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48 Calculations for the Example 1 headway data are shown in Table 6. Passenger waiting time distribution for Table 5. These calculations use a three-bin format, with thresh- Example 2. olds of 9 and 11 min, which are 1 min and 3 min beyond the scheduled headway, respectively; the waiting times represented Waiting Time by these bins might be interpreted as "normal,""excessive," and Case Normal Excessive Unaccept. Total 0 to 9 min 9+ to 11 min > 11 min "unacceptable." The last observed headway in the table, which is 13 min long, best illustrates the calculation. Of the passen- a 84.1% 8.2% 7.7% 100% gers arriving during the 13 min, those arriving in the first b 94.8% 4.4% 0.8% 100% 2 min of the headway wait 11 min or longer; those arriving in the next 2 min wait 9 to 11 min; and those arriving in the last 9 min wait between 0 and 9 min. Overall, this table shows that Therefore, distribution of schedule deviation is a measure of 4.2% of passengers wait longer than 11 min. operational performance that relates closely with passenger This format can also offer insights when comparing wait- experience. For interpreting schedule adherence as passenger ing time distributions. Table 6 compares the percentage of experience, it is helpful to distribute schedule deviations into passengers with excessive and unacceptable waiting times for more than just the traditional three bins of "early," "on time," Example 2, Cases a and b. As Table 6 shows, the improved reg- and "late." For example, the user might specify thresholds at ularity of Case b has dramatically reduced the percentage of -1, 0, 3, 5, and 10 min and determine the percentage of trips, and therefore the percentage of passengers experiencing passengers with excessive and unacceptable waiting times. trips, in bins interpreted as unacceptably early, less than a minute early, on time (0 to 3 min), a little late, quite late, and 6.3 Long-Headway Waiting unacceptably late. Time Analysis As valuable a measure as schedule adherence is, it still does For routes with long headways, most passengers time their not express an impact on passengers. The researchers were arrival at the platform to make a targeted departure. Sched- able to extend the waiting time framework to long-headway ule adherence or schedule deviation is critical to determining service in order to determine how much excess waiting time passengers' waiting time. Let V equal the schedule deviation is caused by service unreliability. of the trip a passenger is trying to meet, defined by 6.3.1 Waiting Time Components V = departure time - scheduled departure time Passengers are assumed to have a target arrival time and to Early departures are represented by negative values of V. arrive at or before this time. The target is set to give passengers As long as schedule deviations are small compared to the a very low probability of missing the bus. Therefore, the target scheduled headway, the number of passengers using a given is an extreme value at the lower end of the schedule deviation trip will be independent of its schedule deviation or headway. distribution; the researchers use the 2nd-percentile schedule Therefore, unlike with short-headway service, the experience deviation (V0.02), meaning the time by which only 2% of buses of passengers is the same as the "experience" of buses (i.e., if will have already departed. Passengers following this policy 15% of the buses were late, it is fair to say that 15% of the pas- will miss the bus once every 50 trips, or less (depending on sengers had to wait for buses that were running late). how long before the target they arrive). Two components of waiting time are not avoidable and are, therefore, part of ideal waiting time, not excess waiting time. Table 5. Excessive waiting time calculation. Because they depend only on the planned headway, they are not a subject of AVL data analysis; nevertheless, for com- Waiting Time (Bin Width) pleteness, they are mentioned here: Normal Excessive Unaccept. Headway Total 0 to 9 min 9+ to 11 min > 11 min Schedule inconvenience: a well-known form of hidden (9 min) (2 min) (1000 min) waiting time that arises from departures or arrivals not 4.0 4.0 4.0 being scheduled when passengers want to travel. It is man- 5.0 5.0 5.0 ifested in passengers who are going to work arriving before 7.0 7.0 7.0 their work start time because the next bus would get them there too late. Likewise, after work, passengers may consis- 9.0 9.0 9.0 tently have time to kill between leaving work and the next 10.0 9.0 1.0 10.0 scheduled departure. 13.0 9.0 2.0 2.0 13.0 Synchronization time: the cost, in equivalent minutes of Total 43.0 3.0 2.0 48.0 waiting, of ensuring that one is at the station by the target Percentage 89.5% 6.3% 4.2% 100% arrival time. To be sure of hitting that target, many passen-

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49 gers will arrive early because of uncertainty in access time, ning potential waiting with platform waiting, summary mea- the limits of human punctuality, and risk aversion. The sures of excess waiting time can be calculated as follows: waiting time between when passengers arrive and the ideal mean excess platform waiting = E[V] - V0.02 arrival time is part of synchronization time. excess budgeted waiting = V0.95 - V0.02 mean potential waiting = V0.95 - E[V] Schedule inconvenience and synchronization time are equivalent excess waiting = (excess platform waiting) unavoidable; they can be "blamed" on planning, as they are + 0.5 (potential waiting) the inevitable consequence of designing a service with a long = 0.5 (excess platform waiting headway. In contrast, random waiting time that occurs after + excess budgeted waiting) the ideal arrival time is due to service unreliability and is, therefore, excess waiting time. Its components, and their 6.3.2 Example Analyses relation to schedule deviations, are shown in Figure 15. Excess platform waiting runs from the passengers' target Example schedule deviation and waiting time summary arrival time to the bus's actual departure time. Excess bud- reports for long-headway service are shown in Figure 16. geted waiting time runs from the target arrival time to the These figures compare two example cases, each based on 100 95th-percentile schedule deviation (V0.95). As in the case of synthesized observations: short-headway waiting, passengers cannot plan on being Case No-OC is a route with relatively poor reliability. picked up at the average departure time; they have to budget Schedule deviation has a standard deviation of 2.2 min, so for an extreme value. The researchers assume they budget that only 72% of departures are in an on-time window of for the 95th-percentile schedule deviation; with that value, 0 to 5 min late. passengers will arrive late 5% of the time. Excess budgeted waiting time, being the spread in the sched- ule deviation distribution between V0.02 and V0.95, can be thought of as the time between the earliest likely and latest likely a. Distribution of Schedule Deviations departure times. The greater the service unreliability, the greater 100% the spread, and the more time passengers have to budget for 90% waiting. Of course, not all of budgeted waiting time is actually 80% spent waiting on any given day; part is spent waiting, and the 70% >10 min late remainder is potential waiting time, just as with short-headway 5+ to 10 min late service. Like schedule inconvenience, potential waiting time is 60% 3+ to 5 min late manifested in passengers arriving earlier than planned at their 50% On time destination. The difference is that potential waiting time varies 40% One minute early from day to day and cannot be counted on in planning daily 30% > 1 min early activities, while schedule adherence is not random, and can be 20% counted on, making it less of a burden to passengers. Using the notation that E[V] equals mean schedule devia- 10% tion, and using the weight of 0.5 suggested earlier for combi- 0% case No-OC case OC b. Waiting Time Summary excess budgeted 9.00 waiting time 8.00 7.00 excess excess budgeted 6.00 synchronization platform potential wait minutes time waiting waiting 5.00 equivalent excess 4.00 wait time 3.00 excess platform 2.00 wait 1.00 pax V0.02 = earliest V = bus V0.95 = budgeted arrival likely bus dep. departure depature time 0.00 time = target time case No- case OC arrival time OC Figure 15. Long-headway waiting time Figure 16. Long-headway schedule components. deviation and waiting time analysis.

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50 Table 7. Waiting time component comparison. Case No-OC Case OC Change Component (min) (min) (min) 2nd-percentile schedule deviation -1.0 0.1 1.1 Mean schedule deviation 3.48 1.88 -1.60 95th-percentile schedule deviation 6.8 4.6 -2.2 Mean excess platform wait 4.5 1.8 -2.7 Equivalent excess wait 6.2 3.2 -3.0 Excess budgeted wait 7.9 4.6 -3.3 Potential wait 3.4 2.8 -0.6 Case OC is the same route, and has the same underlying The mean waiting time calculations do not account for the variability, as Case No-OC, but operational control is applied (small) percentage of passengers who miss their bus (perhaps by holding, and the scheduled departure time has been because the bus was early) and have to wait a full headway for shifted earlier by 2.2 min. With that schedule adjustment, the next one. The reason this percentage is not taken into the earliest 30% of departures will be held; as a result, they account is two-fold. First, the expected penalty for missing the are assumed to have random schedule deviations between bus is 0.02 h, where h is the scheduled headway, and 0.02 is the 0 and 1.5 min. probability of missing the bus. Because this quantity depends on the planned headway rather than on a schedule deviation, Figure 16(a) shows a strong improvement in operational quality reflected in the distribution of schedule deviations. it does not contribute to excess waiting time because it will be The 5% of trips that were early disappear, and the percentage the same regardless of service reliability. (That would not be of trips in the 0- to 5-min window rises from 70% to 97%. the case if the target arrival time were not percentile-based, The percentage of trips in the highest quality category (0 to e.g., if it were set at 1 min before the scheduled departure 3 min late) rises from 39% to 79%. time, for example.) The waiting time summary found in Figure 16(b) expresses Second, while the immediate impact of an early bus is to these results in terms of impact to passengers. Mean platform make a few passengers wait a long time, in the long term the waiting time falls by 2.7 min because passengers do not have impact of early buses is to make all passengers arrive earlier at to arrive before the scheduled departure time and the average the bus stop every day. For example, a passenger who misses schedule deviation has been reduced considerably. Counting a bus on a route with a 20-min headway suffers a 20-min changes in potential waiting time, there is net reduction in waiting time penalty that day. However, for maybe the next equivalent waiting time of 3 min. 100 days, the passenger will arrive at the stop 2 min earlier to Table 7 provides additional detail on how operational con- be sure to not miss the bus again, which costs him 200 min- trol shrinks the spread between early and late schedule devi- utes of additional waiting. As this simple example shows, the ations. The 2nd-percentile schedule deviation rises by 1.1 min impact of earliness on waiting time is accounted for by setting and the 95th-percentile schedule deviation falls by 2.2 min, the target arrival time to the 2nd-percentile schedule devia- shrinking excess budgeted waiting time by 3.3 min. tion, which is sensitive to earliness. Because potential waiting time has most to do with the To reiterate, determining extreme values from data requires upper end of the schedule deviation distribution, actions large sample sizes. A rule of thumb is that there should be at that reduce the upper end (gross lateness) tend to most least 5 observations outside the extreme value estimated. affect potential waiting time, while actions that reduce the Therefore, about 250 observations are needed to make a reli- lower end (earliness) tend to reduce platform time, as in the able estimate of the 2nd-percentile schedule deviation. AVL example. Because platform waiting time affects passengers will provide that kind of sample size, but it may be necessary more strongly than potential, actions that reduce earliness to aggregate over a considerable date range and/or over a are therefore particularly effective. period of the day with several scheduled trips.