Value of Travel Time Reliability in Transportation Decision Making: Proof of Concept—Portland, Oregon, Metro (2014)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

7Overview of DynusT–FAST-TrIPs Integrated Model 2.1 DynusT 2.1.1 DynusT Model Structure As shown in Figure 2.1, DynusT consists of iterative inter­ actions between its two main modules, traffic simulation and traffic assignment. Vehicles are created and loaded into the network based on their respective origins and follow a spe­ cific route based on their intended destinations. The large­ scale simulation of networkwide traffic is accomplished through the mesoscopic simulation approach, which omits intervehicle car­following details while maintaining realistic macroscopic traffic properties (i.e., speed, density, and flow). More specifi­ cally, the traffic simulation is based on the anisotropic meso­ scopic simulation (AMS) model, which simulates the movement of individual vehicles according to the concept that a vehicle’s speed adjustment is influenced by the traffic conditions in front of the vehicle. In other words, at each simulation interval, a vehicle’s speed is determined by the speed–density curve, and density is defined as the number of vehicles per mile per lane with a limited distance—defined as the speed­influencing region (SIR)—downstream of the vehicle (Chiu et al. 2010). After simulation, necessary measures of effectiveness (MoEs) are fed into the traffic assignment module. The traffic assign­ ment module consists of two algorithmic components: a time­ dependent shortest­path (TDSP) algorithm and time­dependent traffic assignment. The TDSP algorithm determines the TDSP for each departure time, and the traffic assignment component assigns a portion of the vehicles departing at the same time between the same origin–destination (O­D) pair to the time­ dependent least­travel time path following a “route swapping” type of traffic assignment procedure. In DynusT, the assignment algorithm adopts the isochronal vehicle assignment method (Nava and Chiu 2012), which aims to maintain a balance between computational efficiency and solution algorithm quality. Innovations in computational efficiency allow DynusT to perform 24­hour assignment, a feature that was critical for estimating daily traffic patterns for the purpose of this study. The computational features include (1) reusing vehicle IDs to commit computer memory only for those vehicles that exist in the network during simulation so that memory usage is not cumulative to the total number of generated vehicles, and (2) assigning vehicles with TDSPs that are solved based on an epoch, which is the time period over which network statistics are collected for solving for the TDSP. An epoch was defined as about one to two hours in length. The memory usage for the TDSP is limited by the length of the epoch regardless of the length of the total evacu­ ation simulation period. Once the assignment of the current iteration is finished, all vehicles are again loaded and moved along their paths in the simulation module to evaluate if the time­dependent user equilibrium (TDUE) condition is satisfied. If so, the algorith­ mic procedure is terminated; otherwise, the next iteration begins. 2.1.2 Anisotropic Mesoscopic Traffic Simulation The AMS model was developed based on two intuitive con­ cepts and traffic characteristics: (1) at any time, a vehicle’s prevailing speed is influenced only by the vehicles in front of it, including vehicles that are in the same or adjacent lanes; and (2) the influence on a vehicle of traffic downstream decreases with increased distance. These two characteristics define the “anisotropic” property of the traffic flow and pro­ vide the guiding principle for AMS model design. Based on such concepts, for any vehicle i, only those leading vehicles present in vehicle i’s immediate downstream area and within a certain distance are considered to influence vehicle i’s speed response. This concept is a similar concept to a stimulus– response type of car­following model, with the distinction that in AMS, the stimulus of a vehicle’s speed response is represented in a macroscopic manner instead of using intervehicle distance or speed as in microscopic models. C h a p T e r 2

8For modeling purposes, the speed­influencing region for vehicle i (SIRi) is defined as vehicle i’s immediate downstream roadway section in which the stimulus is significant enough to influence vehicle i’s speed response. This concept is further depicted in Figure 2.2, in which a multilane, homogeneous roadway segment is considered. The SIR for vehicle i is defined as the area (including the lane in which vehicles reside and all the adjacent lanes) in front of vehicle i, where the traffic con­ dition (represented by the density) affects vehicle i’s speed response. At each simulation clock tick, vehicle i’s speed is influenced by the density in SIR. The upstream and down­ stream traffic outside the SIR does not influence vehicle i. The SIRi length can be assumed to be either equal for all vehicles or variable according to different flow conditions. In this study SIRi length was assumed to be an average value l across all vehicles. The traffic density in SIRi, denoted as ki, is calcu­ lated as the number of vehicles present in SIRi divided by the total lane miles of SIRi. As such, the unit of ki becomes the number of vehicles per mile per lane. At the beginning of a simulation interval t, for each vehicle i, the prevailing speed of vehicle i during the simulation interval t is determined by Equation 2.1, where ℘: k → v is a nonincreas­ ing speed–density relationship function with two boundary conditions: ℘(0) = vf and ℘(kqueue) = 0. The queue density kqueue is defined as the “bumper­to­bumper” density observed in a long, standing­still queue, which is generally greater than the jam density reported in the literature. The algorithmic steps of an AMS model during simula­ tion are as follows. At each clock tick t (the beginning of a simulation interval), each vehicle’s speed vti is evaluated based on its SIR density, which is obtained from the previ­ ous clock tick ki t -1 through the v-k relationship function ℘(kit -1). The SIR density is calculated based on Equation 2.2 or 2.3, depending on whether the SIR spans over the freeway segment with a different capacity. If the SIR spans a homo­ geneous highway section, Equation 2.2 applies; otherwise, Equation 2.3 is used. Vehicle i’s traveling distance at the end of the current simulation interval is obtained by taking the prevailing speed vti times the duration of the simulation interval ∆. (2.1)1( )=℘ −v kit it min , (2.2)1 queue 1 k k N nli t i t =     − − min , (2.3)1 queue 1 1 1k k N mx n l xi t i t i t i t( )= + −     − − − − where i: subscript denoting a vehicle. The index i decreases with vehicles traveling in the same direction on the same link; t: superscript denoting a simulation interval; l: SIR length; vi t: prevailing speed of vehicle i during simulation interval t; xi t -1: distance between vehicle i and lane drop (open) at clock tick t - 1; ki t -1: density of SIR for vehicle i; Ni t -1: number of vehicles present in SIR, excluding vehicle i; vf: free­flow speed in the speed–density relationship; ℘: k → v: nonincreasing speed–density function specify­ ing the v-k relationship, where ℘(0) = vf and ℘(kqueue) = 0; and kqueue: queue density, ℘(kqueue) = 0. During the AMS simulation, each vehicle maintains its own prevailing speed and SIR at the beginning of a simula­ tion interval. The traveling distances of individual vehicles are therefore likely to differ, even though they are on the same link. This feature is different from certain previous models (Jayakrishnan et al. 1994; Balakrishna et al. 2005) in which all moving vehicles on the same link travel at the same speed. This difference characterizes the AMS model as a vehicle­based mesoscopic model having a greater degree Figure 2.1. DynusT model algorithmic structure.

9 of resemblance with car­following–based microscopic mod­ els. The major difference between AMS and car­following models is that in AMS, a vehicle’s speed adjustment at each simulation time interval is governed by the SIR density ki t , which is a macroscopic measure of all the vehicles present in the SIR region, instead of an intervehicle measure between the target and the leading vehicle(s). Because the SIR moves with each vehicle during simulation, it can be anticipated that in the AMS model, the vehicle­advancing mechanism is generally independent of the representation of network structures (i.e., the size or length of the cell, segment, or link) under the uninterrupted flow condition. Each vehicle makes speed adjustment decisions solely based on its SIR den­ sity; the AMS simulation results generally remain stable regard­ less of how link lengths are defined unless the link is shorter than a certain threshold that violates the one required by a general time­based simulation. AMS handles queue formation and discharge in a natu­ ral and straightforward manner. When kqueue is reached, v = ℘(kqueue) = 0; vehicles speed up when the SIR density decreases. This mechanism allows for clear representations of substantial or transient queue formation or discharge. When a free­moving vehicle approaches the end of a queue, its speed gradually approaches the same speed of the queue tail as its SIR density approaches the SIR density of the leading vehicles. Depend­ ing on how the overtaking condition is met, this vehicle may trail at the end of the queue without “jumping over” leading vehicles, or it may stop ahead of the leading vehicle. Equation 2.1 was further extended to simulate traffic flow in uninterrupted flow facilities under various configurations, such as homogeneous highways, nonhomogeneous highways, and temporary blockage, by specifically considering different SIR density ki t calculations corresponding to those conditions. As shown in Equation 2.2, in the case of the homogeneous highway, ki t is calculated as the number of vehicles present in the SIR divided by the total number of lane miles of the SIR (i.e., the SIR length times the number of lanes). When lane drops or lane additions occur within the SIR, the total lane miles of SIR are the sum of the lane miles of separate sections, as shown in Equation 2.3. The lane drop (Figure 2.2b­1) or Source: Chiu et al. 2010. l m lanes n lanesVehicle i (c) Non-Homogeneous Highway (bottleneck discharge) l-xx l-x l x m lanes n lanesVehicle i (b-1) Non-Homogeneous Highway (lane drop) n lanes l-x l x m lanes Vehicle i (b-2) Non-Homogeneous Highway (point bottleneck) l n lanesVehicle i Speed Influence Region SIRi (a) Homogeneous Highway Vehicle i (d) AMS with SIR extending beyond current link boundary 0 Figure 2.2. AMS model concept.

10 point bottleneck (Figure 2.2b­2) (from m to n lanes, n < m) occurs downstream from vehicle i. The total lane miles in the SIR are calculated as mx + n(l - x), and the resulting ki t is the smaller of kqueue and 1 ( )+ − −N mx n l x i t , which is the number of vehicles in the SIR at the beginning of time interval t - 1 divided by the total lane miles mx + n(l - x) in the SIR. In the case of a lane drop or a point bottleneck (n < m), the SIR density of a vehicle gradually increases (and hence speed reduces) as it approaches the bottleneck. When n = 0, a com­ plete blockage occurs that can be applied to either the point blockage or red­light signal indication. In the case of discharg­ ing from a bottleneck, as a vehicle approaches the opening up of the bottleneck, the density reduces and speed increases gradually. Vehicle simulation under the presence of transit vehicles needs to properly differentiate the situation with or with­ out bus pullouts. As illustrated in Figure 2.3a, when a bus pullout is present and a transit vehicle resides in the pull­ out, the SIR area of passing vehicles remains unchanged. Without the pullout (Figure 2.3b), the stopped transit vehi­ cle typically blocks one traffic lane, creating a temporal blockage to the traffic downstream. The SIR area of the approaching vehicles is modified as shown in Figure 2.3b, which leads to a slowdown effect similar to the one due to lane drop in Figure 2.2b­1. The departure from each stop involves different rules for frequency­ or schedule­based transit. For a schedule­based transit operation, a transit vehicle must be held until the scheduled departure time if it is ahead of schedule after boarding and alighting. Such vehicle holding is unnecessary in a frequency­based operation. 2.2 FaST-TrIps For the transit component, FAST­TrIPs is divided into two main submodules, transit assignment and simulation. The transit assignment submodule plays the role of passenger assignment for given O­D pairs considering the capacity constraint of each transit vehicle. For assigning transit passengers for the O­D pairs, a hyperpath model (Noh et al. 2012) is activated by searching a feasible strategic path on each O­D pair. Since each hyperpath has multiple alternative links for each predecessor link, at least one single path, a so­called elementary path, is gen­ erated on a hyperpath. Each passenger is loaded on the specific elementary path according to the probability chosen by a logit model. The assigned passengers, including their path, are given to and simulated through the submodule transit simulation in FAST­TrIPs. During the simulation, experienced arrival and departure times of transit vehicles are used to simulate boarding and alighting of passengers, considering transfers and other components (i.e., walking and waiting). Each passenger’s trajec­ tory (i.e., experienced path) is recorded, and dwell time for each transit route is calculated as a function of the boardings and alightings at each stop. The results of the simulation are used in the next iteration of auto–transit vehicle simulation, and they are also fed back to the activity­based model for updating the demand when the DynusT–FAST­TrIPs integration model converges. 2.2.1 Passenger Assignment For considering passenger strategic choice on a set of routes at each stop, a hyperpath model is introduced on a link­based time­expanded (LBTE) transit schedule network (Noh et al. 2012). To apply the hyperpath model, an LBTE transit SIR Transit Vehicle Transit Vehicle SIR (a) (b) Figure 2.3. (a) SIR area with bus pullout and (b) SIR area without bus pullout.

11 schedule network is prepared. The basic unit of the LBTE transit network is defined on each link, which connects two consecutive stop time runs by an identical transit vehicle. Crucial time information, such as departure and arrival times, is allocated on each schedule link. Like a turn penalty between two successive links, waiting and transfer times are included in the attributes of the LBTE transit schedule net­ work. All time­related information is represented in a gener­ alized cost in the hyperpath search. On the prepared LBTE transit network, a hyperpath search model is applied to generate the optimal hyperpath. The fun­ damental hyperpath model was proposed by Nguyen and Pallottino (1988) and Spiess and Florian (1989) in terms of understanding a passenger’s strategy on a frequency­based transit network. Divergent hyperpath models have been applied on transit schedule­based networks (Hamdouch and Lawphongpanich 2008). In the LBTE transit network, a hyper­ path is generated in a recursive manner by updating the gener­ alized cost from the origin to all destinations by a forward search, or from the destination to all origins by a backward search. For example, consider a network with three links a1, a2, and a3, as shown in Figure 2.4, and assume a backward search from the destination. cˆa2 and cˆa3 are the labels of links a2 and a3, respectively. Assuming that those two labels are given or have been updated, the label of link a1 is updated by a weighting function of link a2 and a3 or hyperlink e3 and the cost of link a1, ca1, so that cˆa1 = ca1 + w(cˆa2, cˆa3), where w(z) stands for a weighting function of the hyperlink e3 with two alternative links, a2 and a3. For the appropriate weighting function on this transit schedule network, Noh et al. (2012) proposed a logsum­type function. For the hyperpath search, we use the acyclic property of a transit schedule network to determine an efficient search algo­ rithm. The acyclic property of the LBTE transit network holds that every transit link is connected to the next successive link only if the first link arrives at or before the departure time of the next link. According to this acyclic property, a hyperpath is searched in order of the latest departure time, from a destina­ tion to all possible origins, by using a backward search. More detail on this algorithm is given in Noh et al. (2012). For better computational performance, a hierarchical hyperpath model was implemented by Khani et al. (2012). In addition, as a back­ ward search is employed, the hyperpath search is assumed to be initiated with a preferred arrival time at the destination. With the hyperpath model, we use a method of successive averages–type assignment model under the assumption of a congested transit network. First, by implementing a logit­based hyperpath, the proposed model is categorized as a stochastic user equilibrium transit assignment model. However, it is pos­ sible to have a certain level of congestion if the number of pas­ sengers exceeds the capacity of a transit vehicle. The order of boarding is strictly dependent on priority, depending on whether the passengers are on board or waiting at a stop; at the stop, priority depends directly on the time of arrival at the stop. To represent this congestion, we introduce a “soft” capac­ ity constraint that increases the travel cost when the vehicle capacity is exceeded. This proposed capacity penalty function is exponential in form and is related to the residual capacity rb of a transit vehicle and the assigned flows for boarding, fab, as shown in Equation 2.4: max 0, (2.4)cap max 0,c f r f eab ab b kb f r fab b kb [ ]= − [ ]( )α − − where fkb is the sum of flows having higher priority than flows fab. To manage the priority of different boarding flows, a diag­ onalization technique proposed by Sheffi (1984) was applied. The proposed algorithm is given in Figure 2.5. The proposed algorithm is separated into inner and outer loops, with the inner loop specified in Step 3 (diagonaliza­ tion) and the outer loop consisting of Step 2 through Step 4. The proposed algorithm starts with all­or­nothing assign­ ment in Step 1. In Step 2, residual capacity is updated only by the flows satisfying the diagonalized equilibrium in Step 3. Step 3 gives a typical method of successive averages–based passenger loading process on the updated network costs. This process continues until the outer loop is converged at Step 4. After the transit assignment, passenger flows are created by assigning each passenger a specific elementary path that is sampled from their optimal hyperpath. 2.2.2 Passenger Simulation The passenger simulation model is a high­resolution model capable of simulating the path taken by individuals in the tran­ sit and intermodal networks. The main inputs are the paths generated in the transit assignment submodule. There are three categories of data inputs to the passenger simulation: 1. Transit network, including stops, routes, and schedule; 2. Transit vehicle simulation results, including the actual arrival and departure of transit vehicles at each stop; and 3. Passenger objects, including information about each pas­ senger and his or her path choice. The simulation model is a combination of an event­based simulation and a time­based simulation. Two main modules Sa1 a2 a3 e3 2 ˆ ac 3 ˆ ac 1 ˆ ac 1a c Figure 2.4. Hyperpath on an LBTE transit schedule network and cost update.

12 capture the behavior of passengers and their interaction with transit vehicles. The first module captures the access, egress, and transfer behavior of passengers and is a time­based simu­ lation over fixed time intervals. In the same way, the simulation captures the movement of passengers from their alighting stop to either their destination or the next boarding stop (in case of a transfer). The detailed information of each passenger’s trip is recorded. The other module takes care of the boarding and alighting of passengers whenever a transit vehicle arrives at a stop. Therefore, an event­based simulation is used for this part, and a transit event is defined as the simulated arrival of a transit vehicle at a transit stop. By looking at different factors such as the number of passengers and type of transit vehicle, a dwell time is calculated for the transit vehicle at the stop. For each transit vehicle, based on the type of route, a capacity is assumed. All the information regarding the boarding, alighting, and passenger load of the vehicles is written in the output files and can be used in the next system simulation and assignment. The model also has a postprocessor for preparing statistics and measures of performance based on the simulation results. 2.3 DynusT and FaST-TrIps Integration 2.3.1 Model Integration The original DynusT–FAST­TrIPs model developed in SHRP 2 C10B was incorporated with a tour­based activity model, DaySim, which produces the demand for a DTA model. Similar models of this interaction of activity­based and DTA models include CEMDAP­VISTA (Lin et al. 2008), TASHA­MATsim (Hao et al. 2010), and OpenAMOS­MALTA (Pendyala et al. 2012). When using the demand from DaySim, three tables provide household, tour, and trip data for model input. Trips, including the origin, destination, estimated depar­ ture and arrival times, and mode of transportation for a single trip, are necessary as input to the DynusT–FAST­TrIPs model. The overall integration model is shown in Figure 2.6. The demand inputs for the DTA model are generated by DaySim, and Google’s GTFS data are used to generate the transit sched­ ule. First, auto trips are assigned to DynusT, and transit­related trips are assigned to FAST­TrIPs. Second, DynusT is initiated Step 1: (Initialization) - Search the least cost path - Load flows on the searched path - n = 0 * Step 2: (Capacity Update) - If subloop (from Step 3 or Step 1), then capacities are fixed and n = n + 1. - Else (from Step 4), residual capacities are changed by new flows and n = 0. Step 3: (Diagonalization) - Update the cost of network - Step 3.1: (Auxiliary Flows) Search the least cost path Load flows on the searched path - Step 3.2: (Flows update: MSA) flowsn+1 = 1/(1 + n) flowsn+1 + n/(1 + n) flowsn - Step 3.3: (Convergence Test) If satisfied, then go to Step 4. Else then go to Step 1. Step 4: (Convergence Test) - If satisfied, then Stop. - Else then go to Step 1. * n: iteration number. Figure 2.5. Transit assignment algorithm, using diagonalized method of successive averages (MSA) with hyperpath.

13 for the assignment and simulation of auto and public transit vehicles. The auto and transit vehicles assignment is run to completion before any input to FAST­TrIPs is generated. Third, the departure times and vehicle trajectories for public transit vehicles are fed into FAST­TrIPs to run the transit passenger assignment and simulation sequentially. As one output of the FAST­TrIPs model, dwell times at each stop are estimated using the number of boarding and alighting passengers at each stop. The estimated dwell times, in turn, will affect the vehicle assignment and simulation in the next DynusT iteration. In this way, the consecutive iterations between DynusT and FAST­ TrIPs reach convergence through the consistency of dwell times between the two submodules. Finally, auto and transit (vehicle and passenger) skim tables are produced and returned to DaySim to reach convergence between the demand and sup­ ply models. For Project L35A, DaySim was replaced with the Portland Metro trip­based travel demand model (TDM), and certain modifications were made to enable such integration. First, time­dependent O­D matrices were produced from the Metro TDM system. Different vehicle­type matrices were produced such as high­occupancy vehicle, single­occupancy vehicle, Figure 2.6. DynusT–FAST-TrIPs integration model considering the upper level activity-based model (DaySim is used as an example). PAT  preferred arrival time. freight, and transit. To enable feedback, five skim matrices were produced to accommodate Metro’s five­time­period model. Although the DynusT–FAST­TrIPs integrated system is capa­ ble of producing skim information at a much finer resolution, such as 15­min or hourly resolution, aggregation to accom­ modate a time­of­day model in the trip­based framework was a necessary step. 2.3.2 Convergence In the DynusT–FAST­TrIPs integration model, three types of convergence apply, as shown in Figure 2.6. Each submodule, DynusT and FAST­TrIPs, has its own individual convergence method for reaching user equilibrium (using a relative gap measure). DynusT uses the simulation­based relative gap mea­ sure. The convergence of FAST­TrIPs is estimated by the relative gap of generalized total travel cost, including link travel cost and capacity cost, from one iteration to the next, typically in the transit assignment submodule. Finally, for the combined DynusT–FAST­TrIPs model, we propose a relative gap measure using dwell times to compare dwell time changes from one iteration to the next iteration.