Half-(head)way there: Comparing two methods to account for public transport waiting time in accessibility indicators

Abstract

Various methods have been developed to account for travel time variability and uncertainty when analyzing public transport networks and computing related accessibility indicators. In this paper, we establish some convergence characteristics of one such method, implemented in the R5 routing engine, yielding guidelines for the minimum number of randomized schedules. This parameter has implications for result stability, analysis turnaround time, and computation costs. We also confirm that for travel time and accessibility results, there are spatially varying differences between our method and the conventional method relying on the assumption of half-headway waiting times. The conventional method appears to understate the benefits of transit in certain locations, particularly those served by multiple lines. Researchers and planning practitioners may find the R5 method preferable when analyzing complex networks or comparing transit scenarios where routes are specified in terms of headways or frequencies, rather than complete schedules with exact departure times for each trip.

Keywords

cumulative opportunities accessibility public transit headway R5

Introduction

Several recent articles describe new interactive sketch planning tools based on cumulative opportunities accessibility indicators (Conway et al. (2017); Conway and Stewart (2019); Stewart and Byrd (2021)). The open-source network analysis software underlying these tools, R5, has been widely reused (e.g., Pereira et al. (2021)) and favorably compared to other routing software in terms of features and performance (Higgins et al., 2022). A key goal driving the development of R5 has been to characterize the variability and uncertainty in public transit travel times (and therefore accessibility) for riders departing at different times of day, where some or all transit routes are described in terms of headways or frequencies, rather than completely specified timetables. Such headway-based routes are a common way to represent long-term visions for transit network growth and change in scenario- and sketch-planning exercises.

The core methods in R5 treat travel times as random variables rather than single values. The underlying distributions can take on complex forms where multiple rides are chained together, or where multiple alternatives provide service to the same destination, particularly where headway-based sketch planning routes are overlaid onto a baseline network composed of fully timetabled routes. To address this problem, Conway et al. (2017) developed a Monte Carlo approach, sampling a large number of possible system-wide schedules meeting the scenario’s headway constraints, while also examining all departure times in a multi-hour window at one-minute resolution.

This approach allows rapid computation of travel time distributions to millions of destinations from one origin and when carefully optimized can provide region-wide accessibility figures from millions of origins in a matter of minutes. However, as a randomized sampling approach it does not exhaustively consider every possible combination of route schedules, and the results are expected to show variation between runs. The question then arises of how many randomized schedules, or Monte Carlo draws, must be examined to mitigate statistical noise and obtain stable, reliable results for use in a planning or decision process.

Conway et al. (2018) focus on quantifying the uncertainty in results and providing confidence intervals for this method. In everyday practice, however, constraints are placed on computation budget and turn-around time. It is preferable to allocate scarce computation resources to a higher number of draws yielding results with minimal error, rather than exactly characterizing the larger amount of error present with a lower number of draws. To make an informed trade-off between computation requirements and accuracy, we need practical guidelines for setting a sufficient number of draws before an analysis is begun.

In this paper, we also compare two methods to account for waiting time on transit routes that lack completely specified schedules (i.e., routes without exact departure times specified for each trip):

The half-headway (hereinafter HH) method assumes a rider always waits half the headway of a route for a vehicle to arrive on that route. This conventional approach attempts to find a central tendency in total travel time by using the central tendency of wait time at each boarding event. For routes running at constant frequencies, waiting time does not vary at different departure times. This method is intuitive and perfectly appropriate for journeys that involve a single route. But it is decreasingly appropriate for journeys that could be made with multiple alternative routes or that require transfers between routes.

The Monte Carlo (hereinafter MC) method in R5 evaluates many alternative system-wide schedules (draws), each of which coherently respects the headway parameters of every route in a scenario. Each MC draw is a combination of full timetables for the entire network. These timetables are provided by input data for schedule-based routes and generated randomly for headway-based routes. Waiting time and travel time are then derived for each draw and each minute of a specified departure window. When selecting a representative travel time, lower percentiles of the derived travel times represent both the rider choosing a better departure time from within a given window and better timing of the network as a whole with respect to that rider’s origin point and the spatial distribution of destinations.

MC is more complicated than HH, both to implement and interpret. Interpretability is an important criterion for accessibility indicators (Geurs and Van Wee (2004)); trade-offs between ease of interpretation and quality of results should be weighed carefully. In this article, we evaluate stability characteristics of the MC method and demonstrate that it provides better results than the HH method for journeys involving multiple routes, justifying its relative complexity.

This paper is structured around two topics: the convergence of travel time and accessibility values produced by the MC method as the number of randomized schedule draws increases; and second, the spatially varying differences in the indicator values produced by the MC method relative to the more conventional HH method.

Convergence

We expect travel time and cumulative-opportunity accessibility values to converge as the number of MC draws increases. More specifically, we expect that over a given number of trial repetitions with MC draws, the range and root-mean-square deviation (RMSD) of travel time for any specific origin-destination pair, and of accessibility at any specific origin, will be smaller with larger numbers of draws. We investigate whether results stabilize after a certain number of draws. If results indeed stabilize for a network known to present a high degree of uncertainty (e.g., arising from many frequency-based routes), the number of draws at which they stabilize could serve as a guideline for ensuring convergence in other, less uncertain networks. Finally, for a given number of draws, we expect lower variation across trials in locations served by high-frequency routes that do not require transfers to access the destinations of interest.

Monte Carlo versus half-headway method

Compared to the HH method, we show the MC method yields shorter travel times and higher accessibility values for many origins in a real-world network, even when selecting travel times at or above the 50th percentile. But in peripheral areas where journeys involve multiple transfers and lower-frequency routes, the MC method yields longer travel times, suggesting that the HH method may misestimate travel times in a spatially biased way.

We expect the differences between these methods to be smaller for journeys served by a single high-frequency route and larger for journeys served by multiple route alternatives with lower-frequency segments. This result relates to the common line problem (Chriqui and Robillard (1975)), as discussed below.

Empirical case: Santiago de Chile

We test these methods using Santiago de Chile as a case study. Santiago’s integrated transport network includes approximately 11,300 bus stops and 120 rail stations, served by the privately operated bus system formalized in 2007 (see Muñoz et al. (2014) for an overview) and Metro. Bus routes are loosely categorized as trunk routes, which traverse the city and often overlap on key corridors, and feeder routes, which are shorter and generally run with lower frequencies. Two characteristics of Santiago’s land use and public transport make it a suitable for highlighting differences between the MC and HH methods.

First, jobs are highly concentrated, which makes cumulative opportunity measures of access to jobs especially sensitive to the selected travel time limit and other parameters. Garreton (2017) describes Santiago’s evolution and spatial development: residential locations reflect a clear economic gradient; high-income households are clustered northeast of the city’s historic core, and lower-income households tend to live in the southern and western sectors. Government programs in the 1980s resettled thousands of poor households from central areas to sprawling housing developments south of the city. Given the concentration of jobs in the historic core and along an axis northeast of Tobalaba (see Figure 1), today many low-income households endure long transit trips with multiple transfers to access jobs.

Figure 1.

Public transport network and example job locations, Santiago de Chile.

Second, Santiago’s Metropolitan Directorate of Public Transport (DTPM) does not generally specify complete timetables; it instead publishes programmed route frequencies. The private operating companies responsible for dispatching buses have financial incentives based on route-level performance indices for frequency compliance and regularity compliance (Beltrán et al. (2013)). They therefore seek to operate individual route patterns with even headways, but there are no incentives for coordinated timetables between routes, even in corridors with overlapping routes. The lack of complete timetables implies a high degree of uncertainty across the network, which serves to clarify convergence characteristics, and differences between the MC and HH methods, of interest in this research.

In addition to these methodological reasons, transit unreliability is a salient issue in Chile. Protests against the transit system in 2019 spiraled into mass fare evasion campaigns, a state of emergency, and eventually a process for drafting a new national constitution. Underlying causes of this unrest include dissatisfaction with transit unreliability and decades of widening inequality (Rodríguez Mega (2019)). The analysis in our paper assumes all service operates as programmed, with deterministic headways and running times—holding fixed the foremost operational aspects of unreliability that could be captured by methods of estimating day-to-day variability in travel times and accessibility (e.g., Wessel and Farber (2019); Arbex and Cunha (2020)). Even assuming service operates as programmed, however, the uncertainty implied by the lack of complete schedules may play a role in perceived unreliability and users’ dissatisfaction.

Past work

Typical transit-focused accessibility calculations and sketch planning applications rely on building a shortest-path tree from a selected origin to all destinations in a region. These trees are built by optimizing a single variable, such as travel time or generalized cost. Estimating waiting time in these methods is a challenge. Route choice models add nuance. But they are generally applied in assignment problems for a known origin and destination stop, rather than to build a tree from an origin to all destinations. Liu et al. (2010) offer a review of the extensive literature about such models.

Estimating waiting time

A “common approach” to estimating transit waiting times is the half-headway (HH) method (Curtis and Scheurer (2010)), used by Farber and Grandez Marino (2017); Mamun et al. (2013); Painter et al. (2018), among others. This approach considers the average wait time to be sufficiently representative of passenger experience. This may be the case for rides on a single headway-based route, but is decreasingly appropriate as alternative routes and transfers involving scheduled routes are taken into consideration.

When a transfer is required, the travel times reported to reach each destination are likely not all achievable within a single system-wide schedule; depending on the structure of the network, a schedule allowing riders to experience half-headway wait times at one point will often produce much worse wait times at some other points. The fact that a passenger boarded with half-headway wait, or minimal wait at one stop, may (together with the inter-stop travel times) necessarily mean that he or she cannot experience half-headway or better when transferring to another route at a downstream stop.

Finally, the half-headway assumption is problematic in the case of overlapping “common” lines in a corridor (see Chriqui and Robillard (1975)), or more generally in cases where multiple paths are available (as discussed in Conway et al. (2017)). Arriagada et al. (2019) evaluate rider behavior in the presence of common lines in Santiago, finding that passengers “use common lines as a unique alternative in 80% of the route sections” and that only 10% of passengers stick to one route when common lines are available.

To address the common line problem, Arriagada et al. (2019) use hyperpaths. This approach identifies an attractive set of routes and calculates waiting time based on an assumed headway distribution for each route in the set, weighted by probability of boarding it. In the foundational work on hyperpaths, Spiess and Florian (1989) start with the half-headway assumption (p. 86) and then extend to more general cases; they recognize that mechanisms for “constructing the links and the nodes of the generalized network depend very much on the particularities of the transit network and the degree of aggregation considered.” In other words, a general method for enumerating attractive sets in large-scale networks may be computationally difficult. Li et al. (2015) estimate that solving an average all-to-one optimal hyperpath problem for the bus network in Chicago, which is similar in scale to Santiago (on the order of 10,000 stops), takes over 12 min of CPU time; they propose a heuristic alternative to identifying the attractive set, which reduces computation time substantially, but it is not guaranteed to be correct unless headways are assumed to follow an exponential distribution. In short, for rapid-turnaround, interactive sketch planning for large-scale transit networks, the assumptions, and computation time that the hyperpath approach requires may be impractical.

Another strain of research in the transit assignment literature incorporates exact timetables—models for full schedule-based networks (the subject of Wilson and Nuzzolo (2008)) or mixed frequency- and schedule-based networks (Conway et al. (2017)). Such models can allow “adaptive path choice” Cats et al. (2011). For example, Hickman and Wilson (1995) consider a “‘clever’ passenger” who “waits at the origin stop until a vehicle arrives and then makes a decision whether to board,” using a case study with three schedule-based bus routes (albeit with stochastic headways and running times) and a frequency-based rail line.

Monte Carlo approach

Conway et al. (2018) also consider a “clever” passenger who must contend with variability due to departure time regardless of whether service is schedule-based or frequency-based, as well as uncertainty from under-specified service when routes are frequency-based. Such routes have a specified frequency, but the offset of departure times from the top of the hour (i.e., phase) is treated as a random variable.

The MC approach handles these complexities, using a single parameter (the percentile of the overall travel time) to handle the variability due to departure time and the uncertainty due to phasing. Simulating possible system schedules, and sampling travel times and paths throughout a departure time window, obviates the need to enumerate attractive sets for hyperpaths a priori. Other advantages of the MC approach include robust handling of services that only run during part of the departure time window, robustness against outliers, and ability to compare well-planned and ill-planned journeys and timetables. For a detailed example in a simple corridor, see the Supplemental Materials.

Measuring error due to sampling

Like our work here, Owen and Murphy (2019) also deal with the error characteristics of sampling methods in cumulative opportunities accessibility analysis over time windows. However, they focus on sampling rider departure times, while our work here focuses on sampling the space of all possible operational choices (route phases implying vehicle departure times) that satisfy a scenario with headway-based routes. We consider the problem of rider departure time sampling to be solved, as optimizations in R5 allow exhaustive examination. At one-minute resolution, there are only a few hundred distinct times in a morning peak period. In contrast, the space of all possible vehicle departure times grows exponentially with the number of routes and can easily reach the order of 10⁵⁰, necessitating a sparser sampling approach. Owen and Murphy (2019) use normalized root mean square error (NRMSE) to measure the quality of sampling strategies, applying a local indicator of spatial association (LISA) to identify discrepancies in proximity to transit infrastructure. Our measurement approach differs somewhat in its aggregation of multiple origin points and consideration of spatial patterns, but is generally comparable.

Methodology and data

Our overall approach follows Conway et al. (2017): we build a routable baseline network from street data (from OpenStreetMap) and public transport schedules (from GTFS feeds), optionally modify or augment that network with scenario layers, calculate travel times to all destinations, select specific percentiles of those travel time distributions (often represented as isochrone curves), and derive location-based cumulative opportunities accessibility indicators at all origin points throughout the study region.

The walking, biking, or driving portion of the routing uses actual streets, and the public transit portion of the routing uses the actual published timetables when available. If published timetables are not available, as may be the case for routes with headway-based dispatching or for future scenarios at a sketch planning stage, we generate a large number of schedules satisfying all headway constraints in the MC approach, holding departure frequencies constant while randomly setting the phase of each route (i.e., the offset of its departure times from the “top of the hour”).

This process is performed by version 51f0702 of the R5 multi-modal routing engine¹, which requires certain parameters to be specified. For example, our analyses specify walking access/egress to/from transit stops, with a 20-min limit per access/egress leg. We use a departure time window of 1 hour, calculating travel times for trips starting every minute within the window. We consider a range of travel time percentiles, including the 95th percentile—at this level, a commuter would complete a weekday morning commute late once every 4 weeks (see Furth and Muller (2006)).

Data and example locations for empirical case

Source data for the full case study include an OpenStreetMap extract ∗ and the 5 July 2019 version of GTFS published by Santiago’s DTPM †. In this GTFS feed, all routes are frequency-based and lack explicit timetables except Metro Line 3 and the Metrotren, which for the purpose of this study are also modeled using headways rather than full timetables.

Population density data are from Chile’s 2017 Census ‡. The example jobs are derived from locations coded as the destinations of trips to work in Greater Santiago’s 2012 origin-destination survey §. There are 966 example jobs, a relatively small sample that should heighten the sensitivity of cumulative opportunity job accessibility indicators to travel time fluctuations.

We selected six specific origins across the region for detailed results (Figure 1). The diverse network and land-use characteristics of these origins are described in the Supplementary Materials.

Results and discussion: travel time

This section discusses travel times from the six example origins to each cell in a 300 m by 300m raster grid (30,687 destinations per origin). We conducted 20 trials for each origin, five percentile values, and five numbers of MC draws (60, 120, 480, 960, and 1200). These combinations yield approximately 92 million travel time values, or 5.5 billion when considering each departure minute separately. The statistics for convergence only include destination cells that were reachable in less than 2 hours in all trials. For comparisons with the HH method, we used the mean travel-time values across the 20 trials.

Convergence

To illustrate travel time variability, isochrones were mapped from specific origins for fifth, 50th, and 95th percentile travel times. Large gaps between the 5th and 95th percentile isochrones indicate areas of high variability (i.e., where travel time can vary widely depending on specific departure time and vehicle operating schedules). As expected, the 5th and 95th percentile isochrones are farther apart for destinations connected to the origin by multiple infrequent alternatives requiring transfers. These maps also superimpose results from the 20 trials to illustrate the statistical noise arising from the MC method. As expected, the isochrones are much less “fuzzy” with 1200 trials than 60 trials, indicating a reduction in statistical noise. Maps and a detailed discussion of these results are available in the Supplementary Materials.

Figure 2 summarizes the maximum per-cell difference in travel time values for these six origins across 20 separate trials, each with the same number of draws. By 960 MC draws, the travel time from any of these six origins to any destination does not differ by more than 3 min, except in a few cases at the 5th and 95th percentiles. Assuming deviations are symmetric, travel times at the worst-case destinations in these trials were within ±1.5 min of the true limit value. In a few cases, the range does increase slightly when the number of MC draws is stepped up. This is not totally unexpected: even if the range is monotonically decreasing with a very large number of trials, any small finite number of trials is still varying around that limit, and that variance may be larger than the amount of convergence at the next higher number of MC draws.

Figure 2.

Maximum range of travel time.

Figure 3 summarizes the mean across reachable raster cells of root-mean-square deviation (RMSD) of travel time in the 20 trials. For all origins except one, RMSD is on average less than 30 s by 480 MC draws. The single low-frequency route (24-min headways) serving Lo Barnechea makes it an outlier among the six example origins. RMSD for Lo Barnechea is generally higher than for other origins, and it peaks at 50th percentile travel time, while travel times from other origins tend to have higher RMSD at extreme percentiles.

Figure 3.

RMSD of travel time, mean across all reachable destination raster cells.

Monte Carlo versus half-headway method

The previous results suggest reasonable convergence in travel time values by 1200 MC draws. In this sub-section, mean travel times for the 20 trials with 1200 MC draws are compared with travel times based on the HH method.

Figure 4 compares MC and HH travel times. The first row of this figure shows travel from an origin in the center of the region, Plaza de Armas. From this origin to most destinations, MC 50th percentile travel times are 1–5 min shorter than HH travel times. The differences between the two methods are less than a minute for destinations near stations along Metro Line 5; fast, frequent service from the Metro station adjacent to the origin means there are few other competitive routes from this origin, mitigating the common line problem.

Figure 4.

Difference in travel time, MC versus HH method, from six example origins at five travel time percentiles.

Results are mixed at 95th percentile travel times. At high travel time percentiles, the MC method reflects long waiting times, while waiting times for frequency-based routes in the HH method remain at half the headway. This difference explains why the upper right map of Figure 4 shows MC times approaching 10 min longer than HH times for travel to outlying areas served by single, low-frequency routes. On the other hand, some destinations for which there are many route alternatives (such as the Santa Rosa corridor running due south of downtown between Metro Lines 2 and 5), triggering the common line problem, have 95th percentile MC travel times that are shorter than HH travel times.

Similar patterns are present for most of the remaining five origins. Again, the origin served by a single route with a 24-min headway (Lo Barnechea) is an outlier. HH travel times reflect a 12-min wait for this route; fifth percentile MC times reflect a very short wait (approaching 12 min shorter than HH), and 95th percentile travel times reflect a wait close to the full headway (approaching 12 min longer than HH).

Summarizing at a high level, most destinations shown in Figure 4 switch from blue (MC faster) to orange (HH faster) between the 75th and 95th percentiles, though there is substantial variability. This result is generally consistent with the simple corridor results discussed in the Supplementary Materials, where HH waiting times fell between the 75th and 95th percentile travel times.

Results and discussion: accessibility

Accessibility indicators are affected by the travel time considerations discussed in the previous section, as well as the interaction between the spatial distribution of opportunities and the chosen travel time limit or decay function. We expect accessibility values to converge, resulting in smaller ranges across multiple trials, as the number of MC draws increases.

As an initial indicator, we consider the number of example jobs reachable within 60 min at 50th percentile travel times. The spatial distribution of these 966 example jobs is shown in Figure 1. Maps in this section depict a number of jobs reachable from each origin in the region, using the 300m by 300m raster grid described above as origins.

Convergence

Figure 5 shows the maximum per-cell range in this accessibility indicator across 10 trials, using 60 MC draws (left) and 1200 MC draws (right). With 60 MC draws, the accessibility range for most cells is between 5 and 50 example jobs (0.5 and 5.0% of the total), and some cells have a range of approximately 200 example jobs (20% of the total). Assuming symmetrical distributions, this suggests that results are with ±10% of a stable value. With 1200 MC draws, the range for most cells is less than 5 example jobs (0.5% of the total).

Figure 5.

Range of accessibility values over 10 trials, for 60 MC draws (left) and 1200 MC draws (right).

In both cases, the origins with the largest range are close to the boundary of a 60-min commute from the cluster of jobs downtown; the darkest cells in Figure 5 are generally coincident with the 60-min contours from Plaza de Armas in Supplemental Figure S1.

Monte Carlo versus half-headway method

The travel time results in Figure 4 suggest that for many trips in Santiago, MC median travel times are shorter than HH travel times. Given this result, the result in Figure 6 follows naturally: accessibility given median travel time is generally higher using the MC method. Again, the difference between the two methods is lower around Metro lines (corridors served by a single attractive route, mitigating the common line problem) and higher at origins from which multiple lower-frequency route alternatives provide access to job centers.

Figure 6.

Example jobs reachable within 60 min given median travel times (mean of 10 trials, left) and difference from HH values (right).

Conclusion

Travel times converge on stable values within a computationally tractable number of draws. It follows that the accessibility indicator values derived from these travel times also converge, though accessibility converges more slowly at origin locations situated precisely at the accessibility indicator’s travel time threshold relative to large clusters of opportunities. In this respect, the fringe of more slowly converging areas in Figure 5 shares conceptual similarities with the “ring of unreliability” in Cui and Levinson (2016). We do not account directly for unreliability arising from poor schedule or headway adherence in actual operations, but the MC method can give some insights into variability that passengers may experience. Overall, the convergence characteristics described in this article confirm that the MC method is a practical approach for obtaining stable measurements of scenario impact without incurring excessive computing costs.

Travel times resulting from our MC method differ noticeably from those produced by the classic HH method, in a spatially varying fashion that a simple calibration factor would not capture. Accessibility indicators derived from the MC travel times also show perceptible differences relative to indicators derived from HH travel times, especially in areas served by multiple low-frequency lines. This confirms the relevance of the more computationally demanding MC approach in estimating the impact of changes to complex networks where some or all lines are modeled in terms of frequencies rather than complete schedules.

Implications for practice

This research validates the original motivation for developing the MC approach: the classic HH approach understates the benefits of transit for certain places and overstates the benefits for others. Planners should question the validity of the half-headway assumption when making nuanced decisions between alternative scenarios specified in terms of headways, particularly where multiple lines or combinations of lines provide connections to areas of high opportunity density. In such situations, the HH method tends to overestimate waiting times, which artificially reduces the estimated accessibility benefits of transit. As common trunks and grid-like networks are widespread in both existing transit networks and prospective network redesigns, planners may find the MC approach tested in this research more suitable. Though more computationally intensive, a well-optimized implementation of MC still allows for near-instantaneous computation of travel times from one origin to all destinations and can compute accessibility indicator values for every location in a metropolitan region in a matter of minutes.

This work provides initial empirically derived guidelines for setting the number of draws or “simulated schedules,” a user-specified parameter with implications for analysis turnaround time and computation costs. Satisfactory convergence of both median travel times and the derived accessibility indicators was achieved with 1200 draws; accessibility values converged to within ±2.5% of the total number of example jobs, for almost all origins (see Figure 5). This result was obtained using a sparse, concentrated example set of jobs and a network with a high degree of uncertainty due to frequency-based specification of routes—both factors should hinder convergence. We therefore expect 1200 draws to be sufficient in other networks with generally similar frequencies, but less uncertainty (i.e., more routes represented with scheduled service).

This paper further characterizes the benefits of the approach described in Conway et al. (2017) for assessing accessibility in mixed schedule- and frequency-based networks. Santiago’s DTPM has recently implemented a number of exact timetables for relatively low-frequency routes, including 16 that operate during the day and 21 that operate at night. The approach we describe could be useful in network planning as more of these routes are integrated with the frequency-based service in the rest of the network, or in cases where future scenarios are being developed.

Future research

Our results and conclusions suggest several avenues for further exploration.

Although we have determined a sufficient number of MC draws to achieve convergence on this particular transit network, in order to establish more general convergence guidelines more comparisons are needed between networks with different characteristics. We plan to perform a similar convergence analysis on networks with different ratios of frequency and fully scheduled lines, with higher and lower frequencies and differing network structures.

In place of the simple random sampling strategy used here, we intend to experiment with low-discrepancy sequences (quasi-Monte Carlo methods), which are expected to provide better error characteristics and more rapid convergence.

As mentioned in the conclusions above, convergence of accessibility indicators is slower where large clusters of opportunities are situated within the threshold isochrone’s band of uncertainty. This problem is tied to the use of a hard-edged travel time threshold for inclusion of opportunities. Small shifts in travel time of even a few seconds can cause large clusters of opportunities (like large office buildings) to fall in and out of the isochrone from one trial to the next. This effect could be alleviated by applying a softer roll-off (e.g., gravity decay or sigmoid), effectively diffusing this localized error in the accessibility value across adjacent cells, over a distance controlled by the steepness of the threshold curve. We are currently integrating these changes and plan to compare RMS error figures for hard versus soft thresholds.

As Liu et al. (2010) paraphrase Hall (1982), “a transportation engineer can be very successful at reducing the travel time for ideal travellers, yet fail at improving the actual travel time seen by real travellers.” Our work could be extended by integrating recent interdisciplinary research into characterizations of rider behavior and boarding strategies (e.g., Viggiano et al. (2014); Nassir et al. (2017); Ingvardson et al. (2018)).

Finally, the travel time distributions from which we select our chosen percentiles contain variation due to two separate factors: the departure time (which we vary uniformly across a multi-hour time window) and the uncertainty in vehicle arrival times. The latter is randomized in the MC approach but not the HH approach, while the former source of variation is present (and identical) in both the MC and HH approaches. It would be interesting to repeat these analyses with the departure time held constant, to completely isolate the effect of transit schedule randomization. However, arguably the true impact of different possible schedules (with their different implications for wait and transfer times) is not felt unless many different rider departure times are also considered.

Supplemental Material

Supplemental Material - Half-(head)way there: Comparing two methods to account for public transport waiting time in accessibility indicators

Supplemental Material for Half-(head)way there: Comparing two methods to account for public transport waiting time in accessibility indicators by Anson F Stewart and Andrew M Byrd in Environment and Planning B: Urban Analytics and City Science

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Anson F Stewart

Supplemental material

Supplement material for this article is available in online.

Notes

Anson F. Stewart, Project Lead for Analysis and Research at Conveyal LLC, works closely with transport professionals implementing innovative methods that center access to opportunities. He also collaborates with various public transport authorities as a Research Scientist at MIT, where he completed PhD and masters degrees and now serves as Deputy Director of the MIT JTL Urban Mobility/Transit Lab.

Andrew M. Byrd is a co-founder of Conveyal LLC which creates tools for joint land use and transportation planning, and is a primary contributor to the architecture, algorithms, and governance of related open-source software components. He holds a master’s degree in urban transportation planning from the Planning Institute of Paris and the Ecole des Ponts ParisTech.

Notes

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