t (4) i i i i= i= () The arrival tie for the probe vehicle at bottleneck is where w ( t ) λ c t = t + w + FFTT = t + + FFTT (5) ( t) λ =. c (3) Finally, the end-to-end travel tie fro bottleneck to bottleneck is = + = i + i i= p t w FFTT w (6) In a suary, given the nuber of vehicles on each link, the queue discharge rate and the net flow on each bottleneck, the end-to-end travel tie for a vehicle can be calculated by applying Eqs. (4-6) iteratively fro link to link. At each iteration, first apply Eq. 4 to obtain the nuber of queued vehicles at the bottleneck, then copute the queuing delay and update the end-to-end travel tie up to the bottleneck of interest..3 Illustrative exaple To deonstrate how to use our odel to calculate the end-to-end travel tie and capture the delay propagation along a corridor, a short corridor with 3 bottlenecks (Figure 4) is used as an

11 illustrative exaple. In this exaple corridor, bottleneck is connected with an on-rap and bottleneck 3 is connected with an off-rap. The bottleneck discharge rates for those bottlenecks are 9, 9 and 6 vehicles/in, respectively. The existing nubers of vehicles on each link are 75, 6 and 65, respectively. The inflow rate fro the on-rap to bottleneck is vehicles/in (vp), and the outflow rate to the off-rap is 8 vehicles/in (vp). The free-flow travel tie over each link is 5, 4 and 4.5 inutes, respectively. 3'' 8 vp vp ' Space 3 x 3 (t )=65 veh x (t )=6 veh x (t )=75 veh FFTT w = 3.33 t p FFTT w = 5.4 t Free flow travel tie Waiting tie in the queue 7: 7:5 7:8.33 7:.33 7:7.74 7:.4 7:3.77 p FFTT 3 w 3 = 8.53 λ = 3 λ = λ 3 = 5.89 Figure 4: 3-bottleneck exaple corridor For the probe vehicle in Figure 4 (starting at tie 7: AM), we now have the following calculation process for its end-to-end travel tie. ) Departing at 7:, it takes 5 inutes (free-flow travel tie) for this probe vehicle to reach the point-queue of bottleneck at 7:5. At this tie instance, the nuber of vehicles λ t = 75 (5 in*9 veh/in) = 3 vehicles. With the waiting in the queue is ( ) t 3 p 3 Tie discharge rate of 9 vehicles/in, this probe vehicle will spend w = 3.33 inutes waiting in the queue. Thus, the total travel tie for this vehicle is 8.33 inutes at the end of this bottleneck. ) The probe vehicle enters link at 7:8.33, spends 4 inutes traveling through the freeflow segent, and arrives at the vertical stack queue at t = 7:.33. Fro 7: to 7:.33, there have been.33 in* veh/in = 46.6 vehicles entering this bottleneck fro the on-rap. The nuber of vehicles waiting in the queue at this tie is λ ( t ) = (75 + 6) + (.33*) (.33*9) = With the discharge rate of 9

12 vehicles/in, this vehicle leaves the queue w =5.4 inutes later. The departure tie fro the second bottleneck is 7: ) Following the sae calculation process, the nuber of vehicles waiting at the queue of λ t = ( ) + (.33*) + (-8*.4) (6*.4) = bottleneck 3 is 3( 3 ) 5.89 vehicles and the waiting tie in the queue is w 3 = 8.53 inutes. This vehicle leaves bottleneck 3 at 7:3.77. The total end-to-end-travel tie p 3 is 3.77 inutes..4 Travel tie calculation algorith with deterinistic inputs The algorith for calculating the end-to-end path travel tie for vehicle entering the corridor with M bottlenecks at tie t is suaried below. Input: The specific starting tie t, the nuber of vehicles on each link ( ) flow rate on each bottleneck End-to-end travel tie calculation For = to M net x t, the net f, and the bottleneck discharge rate c, at tie t. Calculate the arrival tie at bottleneck t = t + w + FFTT, where t =, w =.. Use Eq. (4) to calculate the nuber of vehicles ahead of the probe λ t, when the vehicle in the vertical stack queue of bottleneck, ( ) probe vehicle reaches the beginning of the queue at tie t. 3. Use Eq. (5) to calculate the delay experienced by the probe vehicle on bottleneck, w. 4. Use Eq. (6) to update end-to-end travel tie over, p End For Output: The end-to-end travel tie.5 Discussions p fro bottleneck to bottleneck M. M To consider coplex real-life conditions, the proposed analytical fraework ust further use the following approxiation ethods for calculating the end-to-end path travel tie along a corridor with ultiple bottlenecks..5. Approxiating tie-dependent flow rate with average flow rate In Eq. (9) to (), we use the axiu bottleneck discharge rate to approxiate the actual discharge rate. In reality, the traffic flow rates (including the queue discharge flow rates and net flow rates fro and to raps) are highly dynaic and could fluctuate significantly even in a short tie interval, as shown in Figure 5. In this situation, one needs to use the average flow rate (i.e. the dashed line in Figure 5) during the interval fro t to the arrival tie t to approxiate the tie-dependent volue. Although this approxiation ignores the traffic

13 dynaics, it can still give a reasonable estiate about the total nuber of vehicles leaving or entering the bottleneck before the probe vehicles in Eq. (4). Section 4 will exaine the possible ipact fro using this approxiation ethod for representing travel tie distributions. Flow Rate t Tie-dependent flow pattern Average flow rate during tie interval t Tie Figure 5: Tie-dependent flow rate to average flow rate.5. Considering further reduced bottleneck discharge flow rate due to queue spillback The proposed point-queue based odel needs to capture the effects of queue spillback fro a downstrea bottleneck. Essentially, when a queue spillback occurs, the discharge capacity fro the upstrea bottleneck is then constrained by the discharge rates at the downstrea bottleneck. In this case, the proposed ethod should first detect the possible queue spillback, and then use the reduced queue discharge rate for calculating the waiting tie at the bottleneck with queue spillback. As illustrated in Figure 6, the physical queue on bottleneck spills back to bottleneck (-) between tie t and t 5 through the backward wave lines. Interested readers are referred to the paper by Newell, 993. Due to the queue spillback fro bottleneck, the actual discharge ' rate c of bottleneck (-) between tie t and t 5 is constrained by the discharge rate of bottleneck, c, rather than the original discharge rate c. Suppose at tie t (where t > t ), a probe vehicle arrives at bottleneck (-), if the effect of queue spillback is not taken into account, this probe vehicle in the odel will leave bottleneck (-) at tie t 3 after waiting in the queue behind bottleneck (-), using the original unaffected queue discharge rate c. With the reduced discharge rate c < c at bottleneck (-), the actual waiting tie for the probe vehicle will be longer with a departing tie of t 4 > t 3. 3

14 Space - Backward wave t t t3 t4 t5 Figure 6: Queue spillback.5.3 Calculating the net flow rate for on-raps Do not consider queue spillback Consider queue spillback When estiating the net flow at erge or diverge bottlenecks, the flow rates in previous exaples are assued to be known and tie-invariant. However, special attention ust be paid to the following scenario. If the ainline and the on-rap are both congested, (a) the nuber of vehicles that can enter the bottleneck fro the on-rap and (b) the nuber of vehicles that can enter fro the upstrea segent to the bottleneck are constrained by the ainline bottleneck discharge rate. In this case, the available bottleneck discharge rate should be allocated to the upstrea segent and the on-rap proportionally, according to certain rules (Zhang and Nie, 5). One siple rule is to split the ainline discharge rates according to the nuber of lanes associated with each incoing approach..5.4 Considering vehicle overtaking/passing Lastly, the FIFO property assued on each link rules out the possibility that a vehicle can overtake and pass another vehicle. Future research will consider this condition s ipact on endto-end travel tie estiation using this approach. 3. Methods for calculating end-to-end travel tie distributions 3. Assuptions net In the previous discussion, input paraeters such as the net rates f at erge and diverge, and the bottleneck discharge rate c are assued to be deterinistic. In this section, we will further consider the variations or uncertainty in the input paraeters, especially in the following two applications: () day-to-day travel tie variability estiation by considering flow Tie 4

15 variations at the sae tie period, and () real-tie travel tie reliability prediction, where the near-future traffic flows are predicted fro different sources of data with various degrees of prediction uncertainty. Ephases are placed on how to calculate the end-to-end travel tie distribution based on the stochasticity of the rando input paraeters. 3. Iportant observations on path travel tie 3.. Siple corridor without erge and diverge net Consider a siple two-bottleneck corridor with no on-rap and off-rap, that is, f net and f are equal to. According to Eqs. (4-6), the end-to-end travel tie to bottleneck for probe vehicle entering link at tie t is: ( ) ( ) ( ) ( + ) x t c t x t c t FFTT p = t + w = t + FFTT + = t + FFTT + c c x t x t = t + FFTT + t + FFTT = ( ) ( ) c c And the end-to-end travel tie to bottleneck is: By coparing ( ) ( ) x t + x t c t p = p + FFTT + w = p + FFTT + c x( t) + x( t) c ( p + FFTT) = p + FFTT + c x ( t ) + x ( t ) x t + x t = p + FFTT + p + FFTT = ( ) ( ) ( ) c c ( ) x t p = and c ( ) + ( ) (7) (8) =, we can ake the following c x t x t p iportant observation. That is, the proposed forula can correctly capture the correlations between the end-to-end travel ties p and p, as both values are dependent on the nuber of vehicles on link, x( t ). If x( t ) and x( t ) are assued to be deterinistic, the distributions of p and p are further dependent on the distribution of the bottleneck discharge rates, c and c, respectively. 3.. Siple corridor with erge and diverge If we further consider situations where a erge and diverge occur at both bottlenecks, the path travel tie forulas are expressed as follows. ( ) + ( + ) net = + = (9) c x t f t FFTT p t w 5

16 p net net ( ) + + ( ) + ( + ) x t f FFTT x t f p FFTT = () c The above equations introduce ore coplex dependencies for both p and p, and no additive forula or decoposed eleents can be easily constructed to siplify these coplicated equations. This observation reinforces any previous research results, that is, the analytical quantification of the end-to-end travel tie distribution is extreely challenging. 3.3 Monte Carlo siulation Monte Carlo siulation is widely used to siulate the behavior of various physical and atheatical systes, especially for those probles with significant uncertainty in inputs. This research ais to apply a Monte Carlo siulation ethod to investigate the end-to-end travel tie distribution based on the proposed travel tie calculation fraework. In each siulation run, a realiation of rando input paraeters will lead to a realiation of rando path travel tie outputs, which can be regarded as estiates of the true end-to-end travel tie variable. A sufficient nuber of siulations then provide a good representation of the travel tie distributions under various traffic conditions and uncertainties. The following procedure assues all rando variables are log-norally distributed, and calculates travel tie distribution through K siulation runs. Input: The specific starting tie t, The distribution of the nuber of vehicles on each link ( ) ( ) (, ) x t LN µ σ x x x t, where The distribution of the net flow rate on each bottleneck f net, where ( net, net ) f LN µ σ net f f The distribution of the bottleneck discharge rate c on each bottleneck, at tie t where c LN ( µ c, σc ) Link free-flow travel tie FFTT, assued to be constant. Nuber of siulations = K. For k= to K, For = to M : Based on the underlying distribution paraeters ( µ and σ ) of the individual inputs, generate a set of rando saples for the following key variables: the nuber of vehicles on the link, the bottleneck discharge rate and net flow rates. : Call the algorith introduced in Section.4 to calculate the predicted end-to-end travel tie for siulation k: p [ k ] fro this set of rando saples. 6

17 End For End For Output: Calculate the histogra, ean and variance for the end-to-end travel tie fro the results over K siulation runs. 3.4 Nuerical experients Monte Carlo siulation For the sae exaple corridor (with three bottlenecks) in Section.3, Monte Carlo experients were conducted to calculate the end-to-end travel tie by assuing that the bottleneck discharge rates, inflow/outflow rates on raps and existing nuber of vehicles on the link are all lognoral variables. K = siulation runs were perfored with different scenarios of stochastic input paraeters. Figure 7-(a) and (b) shows the distributions of the siulated end-to-end travel ties p and p 3 for probe vehicle through bottleneck and through bottleneck 3, respectively. Obviously, the ean travel tie based on p3 is larger than that of p. In addition, a clear pattern of randoness propagation can be observed, as p 3 has higher variance than p. It should be noted that, by using different input distributions for flow discharge rates and the prevailing nuber of vehicles on the road, the resulting travel tie distributions will vary. This deonstrates the advantage of the proposed odel in recogniing the ipact of capacity and congestion levels on travel tie reliability. Frequency Bottleneck Histogra (a) 3 36 Frequency 4 44 End-to-end travel tie (inutes) Cuulative % % 9% 8% 7% 6% 5% 4% 3% % % % 7

18 Frequency 4. Model validation using NGSIM data (b) Figure 7: End-to-end travel tie distribution This section uses vehicle trajectory data available fro the NGSIM (Next Generation SIMulation) project (FHWA, 6) as ground-truth data to verify the proposed ethodology and exaine the sources of prediction error. 4.. Data descriptions Bottleneck 3 Histogra End-to-end travel tie (inutes) Frequency Cuulative % The NGSIM vehicle trajectory data used in this study coe fro the I-8 dataset, which were collected by a video caera located at Eeryville, California. This data collection point is located adjacent to I-8, as shown in Figure 8. The site was approxiately,65 feet in length, with an on-rap at Powell Street (indicated in Figure 8 by the red circle). The freeway segent covered in the dataset includes six lanes, nubered increentally fro the left-ost lane (HOV lane). Video data are available for three tie intervals: 4: p.. to 4:5 p.., 5: p.. to 5:5 p.. and 5:5 p.. to 5:3 p.., on April 3, 5. Coplete, transcribed vehicle trajectories are available with a tie resolution of. seconds. % 9% 8% 7% 6% 5% 4% 3% % % % 8

19 Figure 8: Scheatic illustration of NGSIM study area Source: Adapted fro Figure, NGSIM I-8 Data Analysis Suary Report (Cabridge Systeatics, 5) 4.. Data extraction fro NGSIM dataset. To extract vehicle flow counts data fro the NGSIM dataset, we first construct a node-link structure to represent the freeway segent in Figure 8. This stretch of 9

20 freeway is divided into two links, as shown in Figure 9, with the on-rap connected with node.. In order to obtain the flow rate at the node/bottleneck, this study introduces a set of virtual detectors at node and at node, respectively. Meanwhile, another virtual detector is placed on the on-rap link so that inflow vehicles fro the rap are also counted. In addition, video caeras are assued to be installed on both links to provide link snapshots (for probe vehicle data). 3. The vehicle trajectory data is divided into 5-inute intervals for counting vehicles. An exaple of one 5-inute span of vehicle trajectories on one lane is shown in Figure to illustrate how the vehicle counts are collected. As entioned before, two sets of virtual detectors A and B are placed at nodes and (shown as triangles in Figure ), and video caeras C and D are also installed on both link and link. Vehicles are counted along the vertical line drawn at the given tie t. At tie t =, probe vehicle = enters link. At this tie step, two vehicles are observed on link and five vehicles are observed on link by video caeras C and D, that is, x( t ) = and x( t ) = 5. Siilarly, probe vehicle = 5 enters link at tie t. At this oent, x ( t ) = and x ( t ) = 4. Probe vehicle = 8 is worth entioning, which enters link at tie t'. However, at tie t'', this vehicles leaves this lane to another lane. This is shown in Figure as an incoplete vehicle trajectory. During this 5-inute interval, vehicles are counted by detector A, including two vehicles entering before probe vehicle =, but excluding probe vehicle = 8. Meanwhile, 3 vehicles are counted by detector B. This count includes seven vehicles before probe vehicle =, but excludes probe vehicles 7-, which have not yet departed fro link. Direction of traffic flow 44' ' On-rap Figure 9: Node-link representation of NGSIM network

21 D C B A 4.3. Model validation t t t' t'' Figure : Vehicle trajectory on a lane Two end-to-end travel tie prediction approaches are investigated using our odel. The first approach is a lane-based approach, and the second is link-based approach. 9 Tie Let us first define soe notation to represent lane-specific paraeters. Z: nuber of probe vehicles; n: index identifying a lane; t : starting tie for probe vehicle ; n : starting lane nuber for probe vehicle ; (, ) x tn : nuber of vehicles on lane n at tie t; ( t, n ) λ : lane n specific nuber of vehicles behind bottleneck ; c n : lane n specific discharge rate of bottleneck ; ( ) net ( ) f n : net flow rate fro or to raps by lane n; ( n) θ : vehicle distribution rate fro on-rap to lane n; ( ) w n : waiting tie for probe vehicle on bottleneck on lane n; ( ) p n : lane-based end-to-end travel tie for probe vehicle through lane n; The following procedure is used to calculate the lane-based travel tie distribution.

22 For = to Z on the link Obtain arrival tie t and starting lane nuber End For Obtain the lane-based nuber of vehicles ( ) Obtain the lane-specific discharge rate ( ) x,n t ; c n ; n for each probe vehicle. net net Calculate net flow rate f ( n ) fro the on-rap by applying ( ) Calculate the nuber vehicles behind bottleneck ( t, n ) Calculate w ( n ) based on c ( n) and ( ) x t n ; Update the end-to-end lane travel tie ( ), p n. λ ; Output: Lane-based path travel tie distribution based on p ( n ) Lane-based end-to-end travel tie prediction θ n f ; The distribution of the predicted end-to-end travel ties for each 5-inute interval, calculated over the 3 tie periods with available data (4: p.. to 4:5 p.., 5: p.. to 5:5 p.. and 5:5 p.. to 5:3 p..) are plotted in Figure with the ground truth end-to-end travel tie obtained directly fro the NGSIM data. As it can be observed, the distribution of the predicted end-to-end travel tie is very close to that of the ground truth end-to-end travel tie. This deonstrates that our odel is able to accurately predict the end-to-end travel tie distribution.

23 Frequency Title Lane-Based End-to-End Travel Tie Distribution (4: p :5 p..) Travel Tie (sec) Title (a) Lane-Based End-to-End Travel Tie Distribution (5: p..-- 5:5 p..) (b) Ground Truth Travel Tie Predicted Travel Tie Ground Truth Travel Tie Predicted Travel Tie 3

24 Frequency Lane-Based End-to-End Travel Tie Distribution (5:5 p :3 p..) Travel Tie (sec) (c) Figure : Lane-based end-to-end travel tie distributions Link-based end-to-end travel tie prediction One coon practice is to use link-based flow rates or density to predict travel tie variability. We replace the lane-based variables in the previous approach with link-based variables x ( t ) and c. That is, x ( t ) is the existing nuber of vehicles on all the lanes on the link and c is the link discharge rate. The distribution of the predicted end-to-end travel tie and true end-to-end travel tie for different tie intervals are shown in Figure. Title Link-Based End-to-End Travel Tie Distribution (4: p :5 p..) Title (a) Ground Truth Travel Tie Predicted Travel Tie Ground Truth Travel Tie Predicted Travel Tie 4

25 Title Title Link-Based End-to-End Travel Tie Distribution (5: p :5 p..) 6 6 Title 5 (b) Link-Based End-to-End Travel Tie Distribution (5:5 p..-- 5:3 p..) 6 6 Title (c) Figure : Link-based end-to-end travel tie distributions Ground Truth Travel Tie Predicted Travel Tie Ground Truth Travel Tie Predicted Travel Tie As it can be observed, the distribution of the predicted link-based end-to-end travel ties fails to capture the wide-spread distribution in the ground truth travel ties. This is explained by the fact that link-based input variables would yield the sae predicted end-to-end travel ties for those vehicles entering the link at the sae tie, regardless of their driving lanes. In order to understand the extent and sources of the lane-by-lane travel tie variation, we use the tie period between 4: p.. and 4:5 p.. as an exaple. Figure 3 shows the lane discharge rate, the existing nubers of vehicles on the lane, and the average true and predicted end-to-end travel ties for each lane for each 5-in interval in the tie period. The lane sequence is sorted by the true end-to-end travel tie. 5

26 Avg. Travel Tie (sec) Avg. Travel Tie (sec) Lane-by-Lane Travel Tie Variability Lane Lane 5 Lane 6 Lane 3 Lane 4 Lane (a): 4: p.. 4:5 p.. Lane Lane 6 Lane Lane 5 Lane 3 Lane 4 (b): 4:5 p.. 4: p.. Existing # of Vehicles On the Lane Average Predicted Travel Tie Average True Travel Tie Flow Rate Lane-by-Lane Travel Tie Variability Existing # of Vehicles On the Lane Average Predicted Travel Tie Average True Travel Tie Flow Rate 6

27 Avg. Travel Tie (sec) (c): 4: p.. 4:5 p.. Figure 3: Lane-by-lane travel tie variability Several observations can be obtained fro Figure 3. Lane (HOV lane) has the lowest existing nuber of vehicles on the lane and has the lowest average end-to-end travel tie. The left-ost lanes (lanes and ) usually have the highest discharge rates while lanes 3 and 4 usually have the lowest discharge rates. In ost cases, lanes 3 and 4 have the highest average existing nubers of vehicles on the lane, as well as the highest average end-to-end travel ties. These observations iply that, due to the variation of the discharge rates and the nuber of vehicles on the lane, the end-to-end travel ties also present strong lane-by-lane variations. As a result, we suggest using lane-based statistics to better quantify the travel tie variability Prediction error sources Lane-by-lane Travel Tie Variability Lane Lane Lane 3 Lane 5 Lane 4 Lane 6 Existing # of Vehicles On the Lane Average Predicted Travel Tie Average True Travel Tie Flow Rate By coparing the predicted results with the NGSIM ground truth data, we can further uncover other possible sources of errors in the proposed travel tie prediction odel.. Aggregation errors: The link/lane discharge rates c and on-rap flow rates used in the calculations are average flow rates over a certain tie interval, e.g., 5-inute rates, while the existing nuber of vehicles on the link/lane x ( t ) is an instantaneous value based on the entering tie of a probe vehicle.. Measureent errors: The nuber of vehicles on the lane observed by the video caera at tie t is assued to be error-free. In fact, there are always vehicle detection errors in NGSIM vehicle trajectory data associated with the underlying video recognition algorith. 3. Modeling errors associated with lane changing: Since the queue odel incorporates the first-in-first-out principle, lane change behavior is not considered in the calculation. This will introduce two types of errors in the odel: 7

28 5. Conclusions a. The odel ay underestiate or overestiate the nuber of vehicles behind the λ t. For exaple, soe vehicles will enter the lane (fro the bottleneck, ( ) other lanes) before a probe vehicle reaches the bottleneck, or soe vehicles x t on the current lane will leave to one of the adjacent originally counted in ( ) lanes, corresponding to a lower value of ( ) x t. b. When a probe vehicle changes lane fro, for exaple, n = to lane n', the discharge rate used in the calculation should be changed to the one associated with lane n'. In this paper, a travel tie estiation fraework is proposed to calculate the end-to-end travel tie, given the nuber of vehicles along the corridor, the bottleneck discharge rates and the on-rap/off-rap flow rates. Monte Carlo ethods and atheatical approxiation ethods are presented to calculate the travel tie distribution if the distributions of the variables are also given. Ground-truth data fro the NGSIM project are applied for odel validations. Several critical observations are obtained fro the odel validations: () Our odel is able to provide a analytical to predict travel tie and its distribution; () Using ore detailed inputs, e.g., lane-level rather than link-level, could yield ore accurate end-to-end travel tie distribution predictions; and (3) The variation of lane-specific paraeters (the nuber of vehicles on the lanes and lane discharge rates) results in lane-by-lane travel tie variation. Nevertheless, the ipacts of downstrea queue spillback on the upstrea travel tie have not yet been considered in our odel. In queue spillback, the discharge rate of the upstrea bottleneck is constrained by the downstrea bottleneck discharge rate. Moreover, it is also desirable to consider the influence of lane changes on the predicted nuber of vehicles waiting in the queue. These iportant features will be investigated ore thoroughly in future research. 8

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