Guide to Establishing Monitoring Programs for Travel Time Reliability (2014)

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181 B METHODOLOGICAL DETAILS INTRODUCTION This appendix augments Chapter 3 by providing additional details about how to ana- lyze travel time reliability for segments, routes, and networks. Except for the next section on the processing steps, which is new, the appendix follows the structure of Chapter 3 to facilitate cross referencing and cross checking. Material in Chapter 3 that is complete as it stands is referenced but not repeated. OVERVIEW OF THE PROCESSING STEPS Figure B.1 portrays the steps in the reliability analysis process as a cascading sequence of steps designed to transform the various types of raw data into observations of travel times and travel time reliability. To elaborate, Figure B.1 shows infrastructure-based point speed data being enhanced through imputation if data points are missing. Inference is used to transform these spot speeds into average segment-level travel times and can be extended through synthesis to develop individual vehicle travel times, if needed. Vehicle-based automated vehicle identifi cation (AVI) and automated vehicle loca- tion (AVL) data provide direct observations of segment-level travel times. One simply needs monuments (real or virtual observation points) at the beginnings and ends of the segments. For AVI data, the sensors are typically located above or adjacent to the roadway, so it is highly likely that the observations pertain to the facility of interest. For AVL data, map matching is required to determine which facilities are most appro- priate to snap the observations to. The global positioning system (GPS) coordinates are not suffi ciently precise to make this linkage clear.

182 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY The segment travel times and rates can then be combined to develop route-level travel times and rates. The combination process is not trivial because strong corre- lations exist among the times observed on adjacent segments, but it is possible to generate these multisegment density functions. If the AVL or AVI data are sufficiently numerous that direct observations of route-level travel times exist, then the travel times and travel time distributions can be observed directly. Nonrecurring event attributes must be added to the segment- and route-level travel time data so that the effects of these conditions can be ascertained (and the effects of mitigating actions assessed). Congestion-level information also needs to be added so that congestion impacts can be seen and assessed. Combinations of congestion level and nonrecurring events form regimes, the principal categories of system operation for which reliability performance is differentiated. Another view of the processing steps is found in Figure B.2, which connects the four types of data feeds. The figure also shows how those feeds have to be processed to generate segment- and route-level travel time probability density functions (TT-PDFs). Figure B.1. Steps in the reliability analysis process. Infrastructure-Based Point Speed Data Vehicle-Based AVI Data GPS AVL Data Nonrecurring Event Data (Incident /Work Zone /Weather) Imputation Analytical Methods Segment Travel Time Map Matching Transportation Network Route Travel Time Calculate Experienced Travel Time Monte Carlo Simulation and Queuing Analysis Route Travel Time Distribution Link Incidence Matrix Travel Rate Data Influencing Factor Analysis 7-Factor Impact Analysis Route Reliability Analysis Reliability Criteria (Planning time index, Semi variance measure) Contributions of Unreliability Sources Intermediate and analysis result data Legend

183 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY From a travel time reliability perspective, the attributes of the four data feeds are as follows: 1. Spot speeds (spot travel rates) measured indirectly at specific locations. This is illustrated by single loops. Occupancy times are used in conjunction with the observed volume and an assumed average vehicle length to estimate an average vehicle speed. The sensors cannot obtain speeds for every vehicle because a fixed average vehicle length is used to estimate the average speed. 2. Spot speeds (spot travel rates) measured at specific locations. This is illustrated by double-loop detectors (often called speed traps), which can be loops in the pave- ment or virtual loops created by video detectors or other devices. These sensors are very common data sources today, although agencies seem to be reducing the number they maintain and increasingly are obtaining the data from private service providers that monitor tag-equipped or Bluetooth-equipped vehicles. Even though field-based detectors sense individual vehicles, the devices that monitor the inputs almost always summarize the inputs across time intervals (e.g., 30 seconds, a min- ute, 5 minutes) and report the average speed observed, not the individual speeds. 3. Vehicle-specific, direct observations of segment travel times but not entire paths. This is illustrated by sensors that detect AVI-equipped vehicles or Bluetooth devices. Point-to-point travel times can be observed between locations where the sensors have been deployed. The path is not observed, and vehicles that have engaged in long trips are included. This type of data can also be obtained from AVL-equipped vehicles that report their location when they pass predefined locations (e.g., monu- ments). A variant of this data feed is one in which the individual vehicle segment Figure B.2. Data feeds for estimation of TT-PDFs. AVI = automatic vehicle identification and AVL = automatic vehicle location.

184 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY travel times can be observed but vehicles cannot be tracked from one segment to the next because their IDs are suppressed. The L02 research team encountered this type of data set in one of the case studies. 4. Direct observations of the entire path. This is illustrated by the data reported by AVL-equipped vehicles. Such vehicles are observable for their entire trips. Many can report their location, speed, direction of movement, and other information every few seconds. Although such vehicles are uncommon today, they will be more common in the future. The travel time reliability monitoring system (TTRMS) needs to be prepared for them. A third perspective on reliability analysis processing is shown in Table B.1. TABLE B.1. TASKS INVOLVED IN CREATING PDFS OR DEVELOPING MEASURES OF TIME FRAME INTEREST Enhancement or Metric Data Type Type 1 Single Loops Type 2 Double Loops Type 3 AVI Type 4 AVL Passage times na na Use signal strength or bounce-back time Use passage times for latitude–longitude locations Average spot rates Use occupancy, flow, and assumed vehicle length Directly computed by the sensor Not needed Not needed Spot rates for individual vehicles Cannot be obtained Could be obtained Use signal strength or bounce-back times Use GPS speeds at latitude–longitude locations Average times or rates for segments Combine adjacent sensor spot rates Combine adjacent sensor spot rates Determine from adjusted IV-PDFs Determine from adjusted IV-PDFs Segment IV-PDFs Use average times or rates and IV-PDF typical of the traffic conditions Use average times or rates and IV-PDF typical of the traffic conditions Adjust the observed IV-PDFs to account for unequipped vehicles Adjust the observed IV-PDFs to account for unequipped vehicles Incidence matrices Base on field studies or similar segment- to-segment flow conditions Base on field studies or similar segment- to-segment flow conditions Use equipped vehicles on adjacent segments Use equipped vehicles on adjacent segments AVG-PDFs for segments or routes Add estimated segment or route times or rates Add estimated segment or route times or rates Compute from segment or route IV-PDFs Compute from segment or route IV-PDFs IV-PDFs for routes Simulation based on IV-PDFs and coincidence matrices Simulation based on IV-PDFs and coincidence matrices Use equipped vehicles or simulation based on IV-PDFs and coincidence matrices Use equipped vehicles or simulation based on IV-PDFs and coincidence matrices Note: na = not applicable; IV-PDF = individual vehicle TT-PDF or travel rate PDF (TR-PDF) for the time frame of interest (e.g., the a.m. peak); AVG-PDF = average TT-PDF or TR-PDF for a segment or route for the time frame of interest (e.g., a year or the winter).

185 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Highlights from Table B.1 include the following: • It is important to develop passage times (time stamps) for data Types 3 and 4. These are the time values on which the segment travel times are based. Interactions between the sensor and the device can occur more than once, and one of those interactions (or some time stamp derived from them) needs to be recorded as the passage time. For the other two data types, this measure is not relevant. • Spot rates for individual vehicles would be valuable if they could be obtained from data Types 1 and 2. This would significantly improve the ability of these data types to provide quality estimates of the travel times for individual vehicles across seg- ments. In the case of single loops, this is not possible except with more advanced detectors. In the case of double loops, the individual vehicle speeds are observed, but they are not reported by the sensor. For data Types 3 and 4, spot rates can be developed, but they are not needed to generate segment or route travel times. • Average spot rates (spot speeds) are the data reported by sensors of Types 1 and 2. They are useful in estimating travel times and rates for segments. For inputs of Types 3 and 4 they are not needed. • Average times or rates for segments are critical in developing PDFs for data Types 1 and 2. For data Types 3 and 4 they can be estimated from the individual vehicle travel time observations, but they are not needed to generate the segment or route PDFs. • Segment-level PDFs for individual vehicles are directly observable from AVI or AVL data, but for single and double loops this PDF has to be synthesized using the average times (or rates) for the segment and a distribution of individual travel rates that are consistent with the traffic conditions that exist on the segment (e.g., the load on the segment as reflected by vehicle miles traveled per lane mile, volume-to- capacity ratio, or another similar metric). • Incidence matrices are needed in all cases to stitch together the segment travel times to produce route travel times (or rates). For AVI or AVL data, these incidence matrices can be observed directly from the travel times for identical vehicles on successive segments. For single and double loops, the matrices have to be devel- oped from field studies or inferred on the basis of incident matrices for similar facilities at other locations (e.g., defaults). • PDFs for average travel times (or rates) on routes or segments (effectively across some extended time span like a year or a season) can be developed directly from the average segment travel times estimated by the single- and double-loop detec- tors assuming that the averages can be added across successive segments (which is very likely to be true). For AVI or AVL data, these averages and trends in their values can be obtained by analyzing the segment- and route-specific individual vehicle travel times. • PDFs for individual vehicles traversing routes can be obtained through Monte Carlo simulation (described in Chapter 3), assuming the segment-level individual vehicle PDFs have been developed and the incidence matrices have been specified.

186 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY On rare occasions when a large number of AVI- or AVL-equipped vehicles make the same trip at the same time (e.g., between interchanges on a freeway), the in- dividual vehicle PDFs can be observed directly. However, in most instances the origin and destination are to and from other locations, and simulation will need to be used because the sample size of equipped vehicles is likely to be small until a significant percentage of the vehicle fleet is AVL equipped. (Even if the vehicles are all AVI equipped, the ability to make direct observations is dependent on the sensors’ locations.) The paragraphs below further describe the analysis steps involved with each of the four data feeds. Type 1 Data Feed: Imputed Spot Speed Observations The simplest of the data feeds shown in Figure B.1 provides spot speeds (spot rates) at locations where sensors are deployed. This data type requires several enhancements to be able to provide segment- and route-level travel times. The first enhancement is a conversion of the occupancy times and counts into an average travel rate. The second enhancement is to estimate average segment-level travel rates based on the spot rates observed (e.g., combining the spot rates of adjacent sensors to estimate the travel rate for the segment between them). The third enhancement is to superimpose a distri- bution of the individual vehicle travel times on top of these average travel times to generate vehicle-level PDFs for the segments. The fourth enhancement involves using simulation and incidence matrices to estimate route-level PDFs from the segment-level PDFs. Care is needed in estimating the segment travel rates. Hu (1) showed that it is possible to estimate segment travel rates by combining the spot rates for the adjacent detector locations. This estimation procedure is discussed below. Type 2 Data Feed: Direct Spot Speed Observations For the data from sensor Type 2, the enhancement step that generates the estimated spot rates can be omitted. The sensors estimate the average vehicle speeds (travel rates) directly based on the speeds (rates) observed for individual vehicles. Hence, the first enhancement is to combine the average rates from adjacent sensors to estimate the seg- ment travel rate. Hu’s (1) methodology can again be used. The second enhancement is to superimpose the distribution of individual vehicle travel rates on top of the average rates to create the individual vehicle PDFs. Incidence matrices and simulation are then used to combine the segment-level PDFs to generate route-level PDFs. Type 3 Data Feed: Vehicle Time Stamps at Locations For the data from sensor Type 3, time stamps can be collected for individual vehicles at specific locations where sensors have been installed. Successive time stamps can be stitched together to create segment- and route-level travel times. One problem is that the paths are not known. Hence, multiple paths, as well as stops, may be repre- sented in the data. On freeways the issue of multiple paths is typically not significant, but stops may be, especially if the segment contains a rest area or weigh station. On arterial networks both multiple paths and stops may be present unless the sensors are

187 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY very closely spaced. The influence of multiple paths and stops needs to be removed by truncation (filtering) or some other mechanism. A new enhancement task arises with this data type. The PDF for the observed vehicles has to be transformed into a PDF for all vehicles. Creating this overall PDF can be important; for example, when the segment is on a toll road and the unequipped vehicles experience a different delay from those paying tolls. In such situations, the density function for vehicles paying the toll (including time spent in queue) has to be added to the density function for the instrumented vehicles to generate the travel time density function for the unequipped vehicles. The two density functions must then be combined to generate the density function for all vehicles. To combine these segment-level PDFs and create route-level PDFs, one can track the instrumented vehicles across successive segments to generate the incidence matrices. These matrices can then be used in the simulation. Type 4 Data Feed: Complete Path Data The fourth type of data feed provides direct observations of travel times across entire trips (routes), as well as segments. The data can be filtered to remove vehicles that stop, as well as ones that follow paths other than the one for which the segment is defined. This data feed is the most interesting and most useful because it directly sup- ports estimation of the TT-PDFs. The ID-based, location-specific time stamps can be used at both the route and segment level to estimate TT-PDFs. In fact, by looking at the GPS tracks in detail, one can determine the source of the travel time variability (e.g., on- and off-ramps, merge and diverge points, or specific intersections). Assuming the outliers have been removed and only the observations on the seg- ment path remain, then these data become equivalent to the observations from the AVI-equipped vehicles. The ID-based time stamps provide direct observations of travel times for entire trips or portions of those trips. Only one enhancement is needed. The observations from the equipped vehicles need to be adjusted to account for any differences in the travel times experienced by unequipped vehicles. In the case of data from sensor Type 3, the unequipped PDFs are likely to be different at locations such as toll plazas; in the case of AVL-equipped vehicles, this difference may not be significant unless the AVL-equipped vehicles are somehow treated differently by the system. To combine these segment-level PDFs and create route-level PDFs, as with the AVI- equipped vehicles, one can track the instrumented vehicles across successive segments to generate incidence matrices. These matrices can then be used in the simulation. Final Basic Thoughts A final set of basic thoughts pertains to defining the context in which the analysis will be performed. The PDFs of interest might be for individual vehicles (or packages) or for the average conditions seen on a segment or route across a given time frame (e.g., a year or a season). Ultimately, the aim is to have observations (or synthesized obser- vations) of similar individual users (e.g., people, vehicles, packages, trucks) that are making similar trips under similar conditions (whether similar is defined broadly or narrowly). Five attributes are associated with each PDF developed:

188 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY 1. The users (e.g., travelers, vehicles, packages, trucks); 2. The segment or route involved; 3. The measure of interest (e.g., travel time, travel rate, or some other measure of reliability such as on-time performance); 4. The condition pertaining (e.g., a 5-minute time slice during the a.m. peak on nor- mal Tuesdays, the a.m. peak on every weekday); and 5. The sample space from which the observations have been drawn (e.g., all the vehicle trips that occur on freeway F from interchange X to interchange Y during the winter for N years). The methodology makes two other important assumptions. The first assumption is that the travel times or rates have been directly observed or they have been developed for the segment or route of interest. This means that for data Types 1 and 2 the spot speeds have to be enhanced so that they provide segment- or route-level average travel times. The second assumption is that if individual vehicle travel times are of interest, that these segment- or route-level average segment travel times have been enhanced further so that synthesized individual vehicle travel times have been created. NETWORK CONCEPTS This material in Chapter 3 is essentially complete. The importance of establishing a topology that makes it easy to monitor travel time reliability cannot be overstressed. It is most important to establish monuments that do not confound the analysis. The easiest way to do this is by siting the monuments at the midpoints of links, which elim- inates confounding that could result from embedding turning movement delays in the travel time observations. The variability introduced by these delays could completely obscure the reliability assessment. OPERATING CONDITIONS AND REGIMES The most important idea here is the notion of a regime. A regime needs to be an oper- ating condition under which the segment, route, or network is consistent in its behav- ior. That is not to say that it will manifest the same travel times or travel time reliability every time it is in that condition, but the phenomenon by which it is producing that travel time should be the same. The regime definition that worked best during the project was based on a combi- nation of the level of congestion (uncongested, low, moderate, and high were used) and the nonrecurring event taking place, including none. IMPUTATION It is important that no voids exist in the data set being used for the reliability analysis. Imputation provides a way to fill those voids. The discussion presented in Chapter 3 is thorough in addressing imputation, and no additional details are needed.

189 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY SEGMENT TRAVEL TIME CALCULATIONS Segment travel time computations lie at the heart of travel time reliability. It is critical that high-quality travel times are developed from whatever data sources are available. Chapter 3 provided the most important perspectives on this topic. This appendix pro- vides additional thoughts and ideas. Developing Distributions of Spot Rates In many instances it is important to understand how the travel times vary among indi- vidual vehicles. Operational treatments like high-occupancy toll lanes, ramp metering, automated toll collection, and signal timing may have different impacts on different vehicles. Moreover, because a major objective is to reduce the longest of the travel times experienced (e.g., as shown by the emphasis on 95th percentile travel times), it is important to understand the distribution of travel times for individual vehicles. For the AVI and AVL data sources, this is not a problem because individual vehicle travel times can be observed. But for single- and double-loop sensors, this is a significant challenge. In the case of single-loop sensors, individual vehicle spot rates cannot typically be developed unless an advanced detector is employed. However, for double-loop detectors, indi- vidual vehicle speeds are observed. At this time they are not reported by the sensor to the traffic management center, but they could be. It would be helpful if this informa- tion were reported by these sensors at some point in the future. This would improve the ability of the TTRMS to ensure that it is developing defensible individual vehicle TT-PDFs for each segment. For AVI and AVL data sources, the situation is very different. For AVI sensors, although these individual vehicle spot rates are not easy to observe, they are not needed because the vehicle-specific segment travel times are directly observable. In the case of AVL-equipped vehicles, the GPS units can often estimate the vehicle’s speed and provide that piece of information for any desired location. But this observation is not needed to ascertain individual vehicle segment travel times. Those times are directly observable. Developing Average Times or Rates for Segments To develop average travel times or rates for segments based on field sensor average spot speeds, Hu (1) has demonstrated that the following two approaches work well. In the first approach, the arithmetic average of the two spot rates is computed and then adjusted by a factor γ as shown in Equation B.1: 2s 1 2τ = γ τ + τ   (B.1) where τs is the travel rate for the segment, and τ1 and τ2 are the upstream and down- stream spot rates, respectively. The value of γ is dependent on the traffic flow condi- tions on the segment, that is, the regime that is extant at the time for which τs is desired (e.g., the level of congestion). The appropriate value of γ can be obtained from a look- up table once the values have been calibrated for the regimes.

190 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY A second option also works well. Equation B.2 uses two parameters α and β to combine the spot rates: s 1 2α βτ = τ + τ (B.2) Again, the values of α and β are dependent on the traffic flow conditions on the segment, that is, the regime that is extant at the time for which τs is desired (e.g., the level of congestion present). They can be obtained from a look-up table once the values have been calibrated for the regimes. Another option that is perhaps more common today is to assume that the single- or double-loop sensors are in the middle of specific segments and that the spot rate observed at the sensor pertains to the entire segment. In this case, a variant of Equa- tion B.1 can be used to ensure that the segment travel times are consistent with the field observations. Assume there are three sensor stations, i, j, and k, with station j in the middle. Further, assume there are distances di, dj, and dk associated with these sta- tions and that those distances can be further broken down into the distances that are upstream and downstream of the sensor: dui, ddi, duj, ddj, duk, and ddk. Then, focusing on the middle sensor j, Equation B.1 can be rewritten and reinterpreted twice as shown in Equation B.3: d d d d d d d d andij ij di i uj j di uj jk jk dj j uk k dj uk τ = γ τ + τ +       τ = γ τ + τ +       (B.3) Once the values of γij and γjk have been computed, the value of γ for segment j can be developed by Equation B.4: d d d dj uj ij dj jk uj dj γ = γ + γ +       (B.4) Developing Individual Vehicle PDFs for Segments For the single- and double-loop sensors, this is the most challenging task. It would be simpler for the double-loop sensors if the distribution of individual vehicle spot rates was reported, but it is not. Perhaps in the future it will be. The sensor would not have to pass back each of the individual vehicle speed observations. Rather, it could pass back the sum of the squares of the vehicle speeds and the sum of the cubes of the vehicle speeds. These two additional pieces of information, in addition to the already reported average and number of vehicles observed, would be sufficient. Assuming that only the average speed and the number of vehicles observed are available, the procedure is as follows. First, ascertain the current regime for the seg- ment based on its average travel time (travel rate) and vehicle flow rate (the count divided by the time interval). Next, select the prestored distribution of travel rates that corresponds to that regime. Finally, adjust the mean rate of the prestored distribution to match the average travel rate observed, and report the result. This procedure can be illustrated based on an analysis conducted using Bluetooth data from the Berkeley Highway Laboratory, a section of I-80 located adjacent to Berkeley, California.

191 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Bluetooth data were available for 4 days in January 2011: Thursday, January 13; Thursday, January 20; Saturday, January 22; and Monday, January 24. One or more periods of congested operation occurred during each of these days, as shown by Table B.2. TABLE B.2. BLUETOOTH DATA USED TO EXPLORE INDIVIDUAL VEHICLE TRAVEL RATE DENSITY FUNCTIONS Date in 2011 Day TIC CON TOC January 13 Thursday 12:00 to 13:00 13:00 to 19:30 19:30 to 20:00 January 20 Thursday 08:15 to 09:15 09:15 to 10:00 10:00 to 10:30 15:30 to 16:30 16:30 to 19:00 19:00 to 20:00 January 22 Saturday 14:00 to 15:00 15:00 to 18:00 18:00 to 18:30 January 24 Monday 14:30 to 15:30 15:30 to 18:30 18:30 to 19:30 Note: TIC = transition into congested operation; CON = congested operation; TOC = transition out of congested operation. As Table B.3 shows, hundreds of observations were recorded for each operating condition. TABLE B.3. ONE SET OF BLUETOOTH OBSERVATION COUNTS BY DAY AND CONDITION Condition Observation Counts by Day January 13 January 20 January 22 January 24 Total Free flow 1,183 1,446 1,727 1,566 5,922 Transition to peak 121 328 160 126 735 Transition from peak 84 310 80 149 623 Peak (congested) 1,099 639 594 552 2,884 Total 2,487 2,723 2,561 2,393 10,164 Note: All dates are from 2011. Figure B.3 shows the TR-PDFs for the free-flow condition overall and for each day on a segment length of about 4,500 feet. The distributions are all similar, and the vari- ances are small. The minimum is about 50 s/mi (about 72 mph), the 50th percentile is at about 70 s/mi (about 51 mph), and the 95th percentile is at about 86 s/mi (about 42 mph). The coefficient of variation is about 0.15. In this instance, this TR-PDF could be used to estimate off-peak PDFs for individual vehicles for all the times during the year when the facility was lightly loaded. In contrast, the TR-PDFs during peak period congestion on the same segment involve significantly larger travel rates, a wider distribution, and much more day-to- day variability, as shown in Figure B.4. The minimum travel rate is about 60 s/mi

192 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Figure B.3. Off-peak travel rates measured by Bluetooth sensors for the Berkeley Highway Laboratory. 0% 5% 10% 15% 20% 25% 30% 50 60 70 80 90 100 110 120 130 140 150 Pe rc en ta ge o f V al ue s i n 50 s ec R an ge (≈ P DF ) Travel Rate (sec/mi) Off-Peak Travel Rate PDFs All 13-Jan 20-Jan 22-Jan 24-Jan (60 mph), the 50th percentile ranges from 150 to 190 s/mi (19 to 24 mph), and the 95th percentile ranges from 180 to 360 s/mi (10 to 20 mph). Two reasonable options are (a) to use the overall PDF for all the days and adjust it to the median travel rate being observed at a given point in time or (b) to select the PDF whose median travel rate most closely matches the extant travel rate and then adjust that PDF to the extant travel rate. Exactly what to do during the transitions to and from peak flow conditions is more challenging, but the data still provide good guidance. Figure B.5 shows the TR-PDFs for the transition to peak flow conditions observed on the 4 days. Evidence of both off-peak and peak conditions can be seen. The density functions appear to be multi- modal (bimodal). The minimums are about 60 s/mi (60 mph), the median ranges from 90 to 130 s/mi (30 to 40 mph), and the 95th percentile ranges from 160 to 400 s/mi (10 to 20 mph). Again, two courses of action seem reasonable. The first is to use the overall PDF and scale the rates to match the extant median. The second is to identify the PDF whose median most closely matches the extant conditions and then scale the rates to match.

193 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY 0% 5% 10% 15% 20% 25% 0 50 100 150 200 250 300 350 400 450 Pe rc en ta ge o f V al ue s i n 50 s ec R an ge (≈ P DF ) Travel Rate (sec/mi) Approximate Peak Travel Rate PDFs All 13-Jan 20-Jan 22-Jan 24-Jan Figure B.4. Peak travel rates as measured by Bluetooth sensors for the Berkeley Highway Laboratory. Storing the regime-specific PDFs is a minor issue. The option that has emerged from the overall project involves storing the percentile values (either all of them or every 5th percentile). The data rarely seem to fit any specific parametric density func- tion, and significant insight is often gained by using nonparametric analyses. Updating the PDFs The L02 research team originally wanted to use a formal Bayesian methodology to update the parameter values for a gamma density function. However, this proved to be cumbersome, and the gamma density function did not always fit well with the data. Hence, the formal procedure was set aside. Instead, the team elected to reestimate the model parameters using techniques like maximum likelihood applied to an updated sample space that includes new, as well as old, observations. Techniques like exponential smoothing can also be employed. Those techniques have been explored less by the research team, so their value is not well-established.

194 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY ROUTE TRAVEL TIME CALCULATIONS The material in this section of Chapter 3 is essentially complete. Described there are procedures for developing average route travel times and distributions of individual vehicle travel times based on (a) Monte Carlo simulation of traffic flow behavior on successive segments tied together with incidence matrices and (b) a lane-by-lane Monte Carlo simulation of a cascading sequence of bottleneck locations. Accounting for Correlation It is clear that correlation exists among segment travel times, especially when the seg- ments are short. This phenomenon affects the manner in which one needs to combine segment-level TT-PDFs to form route-level TT-PDFs. One cannot add the variances by assuming that the TT-PDFs are uncorrelated. To illustrate, Figure B.6 shows scatterplots for individual vehicle travel times on subsequent segments along a 6-mile section of freeway in Sacramento, California. The site is discussed in more detail in Appendix D. The sequence of AVI monitoring Figure B.5. Transition-to-peak condition travel rates measured by Bluetooth sensors for the Berkeley Highway Laboratory. 0% 5% 10% 15% 20% 25% 30% 35% 0 50 100 150 200 250 300 In cr em en ta l P er ce nt ag e in 5 0 se c Bi ns (≈ P DF ) Travel Rate (sec/mi) Approximate Transition-To-Peak Travel Rate PDFs All 13-Jan 20-Jan 22-Jan 24-Jan

195 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY stations is 39, 9, 10, and 11. Travel times are TT3909, TT0910, and TT1011. The scales in minutes for the travel times are shown along the left-hand and bottom bor- ders. The scatterplots above the diagonal show the correspondence between travel times on adjacent segments (TT3909 versus TT0910 and TT0910 versus TT1011) and then two segments away (3909 versus 1011). Each one is then plotted against the overall travel time (TT3909 versus TT39-9-10-11; TT0910 versus TT39-9-10-11; and TT1011 versus TT39-9-10-11). The scatterplots are symmetric about the diagonal. Not only do the travel times on adjacent segments show a significant degree of correlation but the travel times on each segment also are correlated with the overall travel time. The scatter does not increase dramatically as the segments become further separated, which would be the case if the travel times were uncorrelated. In fact, the correlation between the travel times is strong, as demonstrated by the top right-hand scatterplot, which shows the correspondence between travel times on the first segment (TT3909) and the overall travel times (TT39-9-10-11). Only tightly correlated travel times could produce scatterplots that look like this. Figure B.6. Correlations among individual vehicle travel times for a sequence of three segments along I-5 in Sacramento, California.

196 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Developing Incidence Matrices One way to address the correlation among segment travel times is to develop incidence matrices. Hu (1) studied the interdependence between segment PDFs using a VISSIM model of the Berkeley Highway Laboratory section of I-80 East in the San Francisco Bay Area. The facility is five lanes wide and experiences significant congestion during the afternoon peak. Using the model, AVI-like data could be collected for any sample of the total vehicle population between and among any set of selected locations. For two adjacent segments AB and BC, Hu (1) demonstrated that the following procedure could be applied to faithfully estimate the multisegment PDF for AC: 1. Observe the AVI-equipped vehicles traversing AB, BC, and AC, and note their travel times (and rates) on AB, BC, and AC. 2. Create a small number (10) of travel rate bins for both AB and BC. 3. Create an incidence matrix that shows the frequency with which specific bin-to- bin combinations of the travel rates arise (e.g., a travel rate on AB in Bin X and a travel rate on BC in Bin Y). 4. Use Monte Carlo simulation to generate a PDF for the travel rate on AC: a. Select a first random variable x1. b. Select a travel rate τAB based on x1. c. Identify the AB travel rate bin in which τAB belongs. d. Use τAB and the length of Segment AB to determine when the vehicle will arrive at the beginning of Segment BC. e. Select a second random variable x2. f. Identify the BC travel rate bin from which τBC should be obtained based on x2. g. Select a third random variable x3. h. Select the BC travel rate τBC on the basis of the lower and upper bounds for the BC travel rate bin and the value of x3. i. Compute the travel rate τAC using the following expression: τAC = (τAB * dAB + τBC * dBC)/dAC. This process repeats for every pairwise combination of segments in the route. A sufficiently large number of realizations generated in this manner will result in creating a defensible TT-PDF for the route. Table 3.3 in Chapter 3 shows an incidence matrix that was developed using this procedure. Figure 3.23 further shows that that overall travel times developed via this method reproduce the distribution of τAC almost exactly. For real-world applications, this methodology can be applied by doing field studies of individual vehicle travel rates on adjacent segments (e.g., using temporarily deployed Bluetooth equipment) and then using the resulting incidence matrices for the regime conditions to which they correspond.

197 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Developing PDFs for Average Travel Times or Rates Agencies are often concerned with how their facilities are performing across the year (or some other time frame) given the vagaries of the load conditions to which they are exposed, such as incidents, inclement weather, and unexpected demand conditions. To many agencies, travel time reliability is not about individual vehicle travel times, but rather minimizing the variation in facility performance that arises given the various operating conditions to which the facility is exposed. As the data for individual vehicle travel times are largely not available, these analyses focus on the variations in average travel times (or average speeds) that arise across the year. That is, a PDF is developed for the observed average speeds or travel times, and that PDF is studied to see why certain high travel times (travel rates) arise and how they might be prevented. These PDFs can be developed from any one of the four data sources examined earlier by using a four-step procedure: 1. From any one of the four data sources, compute average travel times (rates) by segment for the various load conditions of interest (e.g., every 5 minutes during workdays). 2. Examine the distributions of these average travel times (rates) for individual seg- ments to see why the travel rate varies and identify ways in which its variation can be reduced. 3. Add the average travel times by segment to generate route-level average travel times. The best way to do this is to walk a time–space matrix that samples seg- ment-specific travel rates based on the times at which hypothetical trips (e.g., start- ing every 5 minutes) would be traversing each subsequent segment. The instanta- neous travel rates observed at a single point in time can also be used, but the result misrepresents the average travel time that any traveler would experience because the cascading effects of congestion are ignored. 4. Examine the distribution of these average travel rates for the routes of interest to see why they vary, and determine if there are actions that can be taken, like clear- ing incidents more quickly, to reduce the extent to which they vary. The L02 research team examined the application of this technique in the context of a 10-mile stretch of I-8 westbound in San Diego, California, as shown in Figure B.7. The route begins in La Mesa and ends at the junction with I-5. Although all the major time periods of the weekdays were studied, it is sufficiently illustrative to focus on the a.m. peak, during which the traffic flows are heaviest. The time frame was from November 3, 2008, until February 27, 2009. Only workdays were examined. Holidays and weekend days were omitted. Starting at every 5-minute interval during the time frame (i.e., from 0:00 on November 3 until 23:55 on February 27), the time–space matrix was walked to develop end-to-end (entire route) average travel times. The data set thus comprises 22,464 observations for these route- level average travel times. The intrinsic quality of each average travel time was assessed based on the percentage of segment-level travel times that were actually observed by

198 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY the detectors in the field as opposed to being imputed. In addition, to the extent possi- ble, the external factors that might have influenced freeway performance (e.g., weather incidents) were identified and tagged to each travel time observation. Of the 22,464 observations, 20,737 of them were for a condition that was deemed normal; that is, no significant external factor could be identified. For 1,022 observations, a weather event was identified; for 556 of them, an incident; for 96 of them, a demand event; and for 53 of them, a special event. During the a.m. peak, the differences in the cumulative density functions (CDFs) are striking for different external factor conditions. Figure B.8 shows the travel time CDFs for conditions for which sufficient observations were available to create a CDF: normal operation, weather, and incidents. It is immediately apparent that the incident and weather conditions not only changed the higher-percentile travel times, but the lower-percentile values, as well. It is also interesting that weather did not produce the largest travel times; the incidents did. However, the weather events produced a major increase in the lower-percentile travel times, but incidents did not. If these findings were indeed indicative of the way this facility performs across the year, the findings could be used to identify actions that would improve travel time reliability for this facility. It would seem useful to ensure that the highway can tolerate adverse weather (i.e., that it drains well so that heavy downpours do not interfere with traction and that it has high-quality reflective lane markings so poor visibility does not adversely affect its performance) and that significant attention would be given to clearing major incidents (not so much minor ones) so that the tail of the incident-based PDF is reduced. Figure B.7. Route on I-8 westbound in San Diego studied for the reliability of its average travel times. Map data © 2012 Google.

199 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 7 9 11 13 15 17 19 21 23 25 Cu m ul ati ve P ro ba bi lit y Travel Time (min) AM Peak Travel Time CDFs Weather Incident Normal All Figure B.8. Travel time CDFs for various conditions in the a.m. peak on I-8 westbound in San Diego. INFLUENCING FACTOR ANALYSIS Much of the motivation behind Project L02 and the interests of operating agencies lie in identifying the reasons why travel times are unreliable or become more unreliable. It is well known that weather and incidents, for example, affect travel times and their reliability, but the question is how much and to what extent. Hence, development of the PDFs and their analysis is inherently focused on deciphering the reasons why travel times are unreliable, or more specifically, become more unreliable under certain conditions. Techniques for performing the factor analysis have been described in Chapter 3. Those discussions are effectively complete.

200 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY CONSIDERATIONS FOR TRANSIT Most of the discussion in this appendix has focused on vehicle (effectively auto) travel times. The figures are dominated by auto travel. The discussions about travel time and travel rates predominantly focus on auto trips, and the commentary about diagnostic ideas relate to auto trips. Transit and freight trips are different. Transit passengers do not control what the vehicles do. They board and alight from the vehicles and make transfers. Their travel times are strongly influenced by the headways at which the vehicles operate and the reliability of the transfers. Freight trips are similar. Packages get picked up and carried from shipper to ter- minal, terminal to terminal, and terminal to receiver. The travel times they experience are heavily influenced by the operating plans being followed by the freight providers and the reliability of their operations. Packages are similar to transit passengers in that they ride on one vehicle after another and their travel time is influenced by the headway between pick-ups (not often thought about that way, but often once a day) and the reliability of the connections between vehicles (i.e., trucks). Unlike transit pas- sengers, packages cannot influence the reliability of their trips. If they get placed on the dock in the loading area for the wrong truck, they cannot move themselves to the area for the right truck. Hence, the reliability of their trip times is likely to be worse than that of the transit riders. On the one hand, freight companies only earn revenues if they deliver packages on time, so they tend to pay attention to whether the packages are being handled correctly. On the other hand, transit agencies are not particularly sensitive to whether the passengers route themselves correctly: if a transit passenger gets delayed or reaches the wrong destination, culpability rests with the passenger as well as the service provider. In spite of these differences, a strong similarity exists between transit trips and package trips in that they are both dependent on the headway between vehicle arrivals and the reliability of connections. The observability of the trips is a different issue. Transit trips are largely unobserv- able. Many transit agencies do not track the movements of their passengers. Even the transit agencies with the most sophisticated data, such as the Washington Metropolitan Area Transit Authority, only know where and when the passengers entered and left the system—akin to AVI-type information. They do not know the path followed unless they were to track Bluetooth devices or cell phones, which they could do. Packages, on the other hand, tend to be tracked carefully by many freight service providers. Public agencies may not have access to this information, but many carriers know where the packages are at all times. In some instances it is because the package’s bar code was just read (i.e., it was picked up or received at a distribution center), or sometimes it is by inference (it was scanned as it was loaded onto the delivery truck, and the delivery truck is en route to the receiver). In this sense, the package data are similar to AVI-type data. In selected (but very few) instances, packages have radio fre- quency ID tags that are read constantly, so the data are AVL-like.

201 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Because carriers rarely share their package-level information except with the stakeholders (the shipper and receiver) who have a need-to-know interest, providing reliable service to freight carriers becomes functionally similar to dealing with reliable travel times for autos. The trucks need to be able to traverse the highway network with reliable travel times. They do not want to be delayed so their deliveries are late. Unlike person trips, however, they often also do not want to be early because they will then have to wait until they were supposed to arrive. Early arrivals mean another activity could have been inserted, which represents a lost opportunity for better efficiency, more cost-effective operation, or more revenue. Hence, this discussion now focuses on transit trips because they are more often under the purview of the agencies responsible for operating the highway system. Case study transit data were only obtained during the San Diego case study. How- ever, those data are representative of the information available to the most progressive transit operators. Selected vehicles were equipped with AVL-like devices that could monitor the latitude–longitude location of the bus in real time, the times at which the bus doors opened and closed, and the number of people who boarded or alighted from the bus. Were all the buses instrumented, then a technique similar to that used to gener- ate the freeway travel times could have been used. It could have been assumed that hypothetical passengers boarded a bus B1 at time T1 at stop S1 bound for stop S2. By simulating a large number of trips from S1 to S2 during different times of day (operat- ing conditions), PDFs of the transit travel times could have been created. For trips on a single line this would have been simple. For trips that involve transfers, the process would have been slightly more complicated. The hypothetical passenger would have boarded bus B1 at time T1 and stop S1, traveled to transfer location X1, alighted at T2, and waited for bus B2 that arrived at X1 at some time T3 > T2. The traveler would then have boarded bus B2, traveled to S2, and alighted at some time T4. The differ- ence T4 − T1 would be the travel time, and the reliability of these trips could also be assessed. In the case of San Diego, where not all of the buses were instrumented, a more complex analysis procedure had to be employed. The process involved two steps: (1) preprocessing the bus trip data to develop information needed to conduct the anal- ysis and (2) generating a synthesized set of hypothetical, representative trips through Monte Carlo simulation. Developing Transit Rider PDFs for Trips Figure B.9 shows the process used to synthesize the trip times. The flowchart at the top of the figure provides an overview. The bottom flowchart provides more detail. The figure is annotated with letters from A to J to provide reference markers for the description that follows. It is also couched in the context of a trip on bus Routes 11 and 7, but the bus route numbers are not relevant to this discussion. It is sufficient to recognize that two separate bus routes are involved with a transfer between them.

202 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY Figure B.9. Analysis flowchart for transit trips involving transfers. The overview starts with Marker A, focused on the initial bus boarding process. A passenger arrives, as does a bus on Route 11. Depending on when they arrive, the pas- senger either gets on the first Route 11 bus or the next (second) one. If the passenger gets on the second bus, the passenger incurs a delay of one headway (what this delay means is described in more detail below). In either event, as shown by the blocks near Marker B, the passenger travels to and arrives at the transfer point, as shown near Marker C. Arriv- ing separately is the first Route 7 bus. An analysis of when that bus arrives relative to when the passenger arrives on the Route 11 bus determines whether the passenger gets on the first Route 7 bus or has to wait for the next (second) one. If he or she gets on the second Route 7 bus, an additional delay is incurred (discussed below). In either event, as shown by the blocks near Marker D, the passenger then arrives at the destination. The detailed description starts with Marker E. Near it are shown the PDFs for the arrival of the passenger (Px) and the first Route 11 bus. Consistent with Bowman and Turnquist (2), the passenger PDF (Δt0) tends to favor early arrivals with a small prob- ability of being late. Separately, consistent with the San Diego data, the Route 11 bus (Δt1) follows a second PDF. The distribution for the bus indicates a small probability

203 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY of departing early (earlier than the scheduled departure time) and a much larger prob- ability of departing late. If the passenger arrives before the Route 11 bus departs, then the passenger boards the first Route 11 bus. If that happens, the descending dashed line toward Marker F indicates that the passenger incurs a travel time (Δt2) to reach the transfer stop and the passenger (on the Route 11 bus) arrives at the transfer stop at t1, which is at some point in time relative to the scheduled departure time (Δt3). (Departure times have been used as the reference because they are worst-case times; it is known for sure that the passenger has arrived when the bus departs.) If the pas- senger misses the first Route 11 bus (because he or she arrives after the first Route 11 bus departs), then a schedule delay (Δt4) is incurred until the next Route 11 bus arrives (to the right of Marker E). A second Route 11 bus arrives (Δt5), the passenger boards, and the Route 11 bus travels to the transfer location (Δt6), shown by Marker G, and the passenger arrives at the transfer stop at t2, which is at some time relative to its scheduled departure (Δt7). Whichever arrival time governs (t1 or t2) becomes the start of the second part of the trip (Marker H). Moreover, the corresponding relative arrival time (Δt4 or Δt7) becomes the basis (Δt8) for determining which transfer bus is caught. If the passen- ger’s relative arrival time on the Route 11 bus (Δt8) is less than the sum of the sched- uled connection time (Δt9) and the relative departure time for the Route 7 bus (Δt10), then the first Route 7 bus is caught. This leads to a travel time to the destination (Δt11), an arrival time (t3), and a relative arrival time compared with the schedule (Δt12) (Marker I). However, if the Route 11 bus arrives late (Δt8), or the Route 7 bus departs early (Δt9 + Δt10), then the passenger may miss the first Route 7 bus, incur a delay (Δt13) until the next Route 7 bus arrives (Δt14), then incur a travel time (Δt15) to the destina- tion and arrive at t4 with a relative arrival time Δt16 (Marker J). Numerical examples help illustrate the analysis. Table B.4 presents four of them. In the first example, no bus is missed. In the second example, the connection bus is missed. In the third example, the first Route 11 bus is missed, but the subsequent connection is made. In the fourth example, both the first Route 11 bus is missed and the first Route 7 transfer bus is missed. In all cases the reference time when t = 0 is the scheduled departure time of the first Route 11 bus. All the values are in seconds. Results obtained from actually working with the transit data obtained in the San Diego case study can be found in that appendix. The first example starts with Δt0 < Δt1 (−120 < 30), which means the passenger gets to catch the first Route 11 bus. The starting time for the trip (t0) becomes −120 seconds (i.e., the passenger arrived 2 minutes before the scheduled departure time, which is the reference point for t = 0). The travel time to the transfer point is Δt2 = 1,570, the arrival time is t3 = t8 = 1,600, and the relative arrival time at the transfer point (relative to the scheduled departure at that location) is Δt3 = Δt8 = 20. Next, the connection is analyzed. The relative arrival time is Δt8 = 20, the transfer time is Δt9 = 240, and the first Route 7 bus is late Δt10 = 50, so the passenger has no problem catching the first transfer bus Δt8 < Δt9 + Δt10. The passenger then departs the transfer stop at t10 = t8 − Δt8 + Δt9 + Δt10 = 1,600 − 20 + 240 + 50 = 1,870, travels to the

204 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY destination Δt11 = 190, and arrives at the destination at t12 = t10 + Δt11 = 1,870 + 190 = 2,060 with an arrival relative to the scheduled arrival time of Δt12 = −10 (10 seconds early) and an overall travel time of tt = t12 − t0 = 2,060 − (−120) = 2,180 seconds (36.3 minutes). TABLE B.4. FOUR NUMERICAL EXAMPLES OF ESTIMATING TRAVEL TIMES FOR TRANSIT TRIPS INVOLVING A TRANSFER Metric No Miss (s) Miss 2 (s) Miss 1 (s) Miss Both (s) Δt0 −120 −90 −30 50 Δt1 30 15 −50 −100 Δt2 1,570 1,730 Δt3 20 350 Δt4 900 900 Δt5 −30 40 Δt6 1,400 1,800 Δt7 −100 400 Δt8 20 350 −100 400 Δt9 240 240 240 240 Δt10 50 −100 70 −100 Δt11 190 210 Δt12 −10 20 Δt13 720 720 Δt14 10 −30 Δt15 180 190 Δt16 −10 30 t0 −120 −90 −30 50 t1 30 15 t3 1,600 1,745 t5 870 940 t7 2,270 2,740 t8 1,600 1,745 2,270 2,740 t10 1,870 2,680 t12 2,060 2,890 t14 2,365 3,270 t16 2,545 3,460 tt 2,180 2,635 2,920 3,410 In the second example, the first Route 11 bus is caught, but the first Route 7 transfer bus is missed. The example starts with Δt0 ≤ Δt1 (−90 ≤ 15), which means the passenger catches the first Route 11 bus. The starting time for the trip (t0) becomes −90. The travel time to the transfer point is Δt2 = 1,730, the arrival time is t3 = t8 = 1,745, and the relative arrival time at the transfer point (relative to the

205 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY scheduled departure time) is Δt3 = Δt8 = 350. The transfer time is Δt9 = 240, and the first Route 7 bus leaves early Δt10 = −100, so the passenger misses the first transfer bus (Δt8 ≥ Δt9 + Δt10, or 350 ≥ 240 + [−100]). Hence, the passenger has to wait for the second transfer bus, which has a scheduled time Δt13 = 720, which is 12 minutes later than the first transfer bus; it arrives a little late Δt14 = 10. This means it leaves at t14 = t8 − Δt8 + Δt9 + Δt13 + Δt14 = 1,745 − 350 + 240 + 720 + 10 = 2,365. The Route 7 bus then travels to the destination Δt15 = 180 and arrives a little early Δt16 = −10 at t16 = 2,545. The overall trip time is tt = t16 − t0 = 2,635 (43.9 minutes). In the third example, the first Route 11 bus is missed and the first Route 7 transfer bus is caught. The example starts with Δt0 > Δt1 (−30 > −50), so the pas- senger misses the first Route 11 bus. The starting time for the trip (t0) becomes −30. The passenger has to wait for the next bus Δt4 = 900, which is a little early Δt5 = −30. The travel time to the transfer point is Δt6 = 1,400, the arrival time is t7 = t8 = 2,270, and the arrival time at the transfer point relative to the scheduled departure time is Δt7 = Δt8 = −100. The transfer time is Δt9 = 240, and the first Route 7 bus leaves late Δt10 = 70, so the passenger catches the first transfer bus (Δt8 ≤ Δt9 + Δt10, or −100 ≤ 240 + 70). The passenger departs the transfer stop at t10 = t8 − Δt8 + Δt9 + Δt10 = 2,270 − (−100) + 240 + 70 = 2,680, travels to the destina- tion Δt11 = 210, and arrives at the destination at t12 = t10 + Δt11 = 2,680 + 210 = 2,890 with an arrival relative to the scheduled arrival time of Δt12 = 20 (20 seconds late) and an overall travel time of tt = t12 − t0 = 2,890 − (−30) = 2,920 seconds (48.7 minutes). In the fourth example, both the first Route 11 bus and the first Route 7 trans- fer bus are missed. The example starts with Δt0 > Δt1 (50 > −100), so the passen- ger misses the first Route 11 bus. The starting time for the trip (t0) becomes 50. The passenger has to wait for the next bus Δt4 = 900, which is a little late Δt5 = 40. The travel time to the transfer point is Δt6 = 1,800, the arrival time is t7 = t8 = 2,740, and the arrival time at the transfer point relative to the scheduled departure time is Δt7 = Δt8 = 400. The transfer time is Δt9 = 240, and the first Route 7 bus leaves early Δt10 = −100, so the passenger misses this bus (Δt8 ≥ Δt9 + Δt10, or 400 ≤ 240 + [−100]) and has to catch the second one. The added wait for the next bus is Δt13 = 720, which is 12 minutes later than the first transfer bus, and that bus arrives a little early Δt14 = −30. This means the departure time from the transfer stop is t14 = t8 − Δt8 + Δt9 + Δt13 + Δt14 = 2,740 − 400 + 240 + 720 + (−30) = 3270. The Route 7 bus then travels to the destination Δt15 = 190 and arrives a little late Δt16 = 30 at t16 = 3,460. The overall trip time is tt = t16 − t0 = 3,460 − 50 = 3,410 (56.8 minutes). These examples illustrate the process used to synthesize the route trip times for transit riders for the general case of a transit trip involving two routes and one transfer. SENSOR SPACING AND SAMPLING FOR TRAVEL TIME RELIABILITY MONITORING Operating agencies have historically created monitoring systems that use sensors placed at strategic locations along their freeway networks. Figure B.10 shows a sec- tion of freeway in California where there are 10 sensors in 5 miles, or a sensor about

206 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY every 0.5 miles. This is a bit dense, but typical. A spacing of a mile or more is common. Of course, putting them at an equal spacing has no particular value; the sensors need to be installed either at locations where congestion rarely occurs (like the first, fifth, ninth, and 10th sensors) so flow rates can be monitored, or at places where bottlenecks arise (like the second, third, fourth, fifth, and eighth sensors) so that queuing can be detected. The advent of vehicle-based sensing technologies, including those that provide speeds for short TMC segments, are revolutionizing these ideas because sensor place- ment becomes less of an issue: nothing has to be installed in the roadway surface. Moreover, actual travel times can be observed if the vehicles are reidentified (e.g., by using their media access control IDs or tag numbers). In addition, and different from sensing the general health or status of the network, which is the purpose for the sensor deployments shown above, monitoring travel time reliability has a different objective. One needs to sense the status of the system (in time or in space) in a way that produces a defensible image of the travel times that are occurring, as well as their changes in time and space. For example, Figure B.11 shows the temporal pattern of AVI-based travel time observations on I-5 in Sacramento, south of US-50, for February 18, 2011, when there was an incident immediately preceding the p.m. peak. The rise and fall in travel times during the incident is dramatic: growing from 5 minutes to 35 minutes in the span of 20 minutes and then dropping back to about 7 minutes in another 30 minutes. The travel times in the p.m. peak, which are typical for this location, rise from 5 minutes (without the incident) to 12 minutes in an hour and a half and then fall back to 5 min- utes in another hour and a half. To adequately observe such transients, especially the first, from the incident, one would have to be sampling the travel times every 1 to 2 minutes so that the rapid rise, as well as the subsequent fall, could be observed. The p.m. peak that follows could adequately be monitored with samples at every 5 to 10 minutes. Of course, a difference exists between how many samples are needed ex post facto to reproduce an observed waveform, like the ones discussed above, compared with monitoring the travel times that unfold in real time. Not only are the rates of change unknown but also latency (how long will it be after the event occurs) becomes an issue. In the examples above, a monitoring rate of every 15 minutes would be too slow to spot the incident in any meaningful way, and it would be adequate but not ideal to observe the p.m. peak. An interval of a minute would be adequate for both. Figure B.10. Typical sensor spacing on a freeway.

207 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY 0 5 10 15 20 25 30 35 40 MP 21:7MP 00:6MP 84:4MP 63:3MP 42:2 Tr av el T im e (m in ) Time of Day Individual Vehicle Travel Times / I-5 Southbound Figure B.11. An example of two travel time transients: an incident followed by a p.m. peak. A sampling rate of 5 minutes would detect both, but would provide a less-responsive and less-accurate representation of the incident-related transient. These data tend to suggest that a sampling rate of 5 minutes or shorter is likely to be adequate. In the spatial domain it is more difficult to understand what is adequate. The challenges are twofold. The first is to observe the vehicle trajectories in a suitable man- ner—in space, not in time—to create defensible travel times. The second is to identify a spacing that allows one to pinpoint the places of reliability trouble, in terms of queu- ing and momentary slow-ups, so that corrective actions can be taken. Fortunately, the objective is not to reproduce the exact vehicle trajectories. To do that would require a sample to be taken roughly every 10 feet because the transient slow-downs or speed- ups span only 30 to 50 feet, and adequately representing them would require five or so observations. Two thoughts are helpful in bounding the lower end of the spatial sampling interval. The first thought is the spatial geometry of highway design, and the second thought is expectations about how long it should take before an incident can be identified. First, ramp lengths and weaving sections are rarely shorter than 300 to 500 feet, so detector spacing shorter than this would be difficult to implement. Second, and in a separate dimension, shockwaves travel at rates in the range of 10 to 30 mph (15 to 45 ft/s), so sensors placed 500 feet apart would be able to detect growing queues 10 to 30 seconds after their formation; at 1,000 feet, it would be 20 to 60 seconds.

208 GUIDE TO ESTABLISHING MONITORING PROGRAMS FOR TRAVEL TIME RELIABILITY A formal technique to estimate minimum sensor spacing and sampling intervals was developed as part of the project and is included in the final report. The conclusions from that work are as follows: • Temporal sampling intervals in the range of 1 to 5 minutes should be adequate for most situations in which both recurring and nonrecurring events occur, although 30 seconds is somewhat better. • Longer sampling intervals can be used when transients are not expected (e.g., off peak) or where separate means exist for detecting incidents. • Spatial sampling intervals in the range of 750 to 1,500 feet are desirable in loca- tions where queuing transients are expected. • Longer spatial sampling intervals can be used when queuing is not expected or a separate means exists for detecting incidents. SUMMARY This appendix has provided additional information about the methods used to de- velop travel time reliability information from data feeds typically available to operat- ing agencies. REFERENCES 1. Hu, J. Estimation of Segment Travel Rates. Master’s thesis. Department of Civil, Construction, and Environmental Engineering, N.C. State University, Raleigh, N.C. 2011. 2. Bowman, L. A., and M. A. Turnquist. Service Frequency, Schedule Reliability and Passenger Wait Times at Transit Stops. Transportation Research Part A, Vol. 15A, No. 6, 1981, pp. 465–471.