Vehicles traveling along interrupted flow facilities often exhibit travel times with bimodal distributions. The characteristics of these distribution s have been studied extensively in the literature, yet the effect of signal control on bimodality have received little attention. Most researchers theorize that the slower group experiences signal delay w hile the faster group does not. We investigate the effect of signal control on bimodal distributions, specifically the difference between average travel times of two groups of vehicles. Testing is performed in simulation and compared against field results of both sampled data using Bluetooth sensors as well as non-sampled data from the NGSIM vehicle trajectories. In coordinated corridors with platooning, we find that red time is a strong indicator of gap between average travel times . Effect of Signal Control on Bimodal Travel Time Distributions 2 INTRODUCTION Travel time is one of the most widely used metrics for road network performance. It has become a standard metric not only in internal assessments, but also for the public in the form of changeable message signs, 511 traveler information systems, and mobile navigation applications. Travel time is collected through periodic tracking of a sample of vehicles in the traffic stream. Examples include positions of fleet vehicles such as taxis, buses, and trucks using data from Global Positioning Systems (GPS), veh icle license plate recognition, and toll tag transponders. Recently, individual vehicles can be re -identified via media access control (MAC) addresses unique to a mobile phone or other Bluetooth (1) or Wi-Fi (2) enabled device s. When mea sured along signalized intersections and corridors , travel times have been observed to demonstrate bimodality. Whereas freeways typically have a clear average travel time, arterials often exhibit two or more “typical” travel times, sometimes separated by 3 0 seconds or more. While there has been much work in establishing the bimodality of travel times and fitting the data to various bimodal distributions , there has been little research into determining how the characteristics of the travel time distributions are affected by signal control. Researchers often assumed that the faster distribution was unaffected by signal control, and that the slower distribution was affected, rarely do research ers establish a firm connection to signal control its impact on travel time distributions. The objective of this research is to investigate the relationship between sign al control and the difference in average travel time between two populations in a bim odal distribution. The remainder of this paper will provide an overview of relevant literature, a discussion of the theory behind travel time bimodality, background of data sources and analysis, findings, and recommendations for future research. LITERATU RE REVIEW While travel times were initially assumed to fit a normal distribution, Wardrop first proposed that they instead followed a skewed distribution (3). Later re searchers suggested lognormal (4), gamma (5), Weibull (6), and Burr distributions (7). As new technologies allowed the easier collection of travel times across short arterial segments, researchers began to observe that data often faile d to fit into unimodal distributions. Davis and Xiong (8) used license plate matching along signalized arterials in Minneapolis, Minnesota, and fo und evidence of bimodal distributions. Susilawati et al. (9) found that bimodal travel times were more likely to occur on shorter links with coordinated signal control, but did not investigate the impact of the traffic control scheme on the distribution. Other research has focused on model development (10), and the relationship between bimodal travel time distributions and travel time reliability (11, 12) and congestion identification (13). Most efforts have demonstrated statistically that arterial travel t imes are better described with a bimodal than unimodal distribution. Several suggest that the bimo dality is due to signal control – that w hile the faster population drives through the corridor without stopping, the slower population encounters at least one red signal (7, 13). Other studies have observed the impact of signal control directly (8, 13). Ji et al. (10) studied the signal timing plans along their test corridor and found that their bimodal travel time model had a greater mixture of its two components when the signal was red for a greater percentage of the cycle. The authors suggest that the percentage of red time is directly correlated to the percentage of vehicles stopped at a red light and therefore appearing in the slower of the two populations. No researchers have studied the relationship between signal control and average travel time differences between the modes, Effect of Signal Control on Bimodal Travel Time Distributions 3 although Young et al. (14) have proposed that the difference is equivalent to the cycle length of the coordinated signal system. THEORETICAL FRAMEWORK Many researchers suggest that the slower population of vehicles represent those that have slowed or stopped due to signal control. This would imply that there is a strong cor relation between some aspect of signal control and the difference between the average tra vel time of each population. Young et al. (14) suggests that this difference in travel times is equivalent to the cycle le ngth of the signal. This seems unlikely. For a vehicle to experience a full cycle length of delay at single signal requires that the vehicle wait through not only the red phase but also most of the following green phase. Uncoordinated Signalized Intersections At an isolated or uncoordinated signalized intersection, where ar rival rates follow a Poisson distribution, vehicles will arrive throughout the red phase . Figure 1 provides an example of this type of arrival. While vehicles arriving during the green phase experience no delay, vehicles arriving during the red phase experience different amounts of delay depending on how early in the cycle they arrived. The maximum amount of delay a vehicle c ould experience is the red phase, startup time, and queue clearance time (and assuming no cycle failure). One would expect the average difference between the vehicles experiencing no delay and those experiencing some delay to be equivalent to half the leng th of the red phase, plus half the average queue clearance time. FIGURE 1 Time -space diag ram of an isolated intersection Ji et al. (10) studied bus movements through a single isolated signalized intersection at two locations in Columbus, Ohio. Their findings seem to support the idea that vehicles in the slower population experience a random sam ple of the red time, and therefore the average delay is close to half the length of the red phase. The slower population in Link 1 experienced 13.7 and 17.1 seconds of delay in addition to what was expected from random arrivals, while the slower population in Link 2 experienced 0.3 seconds of additional delay. Effect of Signal Control on Bimodal Travel Time Distributions 4 TABLE 1 Delay and red phase lengths at an isolated intersection Link 1 Link 2 AM PM AM PM Red phase (s)1 85.0 71.0 63.0 36.0 Expected delay (s)1 56.2 52.6 31.8 18.3 Estimated lost time (delay - 0.5 × red phase ) 13.7 17.1 0.3 0.3 1 From Ji et al. (10) Coordinated Signalized Intersections Travel times are often collected on corridors with coordinated traffic cont rol. On these corridors, signals communicate directly or are set to a common clock to ensure that vehicles receive green phases as they progress through the network at the design speed. Approximately 60% of traffic signals in North America are coordinated, and are often deployed on high volume roads (15). When signals are coordinated, vehicles travel through th e corridor in groups known as platoons. As drivers maintain different speeds, platoon disperse as they move through the corridor, with the slowest ones falling below the design speed and reaching a red signal. Vehicles often encounter the red phase as it i s beginnin g, as demonstrated in Figure 2 . FIGURE 2 Time -space diagram of a coordinated intersection For these signals, in undersaturated conditions and with good platoon formation, the vehicles in the slower population have stopped at the beginning of the red phase, and have experienced the additional delay of decelerating to a stop, waiting for nearly the entire red phase, and accelerating back up to speed. Vehicles in the slower population, therefore, would be expected to h ave travel times that are approximately as long or slightly longer than the red phase. When there are several different red phase lengths along a signalized corridor, the slower population will consist of vehicles that have been stopped at different length red phases. It would be expected that the slower population will have a larger variance in travel time, as they experienced different delays. Over longer corridors, it becomes increasingly less likely for a vehicle to progress through without stopping fo r a red signal. Effect of Signal Control on Bimodal Travel Time Distributions 5 SIMULATION To test various signal control strategies in a controlled environment, simulation was used to measure travel time s of vehicles across a hypothetical network. The test network consisted of a three -lane, one -way road with four traffic signals. The first traffic signal was used as a meter to create vehicle platoons, as would be expected in a standard coordinated traffic signal system. Travel time measurements began after the first signal and concluded after the fourth. Signals were positioned at 2000 -foot ( 610-meter) intervals with coordinated, fixed time control. Travel times were collected for one hour following a five -minute warmup. Each simulation was repeated ten times using different random seeds. The traffic simulation software VISSIM was used, and vehicle behavior followed the default settings. To isolate the effects of signal control, side street traffic was not m odeled. Travel times were sorted into fast and slow populations using k -means clustering as implemented in the GNU Octave programming language (16). In this approach, n observa tions are divided into k sets S, minimizing the sum of distance functions of each point x in the set to a set’s cen troid µ. In this analysis, k = 2. ∑∑‖𝑥−𝜇𝑖‖2 𝑥∈𝑆𝑖𝑘 𝑖=1 Four scenarios were tested, shown in Table 2. The time between cluster centroids, that is, the travel time difference between the two populations, is very close the red time. The time over the red time, about 5.4 seconds in scenarios A -C, represents the time required for a vehicle to accelerate up to the desired speed of 43.5 mi/hr (70 km/hr). From observing the simulation, vehicles exhibit ed tight platooning, with only the rear -most vehicles unable to stay within the green progression band. TABLE 2 Timing Plans and Travel Time Differences in Microscopic Simulati on Scenario Cycle Length (s) Red Time (s) Red Time of Middle Signal (s) Time Between Cluster Centroids (s) A 60 30 30 35.4 B 60 40 40 45.4 C 90 60 60 65.3 D 60 30 40 39.5 Because the simulation ignored traffic from cross streets, pedestrians, parking maneuvers, and other impediments to flow, it represents an ideal environment for coordination. As expected, it exhibits two populations of vehicles, each with very small standa rd deviations. Figure 3 is a histogram of the simulated travel times. Scenario D is of particular interest . In this scenario, the second of three signals used a red time of 40 seconds , whereas the other signals had a red time of 30 seconds. This has the ef fect of a larger variance among the slower population , as some vehicles are stopped at a signal for 30 seconds while others are stopped for 40 seconds. It should also be noted that scenario D used lower volumes to prevent cycle fail ure. Effect of Signal Control on Bimodal Travel Time Distributions 6 FIGURE 3 Histograms of simulated travel times (seconds) FIELD DATA To ensure that findings from simulation are relevant to field conditions, data from real -world arterial networks were collected and analyzed. Because signal timings could not be changed merely to test out different theories, analysis focused on existing signal timing strategies. Two sets of data were analyzed: vehicle trajectories and travel times from the Next Generation Simulation dataset on Lankershim Boulevard in Los Angeles, California, and travel times recorded from Bluetooth reidentification sensors along US -29 in Charlottesville, Virginia. NGSIM Vehicle Trajectories On June 16, 2005 from 8:30 AM to 9:00 AM , the FHWA collected high -resolution video of a 1,600 foot (500 meter) section of Lankershim Boulevard in the Universal City neighborhood of Los Angeles, California (17). As part of the effort, v ideo images were post -processed to determine vehicle positions and speeds ten times per second. Additionally, the status of each signal phase was recorded in real time. Analysis of the data reveals that vehicles traveling southbound through all four signalized intersections exhibited two typical travel times, as indicat ed in the histogram in Figure 4 . Effect of Signal Control on Bimodal Travel Time Distributions 7 FIGURE 4 Histogram of travel times on Lankershim Blvd southbound Using k -means clustering with k = 2 identified two populations of travel times with centroids of 79 seconds for the slower ( n = 354) and 42 seconds for the faster ( n = 358). The cutoff between the two clusters was a travel time of 75 seconds. The difference between the centroid s of each cluster was 35 seconds , which matches a visual inspectio n of the histogram in Figure 4 . Vehicles in the slower cluster were far more likely to stop along the corridor, with 93.2% of vehicles in the slow cluster reaching a speed of 0 mi/hr, while only 24.8% of vehicles in the fast cluster came to a stop. The difference between average travel times of the two populations suggests that the average delay experienced by a stopped vehicle was 35 seconds. As this is a coordinated signal system, vehicles will generally stop towards the beginning of a red phase as the vehicle drops back from the platoon. Table 3 shows the average duration and standard dev iation of red phases along the corridor. These red times seem to support the histogram in Figure 4. Many drivers in the slow population experienced delays of 35 seconds more than those in the fast population, due in large part to the 175 stops at the Lanke rshim Blvd. Off -Ramp intersection’s 31 second red time. Simultaneously, many vehicles experienced an additional 40 to 55 seconds of delay, which corresponds to the high number of stops at Campo De Cahuenga Way / Universal Hollywood Dr. intersection and its 50 seconds of red time. A time space diagram of vehicle movements along the Lankershim Blvd corridor is shown in Figure 5. Effect of Signal Control on Bimodal Travel Time Distributions 8 TABLE 3 Observed Red Times During NGSIM Data Collection Cross Street Mean Red Time (s) Standard Deviation (s) Number of Stops Valley Heart Dr. / James Stewart Ave. 43 13.2 58 MTA Dwy. / Main St. 27 8.8 32 Campo De Cahuenga Wy. / Universal Hollywood Dr. 50 8.8 194 Lankershim Boulevard Off -Ramp 31 7.5 175 FIGURE 5 Time -space diagram of Lankershim Blvd. NGSIM data Bluetooth Reidentification Data Additional data collection was performed using Bluetooth reidentification technology. Bluetooth sensors can record the unique MAC addresses of certain mobile phones and other Bluetooth - equipped devices. When two units are used at different locations and record the same vehicle, the difference between the timestamps can be used to calculated the vehicle’s travel time. Bluetooth sensors were placed along a four -signal, 1 mile (1.6 km) segment of US-29, a high-volume arterial in Charlottesville, Virginia. Units were pl aced mid -block to avoid measuring side street vehicles. Data was collected continuously from October 19 -27, 2016. Analysis focuses on the southbound morning peak p eriod, on weekdays between 6 AM and 9 AM. There are several reasons for this. First, the roadway is near capacity at this time of day, Effect of Signal Control on Bimodal Travel Time Distributions 9 which ensures that most traffic movements are gapping out and therefore the maximum red phase lengths from the timing pla ns are likely the actual red phase lengths experienced in the field. Second, the southbound morning peak period exhibited an obvious bimodal distribution from a visual inspection of the histogram. Travel times were classified into two groups using k -means clustering. The slow group had average travel times of 147 seconds, while the faster group averaged 90 seconds for an average speed of 40 mi/hr (64 km/hr), very close to the 45 mi/hr (72 km/hr) speed limit. The histogram of travel times is shown in Figure 6. FIGURE 6 Histogram of travel times for US -29 in Charlottesville northbound during AM peak period The difference in average travel time between the two groups was 57 seconds. When compared to the maximum red times shown in Table 4, the coordinated signal system behaved as expected. Travel times differences were slightly longer than the average red time , and many slow population travel time differences exceeded 70 seconds, which is slightly longer than the maximum red time at Hydraulic Road of 65 seconds. Effect of Signal Control on Bimodal Travel Time Distributions 10 TABLE 4 Maximum Red Times for US -29 SB, AM Peak Period Cross Street Maxi mum Red Time (s) Hydraulic Road 65 Seminole Court 48 Greenbrier Drive 54 Lenox Avenue 25 Average 48 CONCLUSIONS Travel time is a widely -used performance metric in transportation. Travel times on signalized intersections and corridors often exhibit a bimodal distribution. While researchers have suggested that traffic signal control impacts the form of these distribut ions, no research has investigated the specific role of timing plans on bimodal distributions. This study proposed a theoretical framework for signal control and the differences in average travel time between fast and slow populations. For isolated interse ctions, travel time differences are similar to the average red phase time, but slightly higher given deceleration and acceleration delay. For coordinated signals, vehicles traveling in platoons will encounter a red phase at the beginning of the phase. Vehi cles in coordinated systems must therefore wait out almost all of the red phase. The difference in travel time between this slow group and a faster group should be equal to the red time, plus some additional time for acceleration and deceleration. When a c orridor has multiple signals with different red times, then the slow population will have a wide variance reflecting the experiences of vehicles stopping at different signals. These assumptions were tested in a microscopic simulation, on empirical vehicle trajectories from the NGSIM dataset, and from sampled vehicle travel times collected with Bluetooth reidentification sensors. Results appeared to confirm the assumptions, although additional vehicle trajectory data would be needed to completely confirm th at vehicles are behaving as expected. This paper contributes to a general understanding of the effect of signal control on bimodal travel time distributions. Differences between average travel times between fast and slow groups can be predicted with some certainty, and any deviation from a pre diction can be investigated. Unimodal distributions over short segments, for example, suggests overcapacity, as no vehicles are able to get through without experiencing delay. Travel time differences may also indicate which signals experience disproportion ately more stops. If the travel time histogram shows travel time differences of twenty seconds along a corridor , then the signal with a twenty second red phase is likely the bottleneck. Future research efforts should deploy Bluetooth, WiFi, video, or othe r sensors at multiple points along the corridor to determine how individual vehicles travel within a corridor, rather than relying exclusively on end -to-end travel time. Signal control affects how many vehicles stop at a signal, and for how long. Without i ntracorridor monitoring, it is difficult to develop and validate models. Effect of Signal Control on Bimodal Travel Time Distributions 11 ACKNOWLEDGEMENTS For the US -29 travel time data, Michael Pajew ski assisted in data collection and Mengxuan Zhang performed preliminary analysis.