WEBVTT 00:00:01.000 --> 00:00:03.340 The following content is provided under a Creative 00:00:03.340 --> 00:00:04.760 Commons license. 00:00:04.760 --> 00:00:06.970 Your support will help MIT OpenCourseWare 00:00:06.970 --> 00:00:11.060 continue to offer high-quality educational resources for free. 00:00:11.060 --> 00:00:13.600 To make a donation or to view additional materials 00:00:13.600 --> 00:00:17.560 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.560 --> 00:00:18.922 at ocw.mit.edu. 00:00:21.954 --> 00:00:24.890 GABRIEL SANCHEZ-MARTINEZ: Let's get started with Lecture 9. 00:00:24.890 --> 00:00:29.170 Today's lecture is very relevant to the homework. 00:00:29.170 --> 00:00:32.590 We'll run through some examples of performance models 00:00:32.590 --> 00:00:35.610 like the ones you have to work on or specify 00:00:35.610 --> 00:00:37.840 an estimate on your homework. 00:00:37.840 --> 00:00:40.730 So we talk about performance, performance models. 00:00:40.730 --> 00:00:41.680 What is performance? 00:00:41.680 --> 00:00:43.510 What do we mean by performance? 00:00:43.510 --> 00:00:49.420 Typically, this is a key word used to describe output 00:00:49.420 --> 00:00:53.860 and some sense of how well the output service was, so 00:00:53.860 --> 00:00:57.620 both in terms of the operator and in terms of the passenger. 00:00:57.620 --> 00:01:02.320 So things like running times, waiting times, headways, things 00:01:02.320 --> 00:01:04.810 that you can measure from observations 00:01:04.810 --> 00:01:08.530 of service being delivered, we call that performance, 00:01:08.530 --> 00:01:09.690 generally. 00:01:09.690 --> 00:01:12.280 And the kinds of models that we're going to look at 00:01:12.280 --> 00:01:15.190 are wait time models, service variation 00:01:15.190 --> 00:01:17.960 along routes where you'll see how it's not always the same. 00:01:17.960 --> 00:01:20.830 You might start off at the terminal one way 00:01:20.830 --> 00:01:23.277 and end up a very different way. 00:01:23.277 --> 00:01:25.610 We'll look at running time models and dwell time models. 00:01:25.610 --> 00:01:29.510 So these are the components of running times. 00:01:29.510 --> 00:01:32.110 Let's start with waiting time. 00:01:32.110 --> 00:01:35.660 Well, before that, actually, let's think about-- 00:01:35.660 --> 00:01:37.840 so here are some kinds of models. 00:01:37.840 --> 00:01:41.280 Why would we be interested in modeling these things? 00:01:45.040 --> 00:01:50.526 What applications can you think of for these kinds of models? 00:01:50.526 --> 00:01:52.510 That's two ways of putting the same question. 00:01:55.690 --> 00:01:58.000 Why would we be interested in having a waiting time 00:01:58.000 --> 00:02:01.830 model or a running time model? 00:02:01.830 --> 00:02:02.955 Just give some examples. 00:02:06.190 --> 00:02:07.290 Yeah, over in the back? 00:02:07.290 --> 00:02:11.135 AUDIENCE: We want to understand how people are waiting exactly. 00:02:11.135 --> 00:02:12.510 GABRIEL SANCHEZ-MARTINEZ: But you 00:02:12.510 --> 00:02:13.884 can go out and measure it, right? 00:02:17.910 --> 00:02:21.990 You can go out and do a survey and observe 00:02:21.990 --> 00:02:23.310 how people are waiting. 00:02:23.310 --> 00:02:27.600 So why would a model be good? 00:02:27.600 --> 00:02:31.670 What would a model do that observations don't do? 00:02:31.670 --> 00:02:32.870 AUDIENCE: For a new service. 00:02:32.870 --> 00:02:33.690 GABRIEL SANCHEZ-MARTINEZ: For a new service, 00:02:33.690 --> 00:02:35.310 OK, we're starting to get some ideas, yeah. 00:02:35.310 --> 00:02:36.610 AUDIENCE: Or changes in service. 00:02:36.610 --> 00:02:37.500 GABRIEL SANCHEZ-MARTINEZ: Changes in service-- 00:02:37.500 --> 00:02:39.830 OK, so can you give me an example of that? 00:02:39.830 --> 00:02:42.900 AUDIENCE: Like [INAUDIBLE] making the Victoria line 20% 00:02:42.900 --> 00:02:43.874 more frequent. 00:02:43.874 --> 00:02:46.290 GABRIEL SANCHEZ-MARTINEZ: OK, so if you increase frequency 00:02:46.290 --> 00:02:51.679 on the Victoria line, what does that do to demand and to dwell 00:02:51.679 --> 00:02:52.220 time and to-- 00:02:52.220 --> 00:02:54.845 AUDIENCE: The Oxford Circuit is now closed for 30 minutes a day 00:02:54.845 --> 00:02:56.520 because can only interchange, not enter. 00:02:56.520 --> 00:02:58.020 GABRIEL SANCHEZ-MARTINEZ: OK, so you 00:02:58.020 --> 00:03:01.790 can try to use them to predict service changes 00:03:01.790 --> 00:03:05.460 or changes in performance due to changes in service. 00:03:05.460 --> 00:03:07.300 What else, any other ideas? 00:03:10.478 --> 00:03:11.926 [INAUDIBLE]? 00:03:11.926 --> 00:03:14.490 AUDIENCE: [INAUDIBLE] service variation, 00:03:14.490 --> 00:03:18.770 that's [INAUDIBLE] to crowded and to crowded on the bus. 00:03:18.770 --> 00:03:22.940 It's depending on how the planned headway 00:03:22.940 --> 00:03:28.100 sort of variance over time will impact how many people end up 00:03:28.100 --> 00:03:29.300 boarding on a bus. 00:03:29.300 --> 00:03:32.420 Let's say, a bus that came substantially 00:03:32.420 --> 00:03:35.730 after the previous bus might be crowded. 00:03:35.730 --> 00:03:38.522 And I won't be able board. 00:03:38.522 --> 00:03:40.800 And you might even have-- 00:03:40.800 --> 00:03:42.565 so denied boardings for-- 00:03:42.565 --> 00:03:44.106 GABRIEL SANCHEZ-MARTINEZ: [INAUDIBLE] 00:03:44.106 --> 00:03:45.170 will you summarize that? 00:03:45.170 --> 00:03:48.480 Maybe I'll try summarizing. 00:03:48.480 --> 00:03:49.970 You could use one of these models 00:03:49.970 --> 00:03:54.150 to fill in things about performance 00:03:54.150 --> 00:03:56.310 that you can't measure directly from data. 00:03:56.310 --> 00:04:00.900 So you gave an example of measuring headways and then 00:04:00.900 --> 00:04:03.870 estimating crowding on the bus based on headways. 00:04:03.870 --> 00:04:05.065 Was that more or less-- 00:04:05.065 --> 00:04:05.690 AUDIENCE: Yeah. 00:04:05.690 --> 00:04:06.260 GABRIEL SANCHEZ-MARTINEZ: --getting at-- 00:04:06.260 --> 00:04:06.420 AUDIENCE: [INAUDIBLE] 00:04:06.420 --> 00:04:08.878 GABRIEL SANCHEZ-MARTINEZ: OK, trying to generalize a little 00:04:08.878 --> 00:04:10.080 bit-- 00:04:10.080 --> 00:04:11.295 any other ideas? 00:04:11.295 --> 00:04:12.690 [INAUDIBLE]? 00:04:12.690 --> 00:04:16.980 AUDIENCE: I mean, you could use this as a [INAUDIBLE] model 00:04:16.980 --> 00:04:18.660 in any other simulation. 00:04:18.660 --> 00:04:20.493 GABRIEL SANCHEZ-MARTINEZ: Simulation models, 00:04:20.493 --> 00:04:22.830 so if you want to do a simulation model of a system 00:04:22.830 --> 00:04:27.270 that has transit in it, you need all these things to-- 00:04:27.270 --> 00:04:28.050 right? 00:04:28.050 --> 00:04:31.260 Because you need to have your [INAUDIBLE] which, 00:04:31.260 --> 00:04:34.620 it could be a bus, or a passenger waiting, or dwelling, 00:04:34.620 --> 00:04:35.860 or moving between stops. 00:04:35.860 --> 00:04:38.190 So you need these things. 00:04:38.190 --> 00:04:40.530 Any other applications, practical, very 00:04:40.530 --> 00:04:42.153 practical applications? 00:04:45.394 --> 00:04:50.840 So for dwell time and running time models, 00:04:50.840 --> 00:04:53.090 we all have our prediction of when 00:04:53.090 --> 00:04:55.490 the bus is coming to the stop. 00:04:55.490 --> 00:04:58.370 So you need some model to take the bus from where it is right 00:04:58.370 --> 00:05:01.160 now, which you can measure, to how long 00:05:01.160 --> 00:05:02.920 will it take me to reach this stop. 00:05:02.920 --> 00:05:06.680 And so the passenger can look up on the phone, how long will 00:05:06.680 --> 00:05:07.640 I have to wait. 00:05:07.640 --> 00:05:10.250 So these are all examples. 00:05:10.250 --> 00:05:11.240 And there are more. 00:05:11.240 --> 00:05:12.890 Let's start with waiting time. 00:05:12.890 --> 00:05:18.620 So we've already seen this first simple model of waiting time. 00:05:18.620 --> 00:05:23.370 We say that the expected waiting time is half the headway. 00:05:23.370 --> 00:05:26.630 And here, we are considering both waiting time and headway 00:05:26.630 --> 00:05:29.610 to be stochastic quantities. 00:05:29.610 --> 00:05:35.060 So if we observe many headways over a long period of time, 00:05:35.060 --> 00:05:37.550 we see a probability distribution of headways. 00:05:37.550 --> 00:05:40.130 And we're saying, the average headway divided by 2 00:05:40.130 --> 00:05:42.320 should equal the average waiting time. 00:05:42.320 --> 00:05:44.420 Now, this is a very simple model. 00:05:44.420 --> 00:05:47.000 And we know there are some problems with it. 00:05:47.000 --> 00:05:52.460 So we are assuming that passengers arrive independent 00:05:52.460 --> 00:05:54.020 of vehicle departure times. 00:05:54.020 --> 00:05:56.840 We are assuming that vehicles are departing 00:05:56.840 --> 00:05:58.830 at equal intervals deterministically, 00:05:58.830 --> 00:06:01.880 so they're sort of evenly spaced, 00:06:01.880 --> 00:06:03.456 and that every passenger can board 00:06:03.456 --> 00:06:04.580 the first vehicle they see. 00:06:04.580 --> 00:06:09.050 So nobody is left behind by a bus that is too full. 00:06:09.050 --> 00:06:12.450 So obviously, these things don't always hold. 00:06:12.450 --> 00:06:16.010 And particularly, the part of vehicles departing exactly 00:06:16.010 --> 00:06:19.110 at equal intervals doesn't hold. 00:06:19.110 --> 00:06:20.040 It rarely holds. 00:06:23.470 --> 00:06:25.250 AUDIENCE: Even if they depart, there 00:06:25.250 --> 00:06:28.510 are a lot of things along the way that impact the variation. 00:06:28.510 --> 00:06:29.360 GABRIEL SANCHEZ-MARTINEZ: Yeah, so but we're 00:06:29.360 --> 00:06:31.880 saying departing from each stop deterministically, not just 00:06:31.880 --> 00:06:33.500 the terminal. 00:06:33.500 --> 00:06:36.140 So obviously, there are some problems with this. 00:06:36.140 --> 00:06:39.020 And so how do we need to-- 00:06:39.020 --> 00:06:44.490 well, if we start taking care of some of these things, 00:06:44.490 --> 00:06:45.950 what will happen to waiting time? 00:06:45.950 --> 00:06:47.390 Will it decrease or increase? 00:06:51.230 --> 00:06:53.534 So if we take this particular one. 00:06:53.534 --> 00:06:55.200 This is the strongest assumption, right? 00:06:55.200 --> 00:06:58.235 So that vehicles depart deterministically 00:06:58.235 --> 00:07:00.510 at equal intervals-- 00:07:00.510 --> 00:07:03.210 and we say, no, they don't, actually. 00:07:03.210 --> 00:07:04.710 Some of them depart late. 00:07:04.710 --> 00:07:06.390 Some of them depart early. 00:07:06.390 --> 00:07:07.687 There is bunching. 00:07:07.687 --> 00:07:09.770 So what happens to waiting time when that happens? 00:07:09.770 --> 00:07:10.500 AUDIENCE: It goes up. 00:07:10.500 --> 00:07:12.041 GABRIEL SANCHEZ-MARTINEZ: It goes up. 00:07:12.041 --> 00:07:15.479 So how do we adjust this model to account for that? 00:07:15.479 --> 00:07:16.270 Let's look at that. 00:07:16.270 --> 00:07:18.066 AUDIENCE: Doesn't it depend though 00:07:18.066 --> 00:07:19.440 when the passengers are arriving? 00:07:19.440 --> 00:07:21.981 GABRIEL SANCHEZ-MARTINEZ: If you look at an average or a long 00:07:21.981 --> 00:07:26.760 period of time, and you'd have the same number of vehicles 00:07:26.760 --> 00:07:30.960 and drivers, so you either-- the best thing you can do if people 00:07:30.960 --> 00:07:32.314 are arriving randomly is to-- 00:07:32.314 --> 00:07:33.980 AUDIENCE: OK, they're arriving randomly. 00:07:33.980 --> 00:07:35.646 GABRIEL SANCHEZ-MARTINEZ: Yeah, so we're 00:07:35.646 --> 00:07:38.380 only tackling that assumption. 00:07:38.380 --> 00:07:42.180 OK, so there are some issues, as we said. 00:07:42.180 --> 00:07:44.190 There is bulk arrivals. 00:07:44.190 --> 00:07:47.730 So you have a bus stop right outside of a train station. 00:07:47.730 --> 00:07:49.020 And the train has just left. 00:07:49.020 --> 00:07:50.700 And a bunch of people get off the train 00:07:50.700 --> 00:07:52.980 and they all want to board the bus. 00:07:52.980 --> 00:07:54.560 That's not being captured by this. 00:07:57.360 --> 00:08:01.490 And we can think of the passenger arrival process 00:08:01.490 --> 00:08:04.350 in steps, from having no information 00:08:04.350 --> 00:08:06.270 to having a lot of information. 00:08:06.270 --> 00:08:09.420 So random arrivals is what we typically 00:08:09.420 --> 00:08:13.200 assume, certainly for high-frequency service, 00:08:13.200 --> 00:08:16.350 and in some cases, for all kinds of service. 00:08:16.350 --> 00:08:17.869 And that's more problematic. 00:08:17.869 --> 00:08:19.410 If you know how long-headway service, 00:08:19.410 --> 00:08:22.410 people are going to try to time their arrival 00:08:22.410 --> 00:08:24.080 at the stop to the schedule. 00:08:24.080 --> 00:08:26.660 But you will see some models that-- 00:08:26.660 --> 00:08:29.280 in the literature, that, for simplicity, 00:08:29.280 --> 00:08:31.520 especially service planning models, for simplicity, 00:08:31.520 --> 00:08:32.740 might assume this anyway. 00:08:32.740 --> 00:08:34.049 You have a question. 00:08:34.049 --> 00:08:36.035 AUDIENCE: Like random, but Poisson distributed, 00:08:36.035 --> 00:08:37.200 or just, like, random? 00:08:37.200 --> 00:08:39.330 GABRIEL SANCHEZ-MARTINEZ: Yeah, Poisson distributed-- sometimes 00:08:39.330 --> 00:08:40.470 it could just be random. 00:08:40.470 --> 00:08:43.650 Typically the assumption made is that it's a Poisson process. 00:08:43.650 --> 00:08:46.020 So inter-arrival times are negative exponential 00:08:46.020 --> 00:08:48.300 distribution. 00:08:48.300 --> 00:08:51.240 OK, then some passengers will time their arrivals 00:08:51.240 --> 00:08:52.500 to minimum waiting. 00:08:52.500 --> 00:08:58.770 So this could be that you have a phone app or a schedule. 00:08:58.770 --> 00:09:01.200 And you show up a couple of minutes before. 00:09:01.200 --> 00:09:04.530 You're trying to minimize your waiting time. 00:09:04.530 --> 00:09:07.560 You're trying to arrive shortly before the vehicle departs. 00:09:07.560 --> 00:09:09.930 And then there is the running to the vehicle 00:09:09.930 --> 00:09:14.080 before it leaves the stop, and therefore, I have no waiting. 00:09:14.080 --> 00:09:15.870 That's everyone's favorite, right? 00:09:15.870 --> 00:09:19.650 So if we look at a graph of expected headway 00:09:19.650 --> 00:09:23.790 on the horizontal axis and expected waiting 00:09:23.790 --> 00:09:25.620 time on the vertical axis, you will 00:09:25.620 --> 00:09:27.650 see that around 10 minutes-- 00:09:27.650 --> 00:09:30.150 some people say 15, some people say 10, but somewhere around 00:09:30.150 --> 00:09:31.260 there, 12-- 00:09:31.260 --> 00:09:37.310 so you will see that the actual waiting 00:09:37.310 --> 00:09:40.670 time that you would observe is much lower than what 00:09:40.670 --> 00:09:42.710 the simple model says it is, which 00:09:42.710 --> 00:09:46.610 is half the headway for headways longer than 10, 12, 15 minutes. 00:09:46.610 --> 00:09:49.430 And I think those people will have some strategy 00:09:49.430 --> 00:09:51.770 to minimize their waiting time. 00:09:51.770 --> 00:09:56.330 So below that amount, people tend to not time their arrivals 00:09:56.330 --> 00:09:56.960 at stops. 00:09:56.960 --> 00:10:02.240 And they tend to arrive randomly, essentially. 00:10:02.240 --> 00:10:04.960 So if you try to take the Red line here in Boston, 00:10:04.960 --> 00:10:06.920 you just show up at whatever time. 00:10:06.920 --> 00:10:09.820 And typically, most people won't be looking at their phone 00:10:09.820 --> 00:10:12.460 and trying to time their arrival, although that 00:10:12.460 --> 00:10:14.230 is happening increasingly. 00:10:14.230 --> 00:10:17.230 So for those people, the model, the simple model 00:10:17.230 --> 00:10:20.940 of headway divided by 2 tends to underestimate 00:10:20.940 --> 00:10:21.940 the actual waiting time. 00:10:21.940 --> 00:10:24.090 And that's due to headway variability, mostly. 00:10:27.850 --> 00:10:33.420 So we have to take care of that assumption. 00:10:33.420 --> 00:10:36.090 Let's look at a formulation that relaxes that. 00:10:36.090 --> 00:10:37.950 Let's say that vehicle departures are not 00:10:37.950 --> 00:10:39.930 regular and deterministic. 00:10:39.930 --> 00:10:44.290 So let's refine the model to take care of that. 00:10:44.290 --> 00:10:47.580 So let's define n as a function of headway 00:10:47.580 --> 00:10:49.500 to be the number of passengers arriving 00:10:49.500 --> 00:10:54.690 at some headway and the mean waiting time of headway, h, 00:10:54.690 --> 00:10:59.360 to be, well, that mean headway of some specific headway 00:10:59.360 --> 00:11:03.260 that we observe, and then g to be the probability density 00:11:03.260 --> 00:11:04.290 function of headway. 00:11:04.290 --> 00:11:06.300 So we observe many of these highways. 00:11:06.300 --> 00:11:09.150 And we have a histogram of headways. 00:11:09.150 --> 00:11:13.740 So if you want to compute the expected waiting 00:11:13.740 --> 00:11:16.050 time across all passengers, what we want 00:11:16.050 --> 00:11:18.300 is the expected total passenger waiting 00:11:18.300 --> 00:11:21.710 time divided by the expected number of passengers 00:11:21.710 --> 00:11:25.110 over many, many observations of vehicles leaving. 00:11:25.110 --> 00:11:29.670 And that is expressed mathematically in the equation 00:11:29.670 --> 00:11:30.540 below. 00:11:30.540 --> 00:11:33.300 We have integrals, in this case, from 0 to infinity, 00:11:33.300 --> 00:11:35.410 because headways can't be negative, 00:11:35.410 --> 00:11:38.460 so we're not going from negative infinity to positive infinity. 00:11:38.460 --> 00:11:43.080 And we're saying, yeah, for any given headway, 00:11:43.080 --> 00:11:44.910 we'll multiply the number of people times 00:11:44.910 --> 00:11:46.920 how much they wait on average times 00:11:46.920 --> 00:11:49.540 the probability that I see that headway. 00:11:49.540 --> 00:11:52.140 And let's integrate over all possible headways 00:11:52.140 --> 00:11:54.930 and divide that by the number of people on each headway. 00:11:54.930 --> 00:11:56.466 Does that make sense conceptually? 00:11:59.660 --> 00:12:02.690 OK, so now let's make some assumptions. 00:12:02.690 --> 00:12:05.900 Let's say that people do arrive uniformly 00:12:05.900 --> 00:12:08.990 with some rate, lambda. 00:12:08.990 --> 00:12:12.560 And so the number of people arriving in a given headway 00:12:12.560 --> 00:12:15.110 is going to be some arrival rate, which we've said 00:12:15.110 --> 00:12:17.520 is constant, times the headway. 00:12:17.520 --> 00:12:20.370 So if we say that 10 people arrive per minute, 00:12:20.370 --> 00:12:23.780 and lambda therefore is 10 passengers per minute, 00:12:23.780 --> 00:12:28.080 then if the headway is 2 minutes, we have 20 people. 00:12:28.080 --> 00:12:29.520 Does that make sense? 00:12:29.520 --> 00:12:32.130 Any questions so far? 00:12:32.130 --> 00:12:33.932 OK, and then-- 00:12:33.932 --> 00:12:36.015 AUDIENCE: Can you repeat just what you said there? 00:12:36.015 --> 00:12:38.120 GABRIEL SANCHEZ-MARTINEZ: Yeah, so lambda 00:12:38.120 --> 00:12:40.140 is the passenger arrival rate. 00:12:40.140 --> 00:12:43.536 And I gave an example of it being 10 passengers per minute. 00:12:43.536 --> 00:12:44.910 So is that something you measure. 00:12:44.910 --> 00:12:46.460 And we're assuming that that amount 00:12:46.460 --> 00:12:52.160 is constant for a specific time of day at a specific place. 00:12:52.160 --> 00:12:54.440 And then we're saying, if you-- once you 00:12:54.440 --> 00:12:57.410 assume that, then you observe a headway, which could be, say, 00:12:57.410 --> 00:12:58.130 2 minutes long. 00:12:58.130 --> 00:13:00.310 Then you multiply by the arrival rate. 00:13:00.310 --> 00:13:04.084 Now we say it's 20 passengers for that particular headway. 00:13:04.084 --> 00:13:06.500 And there might be another headway that is 5 minutes long. 00:13:06.500 --> 00:13:07.730 That's 50 people-- 00:13:07.730 --> 00:13:09.980 1 minute long, 10 people, and so forth. 00:13:09.980 --> 00:13:11.690 And then for each of those headways, 00:13:11.690 --> 00:13:14.210 the average waiting time is going 00:13:14.210 --> 00:13:18.514 to be half of that headway, correct? 00:13:18.514 --> 00:13:19.430 AUDIENCE: Correct. 00:13:19.430 --> 00:13:21.096 GABRIEL SANCHEZ-MARTINEZ: We're assuming 00:13:21.096 --> 00:13:23.690 people arrive randomly, so within a particular headway, 00:13:23.690 --> 00:13:26.756 this is true. 00:13:26.756 --> 00:13:28.380 There's no problem with this assumption 00:13:28.380 --> 00:13:30.213 as long as you are comfortable with assuming 00:13:30.213 --> 00:13:32.840 that people arrive randomly. 00:13:32.840 --> 00:13:37.430 So now we have this equation for expected waiting time. 00:13:42.140 --> 00:13:47.490 Let me write on the board how we get to that from the integrals. 00:14:05.620 --> 00:14:08.770 Can somebody say why-- 00:14:08.770 --> 00:14:15.100 so why is it the case that, when we have headway variability, 00:14:15.100 --> 00:14:17.050 the waiting time goes up? 00:14:17.050 --> 00:14:20.651 What's the concept that leads to that? 00:14:20.651 --> 00:14:21.150 Eli? 00:14:21.150 --> 00:14:22.525 AUDIENCE: There is unreliability, 00:14:22.525 --> 00:14:26.360 and so people need to build in more [INAUDIBLE] time instead 00:14:26.360 --> 00:14:26.860 of-- 00:14:26.860 --> 00:14:28.276 GABRIEL SANCHEZ-MARTINEZ: There is 00:14:28.276 --> 00:14:31.400 something else that is sort of more fundamental happening. 00:14:31.400 --> 00:14:34.906 AUDIENCE: A bus that comes at a larger headway 00:14:34.906 --> 00:14:36.280 than it was planned will actually 00:14:36.280 --> 00:14:37.610 be collecting more people. 00:14:37.610 --> 00:14:41.890 So more people will be waiting more time. 00:14:41.890 --> 00:14:44.440 GABRIEL SANCHEZ-MARTINEZ: So from a passenger's perspective, 00:14:44.440 --> 00:14:46.481 the probability of arriving during a long headway 00:14:46.481 --> 00:14:48.677 is greater than arriving during a short headway. 00:14:48.677 --> 00:14:49.510 What is that called? 00:14:49.510 --> 00:14:52.974 There is a name for that that's important in transportation. 00:14:55.640 --> 00:15:02.200 Random-- random incidence, so this is something you, if you 00:15:02.200 --> 00:15:05.030 took 200, you should know that. 00:15:05.030 --> 00:15:08.350 But it's good to know, random incidence. 00:15:08.350 --> 00:15:11.860 It's called the random incidence paradox, 00:15:11.860 --> 00:15:14.770 because people without-- if you don't think about it too much, 00:15:14.770 --> 00:15:16.020 you say it's half the headway. 00:15:16.020 --> 00:15:19.870 And then somebody says, no, actually the average waiting 00:15:19.870 --> 00:15:21.140 time is much longer. 00:15:21.140 --> 00:15:21.980 Why is that? 00:15:21.980 --> 00:15:24.070 And you have to sort of solve the paradox. 00:15:24.070 --> 00:15:26.840 It isn't really a paradox, but here we are. 00:15:30.895 --> 00:15:33.860 So the expected waiting time-- 00:15:33.860 --> 00:15:38.560 we'll start from the equation, the last equation 00:15:38.560 --> 00:15:40.330 on slide four. 00:15:40.330 --> 00:15:43.180 And we're just going to plug in what our assumption is. 00:15:43.180 --> 00:15:48.500 So we have integral from 0 to infinity. 00:15:48.500 --> 00:15:52.240 And we said we had n of h. 00:15:52.240 --> 00:15:54.520 And we said that that's going to be lambda h. 00:15:54.520 --> 00:16:01.900 So let's just put that in, lambda h. 00:16:01.900 --> 00:16:05.470 And then we said, for waiting time, 00:16:05.470 --> 00:16:12.550 we said it was going to be 1/2, so 1/2 h. 00:16:12.550 --> 00:16:15.900 And then we're going to multiply times the PVF 00:16:15.900 --> 00:16:19.930 the probability of that headway times dh. 00:16:32.990 --> 00:16:36.392 All right, and here we have the same, 00:16:36.392 --> 00:16:37.600 but without the waiting time. 00:16:41.980 --> 00:16:43.600 So that's just substitution, shouldn't 00:16:43.600 --> 00:16:44.805 be anything wrong with that. 00:16:44.805 --> 00:16:50.410 So now, lambda and 1/2 come out. 00:16:50.410 --> 00:16:52.900 They're not a function of h, so they are constants. 00:16:52.900 --> 00:16:54.390 We can take them out. 00:16:54.390 --> 00:17:02.290 And we are left with, let's see, lambda over 2 integral from 0 00:17:02.290 --> 00:17:07.390 to infinity of h squared-- 00:17:07.390 --> 00:17:10.960 because we have two h's here-- 00:17:10.960 --> 00:17:15.819 times dh dh. 00:17:15.819 --> 00:17:19.140 And here we have-- lambda comes out. 00:17:19.140 --> 00:17:26.589 And we integrate from 0 to infinity of h dh dh. 00:17:26.589 --> 00:17:30.140 OK, now the first observation is that lambda 00:17:30.140 --> 00:17:33.770 has come out on both sides. 00:17:33.770 --> 00:17:36.500 So there's just one lambda on the top, one on the bottom, 00:17:36.500 --> 00:17:38.690 so we can cancel them out. 00:17:38.690 --> 00:17:40.670 So that's very convenient. 00:17:40.670 --> 00:17:42.230 What does that mean? 00:17:42.230 --> 00:17:44.030 This happens because we are assuming 00:17:44.030 --> 00:17:45.870 that lambda is constant. 00:17:45.870 --> 00:17:47.720 That wouldn't happen if it weren't constant. 00:17:47.720 --> 00:17:50.390 But that's convenient, right? 00:17:50.390 --> 00:17:52.760 So now we have some quantity that does not 00:17:52.760 --> 00:17:55.580 depend on the arrival rate. 00:17:55.580 --> 00:17:58.820 So the same equation is not affected by that. 00:17:58.820 --> 00:18:03.070 And let's see if we can recognize 00:18:03.070 --> 00:18:04.720 what these quantities are. 00:18:04.720 --> 00:18:08.000 So let's start with the one on the bottom. 00:18:08.000 --> 00:18:11.680 This is the definition of the expectancy. 00:18:11.680 --> 00:18:14.570 So this is just the average. 00:18:14.570 --> 00:18:19.330 So we're essentially saying, take every-- 00:18:19.330 --> 00:18:21.550 take the average headway, so take every headway, 00:18:21.550 --> 00:18:23.920 and multiply times the probability of observing it. 00:18:23.920 --> 00:18:25.610 Add them all up. 00:18:25.610 --> 00:18:28.520 That's what the average is. 00:18:28.520 --> 00:18:36.290 So we have 2 times the expected headway. 00:18:36.290 --> 00:18:40.570 And at the top, we have the expectants of h squared. 00:18:44.180 --> 00:18:47.700 So I think that takes us to where we are 00:18:47.700 --> 00:18:51.450 on this slide, which is great. 00:18:51.450 --> 00:18:55.620 So part of it was just recognizing 00:18:55.620 --> 00:18:58.410 that that was the definition of expectants, so that's great. 00:18:58.410 --> 00:19:05.000 And now, somehow, we go from here to this equation. 00:19:05.000 --> 00:19:12.730 It's not entirely clear, but it's convenient 00:19:12.730 --> 00:19:17.260 that variance is a function of the expectants of h squared. 00:19:17.260 --> 00:19:20.020 So I'll just remind you, if you haven't seen it, 00:19:20.020 --> 00:19:23.179 then introduce you to the definition-- or not quite 00:19:23.179 --> 00:19:24.220 the definition, actually. 00:19:24.220 --> 00:19:27.400 This is you take the definition and you expand it 00:19:27.400 --> 00:19:29.080 and collect like terms. 00:19:29.080 --> 00:19:33.590 And then you get that the variance of h 00:19:33.590 --> 00:19:46.450 is the expectation of h squared minus the expectation of h 00:19:46.450 --> 00:19:51.700 quantity squared, so means of square minus squares mean. 00:19:51.700 --> 00:19:53.280 You might have seen that before. 00:19:53.280 --> 00:19:59.520 So essentially, if we solve that equation for expectation of h 00:19:59.520 --> 00:20:02.930 squared and plug that in here, we can collect like terms 00:20:02.930 --> 00:20:05.580 and get this last equation. 00:20:05.580 --> 00:20:09.170 So now we have that the expected waiting 00:20:09.170 --> 00:20:13.680 time is half the headway times some adjustment factor. 00:20:13.680 --> 00:20:18.170 And that adjustment factor is 1 plus some amount-- 00:20:18.170 --> 00:20:22.340 that amount is what we call the coefficient of variation-- 00:20:22.340 --> 00:20:22.879 squared. 00:20:22.879 --> 00:20:24.920 So coefficient of variation is standard deviation 00:20:24.920 --> 00:20:26.010 divided by mean. 00:20:26.010 --> 00:20:28.360 So we have normalizing the standard deviation. 00:20:28.360 --> 00:20:29.450 You square that quantity. 00:20:29.450 --> 00:20:30.940 You add it to 1. 00:20:30.940 --> 00:20:35.310 And that'll increase your waiting time by some amount. 00:20:38.510 --> 00:20:40.160 Let's stop here for a second. 00:20:40.160 --> 00:20:43.550 Any questions with the derivation and what 00:20:43.550 --> 00:20:44.570 the meaning of this is? 00:20:47.200 --> 00:20:47.790 Yes, Henry? 00:20:47.790 --> 00:20:49.410 AUDIENCE: Can you go back to that step 00:20:49.410 --> 00:20:52.939 where you went from the integral to 2 e of h. 00:20:52.939 --> 00:20:54.230 GABRIEL SANCHEZ-MARTINEZ: Here? 00:20:54.230 --> 00:20:54.890 AUDIENCE: Yeah. 00:20:54.890 --> 00:20:57.670 GABRIEL SANCHEZ-MARTINEZ: So this is a matter of definition. 00:20:57.670 --> 00:21:00.010 So this here, the integral from 0-- 00:21:00.010 --> 00:21:03.000 from negative infinity to positive infinity of some 00:21:03.000 --> 00:21:07.540 amount times the probability of observing that amount over all 00:21:07.540 --> 00:21:09.490 possibilities of the amount-- 00:21:09.490 --> 00:21:12.220 that's why we go from negative infinity to positive infinity-- 00:21:12.220 --> 00:21:15.020 equals the average of that amount, 00:21:15.020 --> 00:21:17.080 or the expectation of that amount. 00:21:17.080 --> 00:21:20.150 So that part-- do you understand? 00:21:20.150 --> 00:21:21.160 AUDIENCE: [INAUDIBLE] 00:21:21.160 --> 00:21:22.520 GABRIEL SANCHEZ-MARTINEZ: Yes? 00:21:22.520 --> 00:21:24.890 And then here, we're just saying, well, 00:21:24.890 --> 00:21:27.800 this is the same thing, except of the amount instead 00:21:27.800 --> 00:21:30.090 of being h is h squared. 00:21:30.090 --> 00:21:34.070 So we're taking the average of h squared. 00:21:34.070 --> 00:21:36.230 This comes from here to here from the definition 00:21:36.230 --> 00:21:39.120 of expectation. 00:21:39.120 --> 00:21:41.540 And the reason we're not going from negative infinity 00:21:41.540 --> 00:21:44.270 to positive infinity is that h can only 00:21:44.270 --> 00:21:46.610 take non-negative values. 00:21:46.610 --> 00:21:48.230 It can be 0 or higher. 00:21:48.230 --> 00:21:51.890 So it's the same thing to integrate from 0 to infinity, 00:21:51.890 --> 00:21:54.310 in that case. 00:21:54.310 --> 00:21:57.030 Does that clear up the question? 00:21:57.030 --> 00:21:58.285 Any other questions? 00:22:01.170 --> 00:22:09.030 So now we have this modified model of waiting time, 00:22:09.030 --> 00:22:10.560 mainly for high-frequency service, 00:22:10.560 --> 00:22:12.310 where people are arriving randomly 00:22:12.310 --> 00:22:15.070 but vehicles are not necessarily departing stops 00:22:15.070 --> 00:22:17.760 deterministically on equal intervals anymore. 00:22:17.760 --> 00:22:20.730 Now these vehicles can come-- 00:22:20.730 --> 00:22:23.300 as long as they come independent of the passenger arrival 00:22:23.300 --> 00:22:26.780 process, it doesn't really matter how they come. 00:22:26.780 --> 00:22:31.260 This model calculates the expected waiting time 00:22:31.260 --> 00:22:32.650 across all passengers. 00:22:32.650 --> 00:22:34.740 So we have some cases that we can look at. 00:22:34.740 --> 00:22:36.360 First case is simple. 00:22:36.360 --> 00:22:38.430 Let's say that the variance is 0. 00:22:38.430 --> 00:22:40.980 That means we go back to the assumption of, 00:22:40.980 --> 00:22:43.740 well, if the variance of headway is 0, 00:22:43.740 --> 00:22:47.120 vehicles are arriving exactly every five minutes, say. 00:22:47.120 --> 00:22:50.890 They're deterministically at equally spaced intervals. 00:22:50.890 --> 00:22:55.610 So we then have that the coefficients of variation is 0. 00:22:55.610 --> 00:22:57.610 And we end up with the same thing we had before. 00:22:57.610 --> 00:23:00.180 So that really checks out with our previous model 00:23:00.180 --> 00:23:03.570 in that special case, but we generalized the model. 00:23:03.570 --> 00:23:06.580 So we can try it with different vehicle arrival or departure 00:23:06.580 --> 00:23:08.150 processes. 00:23:08.150 --> 00:23:11.002 One that we have here is, let's say that vehicle departures are 00:23:11.002 --> 00:23:12.820 a Poisson process. 00:23:12.820 --> 00:23:16.770 So this would mean that they are arriving randomly, essentially, 00:23:16.770 --> 00:23:21.690 and that the time of arrival of a vehicle at a stop 00:23:21.690 --> 00:23:24.270 is independent of any other previous arrival. 00:23:24.270 --> 00:23:27.480 So that would be a good model for a service that 00:23:27.480 --> 00:23:29.610 is controlled very loosely. 00:23:29.610 --> 00:23:32.220 They are hardly controlled. 00:23:32.220 --> 00:23:34.290 So you essentially sometimes see bunching. 00:23:34.290 --> 00:23:35.130 Sometimes you don't. 00:23:35.130 --> 00:23:36.546 Sometimes you have a long headway. 00:23:36.546 --> 00:23:38.080 Sometimes you don't. 00:23:38.080 --> 00:23:43.230 So the definition-- if you go through the Poisson process 00:23:43.230 --> 00:23:46.710 and look at the definitions of the probability distributions 00:23:46.710 --> 00:23:48.720 of inter-arrival times, et cetera, 00:23:48.720 --> 00:23:51.630 you will see that the variance in such a process 00:23:51.630 --> 00:23:53.700 is the square of the expectation, so 00:23:53.700 --> 00:23:55.300 the square of the mean. 00:23:55.300 --> 00:23:58.600 And if we plug that into our equation, 00:23:58.600 --> 00:24:03.120 we get that the expected waiting time is the expected headway. 00:24:03.120 --> 00:24:07.620 So people are essentially waiting about as long 00:24:07.620 --> 00:24:11.210 as your mean headway, which is a bad situation. 00:24:11.210 --> 00:24:13.470 You've doubled their waiting time 00:24:13.470 --> 00:24:17.970 from half the headway to the full headway. 00:24:17.970 --> 00:24:21.150 If you have a system where your vehicles are all bunched 00:24:21.150 --> 00:24:23.460 and you're running pairs of bunches, 00:24:23.460 --> 00:24:28.650 that's more or less what you get, because you have-- 00:24:28.650 --> 00:24:30.410 if you had vehicles every 5 minutes, now 00:24:30.410 --> 00:24:33.672 you have two vehicles every 10. 00:24:33.672 --> 00:24:37.470 Your average headway is five, but your waiting time 00:24:37.470 --> 00:24:40.700 is now five as well. 00:24:40.700 --> 00:24:43.290 OK, what about the third case? 00:24:43.290 --> 00:24:47.070 The headway sequence is 5, 15, 5, 15, 5, 15-- 00:24:47.070 --> 00:24:49.710 thank you, keep going like that, so not quite bunched 00:24:49.710 --> 00:24:52.240 all the way, but there is a lot of variability. 00:24:52.240 --> 00:24:55.260 So I'm saying that the expected headway is 10. 00:24:55.260 --> 00:24:56.836 Does that make sense? 00:24:56.836 --> 00:24:58.120 It's 5, 15, 5, 15. 00:24:58.120 --> 00:25:01.520 The average of that process is 10. 00:25:01.520 --> 00:25:04.840 So we have an average headway of 10. 00:25:04.840 --> 00:25:07.084 And now what's the expected waiting time? 00:25:10.720 --> 00:25:19.630 So it helps to draw a timeline and to divide that timeline 00:25:19.630 --> 00:25:36.910 into pieces, so 5, 15, 5, 15. 00:25:36.910 --> 00:25:47.670 So we have vehicles arriving or departing here, here, here, 00:25:47.670 --> 00:25:49.690 and here. 00:25:49.690 --> 00:25:53.154 And there is a five-minute headway here, 00:25:53.154 --> 00:25:55.570 and a 15-minute headway here, a five-minute headway there, 00:25:55.570 --> 00:25:57.040 and a 15-minute headway here. 00:25:57.040 --> 00:25:58.860 And people are arriving randomly. 00:25:58.860 --> 00:26:01.450 People arrive independent of this process. 00:26:01.450 --> 00:26:05.260 So what you'll see is that, the first thing 00:26:05.260 --> 00:26:09.130 is, if you were to arrive on a five-minute headway, 00:26:09.130 --> 00:26:11.080 the average waiting time for you would 00:26:11.080 --> 00:26:15.470 be half of that five-minute headway, 2 and 1/2 minutes. 00:26:15.470 --> 00:26:18.470 And if you arrive on 15-minute headway, 00:26:18.470 --> 00:26:20.530 then your expected waiting time, given 00:26:20.530 --> 00:26:24.580 that you arrive on a 15-minute headway is 7 and 1/2 minutes. 00:26:24.580 --> 00:26:28.360 So that's the 2.5 and the 7.5 on this equation. 00:26:28.360 --> 00:26:29.980 And then we have the probabilities 00:26:29.980 --> 00:26:32.350 of having arrived on any of those headways. 00:26:32.350 --> 00:26:35.380 Because the 15-minute headway is 3 times longer 00:26:35.380 --> 00:26:38.110 than the 5-minute headway, you are 3 times more 00:26:38.110 --> 00:26:40.930 likely to arrive during a 15-minute headway 00:26:40.930 --> 00:26:42.310 than during a 5-minute headway. 00:26:42.310 --> 00:26:46.600 So therefore, 25% of passengers will arrive 00:26:46.600 --> 00:26:48.280 on the 15-minute headways. 00:26:48.280 --> 00:26:53.320 And you essentially computing a weighted average, 00:26:53.320 --> 00:26:56.560 so 6 and 1/4 minutes. 00:26:59.482 --> 00:27:00.456 Questions on this? 00:27:13.130 --> 00:27:15.380 All right, another assumption that we 00:27:15.380 --> 00:27:17.060 were looking at in the first model 00:27:17.060 --> 00:27:18.770 that we are still having in this model 00:27:18.770 --> 00:27:20.730 is that people can board the first vehicle 00:27:20.730 --> 00:27:22.730 that they see arrive. 00:27:22.730 --> 00:27:26.870 So no vehicle is too full to take passengers. 00:27:26.870 --> 00:27:28.850 No vehicle is leaving people behind. 00:27:28.850 --> 00:27:29.975 That's not true. 00:27:29.975 --> 00:27:34.310 If we look at any process, and we say here 00:27:34.310 --> 00:27:37.880 that w0 is the expected waiting time 00:27:37.880 --> 00:27:40.580 when people can board their first vehicle, 00:27:40.580 --> 00:27:45.290 so when no vehicle is full, and we then 00:27:45.290 --> 00:27:48.630 look at all the expected waiting times for passengers including 00:27:48.630 --> 00:27:52.490 passengers that are left behind, when that capacity is 00:27:52.490 --> 00:27:53.540 being reached-- 00:27:53.540 --> 00:27:55.640 so here on the horizontal, we have rho, 00:27:55.640 --> 00:27:57.820 which is flow over capacity. 00:28:00.209 --> 00:28:02.000 You may have seen this kind of relationship 00:28:02.000 --> 00:28:02.940 from queuing theory. 00:28:02.940 --> 00:28:06.470 So when rho is 0, it means that you will have some supply 00:28:06.470 --> 00:28:08.090 and nobody is using it. 00:28:08.090 --> 00:28:11.270 If rho is 0.5, then your capacity 00:28:11.270 --> 00:28:14.780 is double of the demand. 00:28:14.780 --> 00:28:16.250 That's OK usually. 00:28:16.250 --> 00:28:18.590 If rho is 1, you are-- 00:28:18.590 --> 00:28:20.600 your capacity equals your demand. 00:28:20.600 --> 00:28:23.780 And what we see is that the expected waiting 00:28:23.780 --> 00:28:27.735 time for service will start to approach infinity. 00:28:27.735 --> 00:28:31.130 There is a term that we use in, say, queuing theory, 00:28:31.130 --> 00:28:32.466 for such a system. 00:28:32.466 --> 00:28:33.340 Does anybody know it? 00:28:38.370 --> 00:28:42.930 We say that the system blows up when this happens. 00:28:42.930 --> 00:28:45.000 If you hear that, you'll see that. 00:28:45.000 --> 00:28:49.190 If you take any sort of queuing theory, you'll hear that. 00:28:49.190 --> 00:28:52.980 So as rho equals 1, let's say that you have-- you're 00:28:52.980 --> 00:28:55.620 providing a bus service with capacity 00:28:55.620 --> 00:28:59.550 for 100 people per minute. 00:28:59.550 --> 00:29:03.560 That's pretty high. 00:29:03.560 --> 00:29:07.060 And then your demand equals 100 people per minute. 00:29:07.060 --> 00:29:09.540 So if people are arriving uniformly, 00:29:09.540 --> 00:29:12.450 you might just provide a capacity for everyone. 00:29:12.450 --> 00:29:14.160 And nobody is being left behind. 00:29:14.160 --> 00:29:16.870 But that doesn't happen, because people arrive randomly. 00:29:20.940 --> 00:29:23.810 Some people will leave with some remaining capacity. 00:29:23.810 --> 00:29:26.670 And the next vehicle won't have enough space for that person. 00:29:26.670 --> 00:29:28.890 And then the vehicles themselves will not 00:29:28.890 --> 00:29:31.650 arrive perfectly evenly, so now you 00:29:31.650 --> 00:29:34.710 have a chain reaction of people waiting and not 00:29:34.710 --> 00:29:36.000 being able to board. 00:29:36.000 --> 00:29:38.640 And unless you supply more capacity, 00:29:38.640 --> 00:29:40.410 unless you increase the capacity, 00:29:40.410 --> 00:29:42.600 you're going to have more and more people waiting 00:29:42.600 --> 00:29:43.960 and being left behind. 00:29:43.960 --> 00:29:46.200 So the actual expected waiting time 00:29:46.200 --> 00:29:51.160 would increase as your ratio of flow to capacity approaches 1. 00:29:51.160 --> 00:29:53.220 That's the takeaway here. 00:29:53.220 --> 00:29:58.050 And you could think of two different lines, one 00:29:58.050 --> 00:30:01.260 for low reliability when there is a lot of bunching, 00:30:01.260 --> 00:30:03.540 and therefore, a lot of the capacity that you provide 00:30:03.540 --> 00:30:07.440 is wasted on small headways that are not 00:30:07.440 --> 00:30:11.400 serving as many passengers as you assign them for, and then 00:30:11.400 --> 00:30:13.560 high reliability, where, yes, you still 00:30:13.560 --> 00:30:17.250 have some variability that you can't control. 00:30:17.250 --> 00:30:18.320 You've done what you can. 00:30:18.320 --> 00:30:20.940 And this is relatively high reliability, 00:30:20.940 --> 00:30:23.550 but still you have some variability, 00:30:23.550 --> 00:30:28.020 and therefore, you can't expect to reach that rho equals 1. 00:30:28.020 --> 00:30:29.310 It'll still blow up when you-- 00:30:29.310 --> 00:30:30.770 as you approach 1. 00:30:30.770 --> 00:30:33.870 But it's closer to 1, so that's good. 00:30:33.870 --> 00:30:36.670 Questions on this relationship? 00:30:36.670 --> 00:30:37.170 [INAUDIBLE]? 00:30:37.170 --> 00:30:40.032 AUDIENCE: What do most bus services operate 00:30:40.032 --> 00:30:41.902 at [INAUDIBLE]? 00:30:41.902 --> 00:30:43.860 GABRIEL SANCHEZ-MARTINEZ: I don't have a number 00:30:43.860 --> 00:30:45.370 off the top of my head. 00:30:45.370 --> 00:30:46.370 It varies a lot. 00:30:46.370 --> 00:30:48.390 And it varies a lot of by time period. 00:30:48.390 --> 00:30:54.290 So some services here in Boston operate during the peak leaving 00:30:54.290 --> 00:30:57.230 people behind every day. 00:30:57.230 --> 00:31:00.230 If you look at some of the most crowded stations 00:31:00.230 --> 00:31:03.260 in the tube in London, certainly they have to meter people 00:31:03.260 --> 00:31:04.630 going into stations. 00:31:04.630 --> 00:31:09.970 So they may have approached rho greater than 1, in this case. 00:31:09.970 --> 00:31:11.270 But it's non-stationary. 00:31:11.270 --> 00:31:13.400 So that's during the peak of the peak. 00:31:13.400 --> 00:31:17.120 And then the demand falls, so eventually, those people 00:31:17.120 --> 00:31:21.410 are served, of course, because the supply is kept up 00:31:21.410 --> 00:31:24.350 at some-- at the highest rate that they can keep it. 00:31:24.350 --> 00:31:26.660 And demand it comes down, so it stabilizes. 00:31:30.380 --> 00:31:31.340 Any other questions? 00:31:31.340 --> 00:31:32.298 That's a good question. 00:31:35.880 --> 00:31:37.750 All right, so service variation along 00:31:37.750 --> 00:31:41.060 routes-- so we've been talking about vehicles 00:31:41.060 --> 00:31:44.270 leaving on time or not on time, and bunching. 00:31:44.270 --> 00:31:46.910 So here's what happens. 00:31:46.910 --> 00:31:49.040 This is a space-time diagram. 00:31:49.040 --> 00:31:52.370 Have we seen these diagrams on this course yet? 00:31:52.370 --> 00:31:53.090 AUDIENCE: No. 00:31:53.090 --> 00:31:54.923 GABRIEL SANCHEZ-MARTINEZ: No-- OK, well this 00:31:54.923 --> 00:31:57.200 is a very important diagram in public transportation, 00:31:57.200 --> 00:31:57.380 certainly. 00:31:57.380 --> 00:31:58.540 AUDIENCE: And the homework says to make one. 00:31:58.540 --> 00:31:58.876 So we should probably-- 00:31:58.876 --> 00:31:58.960 GABRIEL SANCHEZ-MARTINEZ: Great. 00:31:58.960 --> 00:32:00.234 AUDIENCE: --learn it. 00:32:00.234 --> 00:32:01.400 I was wondering what it was. 00:32:01.400 --> 00:32:03.691 GABRIEL SANCHEZ-MARTINEZ: This is a space-time diagram. 00:32:03.691 --> 00:32:09.180 You have space on the y-axis and time on the x-axis. 00:32:09.180 --> 00:32:13.040 And what you see are some lines that 00:32:13.040 --> 00:32:16.550 show how a vehicle is moving across space and time. 00:32:16.550 --> 00:32:20.380 So there are variations of this. 00:32:20.380 --> 00:32:23.510 Sometimes people put time on the vertical and space 00:32:23.510 --> 00:32:24.460 on the horizontal. 00:32:24.460 --> 00:32:25.400 That's fine. 00:32:25.400 --> 00:32:29.390 And typically, you see little steps for each stop, 00:32:29.390 --> 00:32:31.490 because if a vehicle is stopped, time is running 00:32:31.490 --> 00:32:35.910 and the vehicle is not moving, so you see little steps. 00:32:35.910 --> 00:32:37.220 So you can see holding. 00:32:37.220 --> 00:32:38.300 You can see dwell times. 00:32:38.300 --> 00:32:40.790 You can see speeds. 00:32:40.790 --> 00:32:42.980 You can see recovery time at a terminal, everything 00:32:42.980 --> 00:32:43.880 on this diagram. 00:32:43.880 --> 00:32:47.001 So it's a great diagram for analyzing service. 00:32:47.001 --> 00:32:49.250 AUDIENCE: When we make, should we make them like this? 00:32:49.250 --> 00:32:49.890 Or like-- 00:32:49.890 --> 00:32:51.890 GABRIEL SANCHEZ-MARTINEZ: The little steps are-- 00:32:51.890 --> 00:32:52.540 AUDIENCE: Little steps-- 00:32:52.540 --> 00:32:54.990 GABRIEL SANCHEZ-MARTINEZ: Yeah, I mean, they're helpful. 00:32:54.990 --> 00:32:56.910 You'll see dwell times, so yeah. 00:32:56.910 --> 00:32:59.760 So this is a conceptual very simplified diagram, 00:32:59.760 --> 00:33:01.850 but here is the idea. 00:33:01.850 --> 00:33:06.230 So you have some scheduled trajectories in dashed lines. 00:33:06.230 --> 00:33:10.130 So we're essentially planning to run service every 10 minutes, 00:33:10.130 --> 00:33:15.540 departing from Stop 1 at 9:00, 9:10, 9:20, et cetera. 00:33:15.540 --> 00:33:17.990 And if things run according to plan, 00:33:17.990 --> 00:33:20.664 you will keep a 10-minute headway throughout the route. 00:33:20.664 --> 00:33:21.830 But that's not what happens. 00:33:21.830 --> 00:33:25.610 If we observe-- let's say that the driver that 00:33:25.610 --> 00:33:27.990 was driving the bus that departed at 9:10 00:33:27.990 --> 00:33:32.780 is a slow driver, so that driver drives slowly. 00:33:32.780 --> 00:33:36.960 So therefore, more time passes covering the same space. 00:33:36.960 --> 00:33:40.130 And then let's say that the 9:20 driver is a fast driver. 00:33:40.130 --> 00:33:42.890 So that driver covers more distance in less time. 00:33:42.890 --> 00:33:44.210 And they bunch. 00:33:44.210 --> 00:33:46.950 They meet somewhere before they reach the end of the route. 00:33:46.950 --> 00:33:49.580 They've platooned. 00:33:49.580 --> 00:33:53.210 And they're running together as a bunch. 00:33:53.210 --> 00:33:54.830 So what does that do to headway? 00:33:59.056 --> 00:34:00.680 AUDIENCE: What does what do to headway? 00:34:00.680 --> 00:34:01.990 GABRIEL SANCHEZ-MARTINEZ: What does this process of bunching 00:34:01.990 --> 00:34:03.172 do to headways? 00:34:03.172 --> 00:34:04.130 AUDIENCE: It increases. 00:34:04.130 --> 00:34:06.060 GABRIEL SANCHEZ-MARTINEZ: Right? 00:34:06.060 --> 00:34:08.800 Headways are increasing, because now you've-- 00:34:08.800 --> 00:34:10.690 well, the average headway remains the same, 00:34:10.690 --> 00:34:12.760 because you have a 0 and a big number. 00:34:12.760 --> 00:34:15.730 But what people actually see are the long headways, 00:34:15.730 --> 00:34:18.580 because the chance of arriving and just catching 00:34:18.580 --> 00:34:22.480 the 30-second headway is much smaller. 00:34:22.480 --> 00:34:24.699 So people are waiting more because these headways are 00:34:24.699 --> 00:34:25.810 longer. 00:34:25.810 --> 00:34:26.310 Yeah? 00:34:26.310 --> 00:34:27.940 AUDIENCE: Well, you probably have more people waiting. 00:34:27.940 --> 00:34:29.108 So there is [INAUDIBLE]-- 00:34:29.108 --> 00:34:30.858 GABRIEL SANCHEZ-MARTINEZ: Yeah, so there's 00:34:30.858 --> 00:34:32.659 a vicious cycle here, right? 00:34:32.659 --> 00:34:34.876 Yeah, we'll get to that on the next slide. 00:34:34.876 --> 00:34:37.001 AUDIENCE: It would be better if the fast driver got 00:34:37.001 --> 00:34:39.030 in front of the slow driver. 00:34:39.030 --> 00:34:41.710 GABRIEL SANCHEZ-MARTINEZ: Yeah, we'll talk about that too. 00:34:41.710 --> 00:34:42.793 We'll talk about that too. 00:34:45.690 --> 00:34:50.436 OK, so do we understand this process of bunching? 00:34:50.436 --> 00:34:52.760 Here is what you were talking about, Ari? 00:34:52.760 --> 00:34:54.909 So there is an issue when that happens. 00:34:54.909 --> 00:34:56.860 So now we have the little steps here. 00:34:56.860 --> 00:34:59.160 So this is Step 2, Step 3. 00:34:59.160 --> 00:35:01.180 And these are the dwell times. 00:35:01.180 --> 00:35:03.055 So nothing happened on bus one. 00:35:03.055 --> 00:35:06.340 It ran exactly as planned in this hypothetical example. 00:35:06.340 --> 00:35:09.990 But because this driver was slow, the headway was bigger. 00:35:09.990 --> 00:35:11.930 And more people arrived during the headway. 00:35:11.930 --> 00:35:14.410 So now the dwell times at Stop 2 are 00:35:14.410 --> 00:35:17.290 going to be longer for that driver that was driving slow. 00:35:17.290 --> 00:35:19.062 Now that bus has more passengers. 00:35:19.062 --> 00:35:21.520 And therefore, the probability of stopping at the next stop 00:35:21.520 --> 00:35:23.200 is higher, because you have more people, 00:35:23.200 --> 00:35:26.620 so the chance that at least one person on that bus 00:35:26.620 --> 00:35:29.980 wants to get off at the next one is higher. 00:35:29.980 --> 00:35:31.779 And if the bus is really crowded, 00:35:31.779 --> 00:35:33.820 the dwell time process is going to be slowed down 00:35:33.820 --> 00:35:38.610 by just friction between passengers. 00:35:38.610 --> 00:35:41.830 So this bus continues to be delayed, not just 00:35:41.830 --> 00:35:46.840 by the driver driving slow, but by the dwell time effect. 00:35:46.840 --> 00:35:49.270 And because that bus is being delayed, 00:35:49.270 --> 00:35:54.460 fewer people are arriving to see the next bus, the third bus. 00:35:57.090 --> 00:36:02.080 The dwell times of that third bus are shorter than planned. 00:36:02.080 --> 00:36:03.700 Fewer people are waiting for that bus. 00:36:03.700 --> 00:36:06.520 And therefore, that bus has a lower probability 00:36:06.520 --> 00:36:07.710 of having to stop. 00:36:07.710 --> 00:36:10.300 And if it stops, fewer people will board it. 00:36:10.300 --> 00:36:12.510 So that bus is going to run fast. 00:36:12.510 --> 00:36:16.374 Even if the drivers are now driving at the same speed, 00:36:16.374 --> 00:36:17.540 there is nothing you can do. 00:36:17.540 --> 00:36:18.837 They will bunch. 00:36:18.837 --> 00:36:21.170 Well, there is something you can-- you can control them. 00:36:21.170 --> 00:36:22.810 And that's a later topic in the course. 00:36:22.810 --> 00:36:27.310 So they pair or they bunch, that's what we're saying. 00:36:27.310 --> 00:36:30.790 OK, so if we think about how if you 00:36:30.790 --> 00:36:33.280 were to survey headways of different points 00:36:33.280 --> 00:36:35.320 along the route, you would see that, 00:36:35.320 --> 00:36:38.450 if you start at a terminal-- 00:36:38.450 --> 00:36:40.670 this is like a probability density function, 00:36:40.670 --> 00:36:41.900 but they're not normalized. 00:36:41.900 --> 00:36:44.800 So don't think about scale. 00:36:44.800 --> 00:36:46.600 So you have headway on the horizontal. 00:36:46.600 --> 00:36:50.600 And the probability density on the vertical. 00:36:50.600 --> 00:36:53.770 And so at the terminal or close to the start of the route, 00:36:53.770 --> 00:36:56.800 you will see something bell-shaped usually. 00:36:59.350 --> 00:37:02.620 It'll be around the scheduled departure time, 00:37:02.620 --> 00:37:05.320 or around the headway. 00:37:05.320 --> 00:37:09.190 And there will be some variability due to drivers 00:37:09.190 --> 00:37:13.210 not being exactly on time when they leave, and supervision 00:37:13.210 --> 00:37:15.220 issues, or whatever it is. 00:37:15.220 --> 00:37:18.650 So maybe the boarding process at the first stop 00:37:18.650 --> 00:37:20.260 introduces some perturbations. 00:37:20.260 --> 00:37:21.990 But it's essentially bell-shaped. 00:37:21.990 --> 00:37:23.930 And it has less variability. 00:37:23.930 --> 00:37:26.290 As you move to the middle of the route, 00:37:26.290 --> 00:37:30.200 you'll see that the bell is being-- 00:37:30.200 --> 00:37:34.930 it's getting fatter and fatter tails, so more variability 00:37:34.930 --> 00:37:37.740 in the headway because of the bunching problem 00:37:37.740 --> 00:37:40.780 that we just described. 00:37:40.780 --> 00:37:43.240 Usually by the time you see bunching, your-- 00:37:43.240 --> 00:37:46.780 which could be at the end, or it could be before the end, 00:37:46.780 --> 00:37:52.750 you'll have this distribution with two spots, 00:37:52.750 --> 00:37:57.190 a lot of vehicles having headway of 0 or very close to 0, 00:37:57.190 --> 00:37:59.290 and a lot of headways longer than that 00:37:59.290 --> 00:38:02.090 by some amount with a lot of variability. 00:38:02.090 --> 00:38:05.470 So these are pairs that are arriving. 00:38:05.470 --> 00:38:09.590 So every time you see a pair arriving, one of them is 0. 00:38:09.590 --> 00:38:11.620 And the other one is some longer headway. 00:38:11.620 --> 00:38:17.524 And they are affecting these two parts of the probability 00:38:17.524 --> 00:38:18.856 distribution. 00:38:18.856 --> 00:38:22.346 Is that understood? 00:38:22.346 --> 00:38:24.292 Any questions? 00:38:24.292 --> 00:38:25.224 [INAUDIBLE] 00:38:25.224 --> 00:38:27.750 AUDIENCE: So the big point you're talking about 00:38:27.750 --> 00:38:29.796 is the lower-- 00:38:29.796 --> 00:38:31.545 GABRIEL SANCHEZ-MARTINEZ: So essentially-- 00:38:31.545 --> 00:38:32.900 AUDIENCE: --the lower curve? 00:38:32.900 --> 00:38:36.200 GABRIEL SANCHEZ-MARTINEZ: --it's just, yes, this one right here. 00:38:36.200 --> 00:38:40.940 So essentially, it's just higher variability of headways. 00:38:40.940 --> 00:38:45.880 So variability of headways tends to be minimized 00:38:45.880 --> 00:38:47.450 at the start of a run. 00:38:47.450 --> 00:38:49.005 And then that degrades. 00:38:49.005 --> 00:38:52.340 It increases as you run the route until you have bunching. 00:38:52.340 --> 00:38:54.560 And when you have bunching, you reach this point 00:38:54.560 --> 00:38:58.430 of having some vehicles that are close to 0 and others 00:38:58.430 --> 00:38:59.810 that are anywhere else. 00:39:02.420 --> 00:39:04.460 So that means the waiting time is 00:39:04.460 --> 00:39:08.610 a function of headway as scheduled, 00:39:08.610 --> 00:39:10.730 also of headway reliability. 00:39:10.730 --> 00:39:13.990 So if you're running a service that is well controlled 00:39:13.990 --> 00:39:17.090 and you have good headways, then you may never reach this point. 00:39:17.090 --> 00:39:18.590 And you may actually have everything 00:39:18.590 --> 00:39:21.530 running somewhat bell-shaped, not too much variance. 00:39:21.530 --> 00:39:24.450 That's your ideal situation. 00:39:24.450 --> 00:39:27.950 And then it's also a function of where you are along the route. 00:39:27.950 --> 00:39:30.554 So if you're closer to a controlled point, which 00:39:30.554 --> 00:39:31.970 could be the terminal, or it could 00:39:31.970 --> 00:39:35.990 be anywhere along the route, if there is control en route, 00:39:35.990 --> 00:39:38.780 then you will have less variability of headways. 00:39:41.620 --> 00:39:45.050 As you run downstream of the last control point, 00:39:45.050 --> 00:39:48.000 then you will see greater variability. 00:39:48.000 --> 00:39:53.510 Great, so what factors affect the headway deterioration? 00:39:53.510 --> 00:39:54.830 Length of route is one. 00:39:54.830 --> 00:39:59.180 So if he no longer route, then it takes an hour and a half 00:39:59.180 --> 00:40:04.340 to cover the whole one-direction run, the half cycle, 00:40:04.340 --> 00:40:07.460 then this deterioration has-- 00:40:07.460 --> 00:40:11.030 that process has more time to act on the route. 00:40:11.030 --> 00:40:16.160 So you will see more headway unreliability, more bunching. 00:40:16.160 --> 00:40:19.440 The marginal dwell time per passengers is another factor. 00:40:19.440 --> 00:40:21.190 If you think of-- 00:40:21.190 --> 00:40:23.270 well, we just saw that the bunching process 00:40:23.270 --> 00:40:27.560 is largely affected by dwell time, 00:40:27.560 --> 00:40:29.150 because if you have a longer headway, 00:40:29.150 --> 00:40:30.191 more people are boarding. 00:40:30.191 --> 00:40:32.030 So if you think of an extreme case 00:40:32.030 --> 00:40:34.650 where everybody boards and alights instantly, 00:40:34.650 --> 00:40:37.010 then you don't have that effect anymore. 00:40:37.010 --> 00:40:40.670 So to the extent that people can board and alight 00:40:40.670 --> 00:40:45.500 fast relative to the time it takes to run the service, 00:40:45.500 --> 00:40:48.350 then this effect diminishes to the extent 00:40:48.350 --> 00:40:50.390 that dwell time is a larger percentage 00:40:50.390 --> 00:40:51.740 of the total runtime. 00:40:51.740 --> 00:40:53.120 Then it increases. 00:40:53.120 --> 00:40:54.650 You have more bunching. 00:40:54.650 --> 00:40:56.942 The stopping probability-- again, 00:40:56.942 --> 00:40:58.400 this has to do with how many people 00:40:58.400 --> 00:41:00.825 are on the bus and the stop spacing. 00:41:00.825 --> 00:41:04.260 If stops are spaced very close to each other, 00:41:04.260 --> 00:41:07.650 then you have a higher variance of where you stop. 00:41:13.880 --> 00:41:16.910 if you have a long distance between stations, 00:41:16.910 --> 00:41:18.890 if you think of a commuter rail line, 00:41:18.890 --> 00:41:20.660 then you're going to stop at every stop. 00:41:20.660 --> 00:41:22.640 And therefore, you decrease the variability 00:41:22.640 --> 00:41:25.280 of this effect of stopping. 00:41:25.280 --> 00:41:29.150 So you have the urban bus route in one hand 00:41:29.150 --> 00:41:32.420 and the commuter rail with long distances between stops, 00:41:32.420 --> 00:41:35.720 and therefore stopping at every stop, on the other hand. 00:41:35.720 --> 00:41:38.030 The schedule headway has an impact. 00:41:38.030 --> 00:41:40.820 If you schedule the headways every 30 minutes, 00:41:40.820 --> 00:41:42.920 it's unlikely you'll see bunching. 00:41:42.920 --> 00:41:46.216 But if you schedule the headways every two minutes, then 00:41:46.216 --> 00:41:47.840 it's very easy for you to see bunching. 00:41:47.840 --> 00:41:49.460 You have to control this very well 00:41:49.460 --> 00:41:52.480 to avoid bunching in that case. 00:41:52.480 --> 00:41:55.590 And driver behavior-- if your vehicles 00:41:55.590 --> 00:42:01.390 are driven by very good drivers or computers 00:42:01.390 --> 00:42:05.770 on the extreme right, driven with driverless trains, 00:42:05.770 --> 00:42:08.170 then you can really sort of try to make up 00:42:08.170 --> 00:42:10.410 for all of these things in driving 00:42:10.410 --> 00:42:12.250 and try to keep things even. 00:42:12.250 --> 00:42:16.150 If they are drivers that are not particularly well trained 00:42:16.150 --> 00:42:19.630 and have not received feedback, for example, on their driving 00:42:19.630 --> 00:42:23.560 speeds and other behavior, and not just that, 00:42:23.560 --> 00:42:26.560 but also being on time at the terminal 00:42:26.560 --> 00:42:29.830 and leaving exactly on time, things like that, 00:42:29.830 --> 00:42:31.360 would all affect this. 00:42:31.360 --> 00:42:35.350 So here is a simple model of how headway will deteriorate. 00:42:35.350 --> 00:42:37.870 So there is also a mistake on the integration. 00:42:37.870 --> 00:42:41.230 That should be p i, not p i minus 1, sorry about that. 00:42:44.330 --> 00:42:48.340 So we have that the headway deviation at some stop-- 00:42:48.340 --> 00:42:50.410 here, we're thinking about a scheduled headway 00:42:50.410 --> 00:42:52.040 and the actual headway we see. 00:42:52.040 --> 00:42:56.530 So if service is running exactly on time, ei is 0. 00:42:56.530 --> 00:43:00.400 If that bus is delayed, then ei is positive. 00:43:00.400 --> 00:43:02.170 And if it's running early, ei is negative. 00:43:02.170 --> 00:43:05.200 So you start out-- 00:43:05.200 --> 00:43:06.790 you go from one stop to the next. 00:43:06.790 --> 00:43:09.440 And the deviation of headway you see at some stop 00:43:09.440 --> 00:43:12.070 is going to be a function of how-- 00:43:12.070 --> 00:43:15.510 what the deviation was at the previous stop. 00:43:15.510 --> 00:43:17.260 So if you're already delayed, your chances 00:43:17.260 --> 00:43:19.720 of being delayed at the next stop are higher. 00:43:19.720 --> 00:43:21.070 That should make sense. 00:43:21.070 --> 00:43:25.510 Then you add the effect of running from that previous stop 00:43:25.510 --> 00:43:26.770 to this stop. 00:43:26.770 --> 00:43:28.990 If you are particularly slow doing that, 00:43:28.990 --> 00:43:33.000 then your delay will increase. 00:43:33.000 --> 00:43:35.800 If you try to drive fast to make up for that, 00:43:35.800 --> 00:43:38.410 then ti will be negative. 00:43:38.410 --> 00:43:43.960 And it will decrease the impact on headway deterioration. 00:43:43.960 --> 00:43:45.084 Yeah? 00:43:45.084 --> 00:43:47.500 AUDIENCE: Do they ever just-- so this seems to imply that, 00:43:47.500 --> 00:43:49.240 like, maybe I could fix my problem 00:43:49.240 --> 00:43:52.080 by just dropping a bus from the schedule. 00:43:52.080 --> 00:43:54.610 Like, let's say I was really behind, 00:43:54.610 --> 00:43:57.400 I could just delay all the other buses 00:43:57.400 --> 00:43:59.276 and, like, take up the schedule of the next-- 00:43:59.276 --> 00:44:01.400 GABRIEL SANCHEZ-MARTINEZ: Yeah, but you will have-- 00:44:01.400 --> 00:44:03.570 you still have a headway, a longer headway, right? 00:44:03.570 --> 00:44:04.030 AUDIENCE: Yeah, [INAUDIBLE]-- 00:44:04.030 --> 00:44:04.750 GABRIEL SANCHEZ-MARTINEZ: So you could make up 00:44:04.750 --> 00:44:06.190 your schedule that way. 00:44:06.190 --> 00:44:11.110 But you're essentially doing an accounting trick by doing that. 00:44:11.110 --> 00:44:13.990 You can make your vehicles look like they're all 00:44:13.990 --> 00:44:16.450 running on time by doing that. 00:44:16.450 --> 00:44:17.840 And you dropped one trip. 00:44:17.840 --> 00:44:20.160 But you still have a longer headway here 00:44:20.160 --> 00:44:23.350 and shorter headways there, and passengers boarding 00:44:23.350 --> 00:44:24.930 mostly that delayed vehicle. 00:44:24.930 --> 00:44:27.280 And so from a passenger's perspective, 00:44:27.280 --> 00:44:30.040 that doesn't do much. 00:44:30.040 --> 00:44:32.909 But then you have to consider the effect of, well, 00:44:32.909 --> 00:44:34.450 you might have-- you might want to do 00:44:34.450 --> 00:44:36.480 that if your drivers need to check out 00:44:36.480 --> 00:44:37.600 at some time in the day. 00:44:37.600 --> 00:44:40.960 And your options are either, you do this now, 00:44:40.960 --> 00:44:45.660 or at some time when they check out, you have to drop trips. 00:44:45.660 --> 00:44:47.680 And that could be during the peak. 00:44:47.680 --> 00:44:49.360 So that would be really bad, right? 00:44:49.360 --> 00:44:53.713 So then you would have to consider that strategy. 00:44:53.713 --> 00:44:55.855 AUDIENCE: And do buses typically post, like, 00:44:55.855 --> 00:44:58.152 a scheduled time that they're supposed 00:44:58.152 --> 00:44:59.110 to arrive at each stop? 00:44:59.110 --> 00:44:59.560 Because normally-- 00:44:59.560 --> 00:44:59.830 GABRIEL SANCHEZ-MARTINEZ: Yes. 00:44:59.830 --> 00:45:01.150 AUDIENCE: --I see, like-- 00:45:01.150 --> 00:45:01.950 GABRIEL SANCHEZ-MARTINEZ: --especially now, 00:45:01.950 --> 00:45:02.590 so typically-- 00:45:02.590 --> 00:45:04.077 AUDIENCE: Like up-to-date minutes. 00:45:04.077 --> 00:45:06.160 GABRIEL SANCHEZ-MARTINEZ: So London will say that. 00:45:06.160 --> 00:45:08.030 So if it's high frequency-- 00:45:08.030 --> 00:45:10.700 and London, the threshold is 12 minutes. 00:45:10.700 --> 00:45:15.760 So anything below, they'll say, AM peak between-- 00:45:15.760 --> 00:45:18.610 expect service every eight minutes, for example. 00:45:18.610 --> 00:45:22.120 And then as soon as the headway increases above that, 00:45:22.120 --> 00:45:23.560 they'll say the times. 00:45:23.560 --> 00:45:27.160 But if you look at service here, for example, here 00:45:27.160 --> 00:45:31.480 in Boston, every bus route here, if you look at the DTFS file, 00:45:31.480 --> 00:45:35.040 has specific times at each stop, even though DTFS 00:45:35.040 --> 00:45:38.260 has the option of giving a headway 00:45:38.260 --> 00:45:42.570 for high-frequency service, they don't do that here. 00:45:42.570 --> 00:45:45.110 AUDIENCE: So only service [INAUDIBLE] time appear? 00:45:45.110 --> 00:45:47.600 GABRIEL SANCHEZ-MARTINEZ: Yeah, but those DTFS file 00:45:47.600 --> 00:45:48.880 has times at every stop. 00:45:51.850 --> 00:45:53.500 So yeah, and that's another thing, 00:45:53.500 --> 00:45:55.250 that if you look at printed schedules that 00:45:55.250 --> 00:45:58.640 are posted somewhere and on paper, 00:45:58.640 --> 00:46:00.140 typically they'll have the terminals 00:46:00.140 --> 00:46:03.119 and some time points in between and not every single stop. 00:46:03.119 --> 00:46:05.410 AUDIENCE: And the time points [INAUDIBLE] approximately 00:46:05.410 --> 00:46:05.910 [INAUDIBLE]? 00:46:05.910 --> 00:46:07.409 GABRIEL SANCHEZ-MARTINEZ: Well, that 00:46:07.409 --> 00:46:09.710 depends on the control policy of that agency. 00:46:12.260 --> 00:46:16.850 So we'll talk more about control in a later lecture. 00:46:16.850 --> 00:46:19.040 So typically, you don't control at every stop. 00:46:19.040 --> 00:46:23.280 You control at, sometimes, just the terminal, and sometimes 00:46:23.280 --> 00:46:26.180 at the terminal and some key points in between. 00:46:26.180 --> 00:46:30.200 And those are timing points for control. 00:46:30.200 --> 00:46:37.760 OK, so back to this model, the headway deviation at some stop 00:46:37.760 --> 00:46:40.460 is, you start out from what it was in the previous stop. 00:46:40.460 --> 00:46:42.140 You add the effect of running time 00:46:42.140 --> 00:46:44.360 to that stop from the previous one. 00:46:44.360 --> 00:46:48.350 And then you add the effect of dwell time, essentially. 00:46:48.350 --> 00:46:51.870 So this pi, not pi minus 1, as I said earlier-- 00:46:51.870 --> 00:46:55.670 and so this is the arrival rate at the stop right now 00:46:55.670 --> 00:46:59.160 and multiplied by the boarding time per passenger. 00:46:59.160 --> 00:47:08.450 So this whole quantity is multiplied by this amount, 00:47:08.450 --> 00:47:10.890 which is the time it takes-- 00:47:10.890 --> 00:47:13.630 it's the deviation of the time it takes to arrive. 00:47:13.630 --> 00:47:16.090 So it's a deviation in headway, essentially. 00:47:16.090 --> 00:47:19.420 So if you're headway is now a minute longer than it used 00:47:19.420 --> 00:47:23.500 to be, then you will have a minute times the boarding 00:47:23.500 --> 00:47:26.860 rate per passenger times the arrival rate of passengers, 00:47:26.860 --> 00:47:30.810 extra people boarding, or extra time, extra time 00:47:30.810 --> 00:47:33.437 of dwell time affecting that bus, 00:47:33.437 --> 00:47:35.020 and therefore slowing it down further. 00:47:39.345 --> 00:47:40.220 Does that make sense? 00:47:40.220 --> 00:47:41.845 Or should I break it down a little bit? 00:47:44.710 --> 00:47:46.710 AUDIENCE: Did not answer that question, but just 00:47:46.710 --> 00:47:47.520 to ask a question? 00:47:47.520 --> 00:47:50.210 AUDIENCE: Yeah, [INAUDIBLE] interrupt anyone else. 00:47:50.210 --> 00:47:52.760 AUDIENCE: So is this accounting for the fact 00:47:52.760 --> 00:47:55.400 that, if there is no one getting on or off this stop, 00:47:55.400 --> 00:47:58.580 the first person to pull into a stop, 00:47:58.580 --> 00:48:01.770 decelerate, and open the door takes a given amount of time. 00:48:01.770 --> 00:48:03.950 And the marginal time to add an extra person 00:48:03.950 --> 00:48:05.832 is quite small, or relatively small. 00:48:05.832 --> 00:48:06.957 Is that accounted for here? 00:48:06.957 --> 00:48:08.623 GABRIEL SANCHEZ-MARTINEZ: No, this model 00:48:08.623 --> 00:48:10.700 is saying that it's a linear effect. 00:48:10.700 --> 00:48:13.100 You're not saying that the first person takes-- 00:48:13.100 --> 00:48:14.930 AUDIENCE: So two would take twice as long as one person. 00:48:14.930 --> 00:48:15.890 GABRIEL SANCHEZ-MARTINEZ: Exactly, yeah, 00:48:15.890 --> 00:48:17.330 so that is a simplification here. 00:48:17.330 --> 00:48:18.967 And it's a point that we will address 00:48:18.967 --> 00:48:20.300 towards the end of this lecture. 00:48:20.300 --> 00:48:21.980 AUDIENCE: OK, [INAUDIBLE] asking questions. 00:48:21.980 --> 00:48:23.270 GABRIEL SANCHEZ-MARTINEZ: Good segues, 00:48:23.270 --> 00:48:24.728 that's, like, the second one today. 00:48:24.728 --> 00:48:26.687 AUDIENCE: This is only extra [INAUDIBLE] right? 00:48:26.687 --> 00:48:28.478 GABRIEL SANCHEZ-MARTINEZ: So yeah, exactly, 00:48:28.478 --> 00:48:30.390 because this is a model of headway deviation. 00:48:30.390 --> 00:48:33.230 So you have this passenger arrival rate and the boarding 00:48:33.230 --> 00:48:35.647 time, but because you're only multiplying it-- 00:48:35.647 --> 00:48:37.730 you're not multiplying it times the whole headway. 00:48:37.730 --> 00:48:40.490 You're multiplying times the headway deviation, which 00:48:40.490 --> 00:48:45.210 is the deviation at this stop when you arrive and this stop. 00:48:45.210 --> 00:48:47.960 It's the deviation when you were arriving 00:48:47.960 --> 00:48:51.380 at the previous stop plus the time it took you to reach 00:48:51.380 --> 00:48:53.030 this stop from that time. 00:48:53.030 --> 00:48:55.730 So now you have adjusted-- now you have the deviation arriving 00:48:55.730 --> 00:48:57.680 at this stop. 00:48:57.680 --> 00:48:59.150 If your headway is a minute longer, 00:48:59.150 --> 00:49:02.660 than you need to add however much extra dwell time 00:49:02.660 --> 00:49:05.155 you have to pick up that extra minute of passengers. 00:49:05.155 --> 00:49:06.780 An that's what that last case is doing. 00:49:11.589 --> 00:49:13.600 AUDIENCE: So at the beginning, the e 00:49:13.600 --> 00:49:15.370 will be 0 at the first stop? 00:49:15.370 --> 00:49:16.870 GABRIEL SANCHEZ-MARTINEZ: Hopefully. 00:49:19.354 --> 00:49:20.770 It depends on your control policy. 00:49:20.770 --> 00:49:25.480 So if you depart your terminal and your driver didn't show up 00:49:25.480 --> 00:49:29.500 on time and they left a minute late from the even headway, 00:49:29.500 --> 00:49:33.350 then you start out with one minute deviation. 00:49:33.350 --> 00:49:36.620 So yeah, hopefully that is 0 at the beginning. 00:49:36.620 --> 00:49:41.030 And then due to effect that we described here, 00:49:41.030 --> 00:49:43.040 it starts deteriorating. 00:49:43.040 --> 00:49:45.230 And this is a mathematical model to-- 00:49:45.230 --> 00:49:49.220 a simple mathematical model to account for that. 00:49:49.220 --> 00:49:50.750 So it's still a deterministic one. 00:49:50.750 --> 00:49:52.580 Here is a probabilistic one. 00:49:52.580 --> 00:49:56.569 The details for this formulation are on this paper. 00:49:56.569 --> 00:49:57.985 If you're interested, let me know. 00:49:57.985 --> 00:50:00.050 I will send it you. 00:50:00.050 --> 00:50:04.260 What I want to highlight here are the quantities. 00:50:04.260 --> 00:50:07.460 So this is a model of headway variance. 00:50:07.460 --> 00:50:09.490 So we are looking at, now, headway 00:50:09.490 --> 00:50:11.360 as a stochastic quantity. 00:50:11.360 --> 00:50:14.870 And we're calculating the variance of headway 00:50:14.870 --> 00:50:16.550 at some stop. 00:50:16.550 --> 00:50:18.680 And again, we see the same pattern. 00:50:18.680 --> 00:50:23.660 We see that it depends on what it was at the previous stop. 00:50:23.660 --> 00:50:26.790 And then we add the effect of the running time, 00:50:26.790 --> 00:50:30.170 so the variance of running times between consecutive stops, 00:50:30.170 --> 00:50:33.920 to the extent that the drivers are different in driving, 00:50:33.920 --> 00:50:36.350 then this variance increases. 00:50:36.350 --> 00:50:38.990 And then you have these two terms-- 00:50:38.990 --> 00:50:42.800 really, three terms, that have to do with dwell time. 00:50:42.800 --> 00:50:44.180 All of these terms have-- 00:50:44.180 --> 00:50:48.680 the q here is the mean number of passengers per bus served. 00:50:48.680 --> 00:50:51.860 And c is the marginal dwell time per passenger-- so different 00:50:51.860 --> 00:50:56.110 notation from the model that we just saw, but same quantities. 00:50:56.110 --> 00:51:03.470 So we have something times that probability is-- 00:51:03.470 --> 00:51:04.970 the bus will skip a stop, et cetera. 00:51:04.970 --> 00:51:08.590 So all these things together are accounting for the dwell time 00:51:08.590 --> 00:51:09.500 effect. 00:51:09.500 --> 00:51:13.580 So the last term here is, again, looking 00:51:13.580 --> 00:51:17.750 at c, which is the marginal dwell time per passenger. 00:51:17.750 --> 00:51:19.700 So this also has to do with dwell time. 00:51:19.700 --> 00:51:24.740 But this one is also multiplied by the covariance with headway. 00:51:24.740 --> 00:51:29.200 So this is that effect of the relationship between headway 00:51:29.200 --> 00:51:32.590 and sort of passenger arrivals, captured here 00:51:32.590 --> 00:51:35.850 as the covariance between those two quantities. 00:51:35.850 --> 00:51:38.720 So the key takeaway from this slide 00:51:38.720 --> 00:51:41.930 is not exactly why this is the right equation. 00:51:41.930 --> 00:51:46.120 If you're interested in that, read the paper. 00:51:46.120 --> 00:51:49.340 I am interested in understanding what 00:51:49.340 --> 00:51:53.810 goes into this equation in terms of what components 00:51:53.810 --> 00:51:56.010 lead to a higher variance at some particular stop. 00:51:59.600 --> 00:52:04.310 OK, we are ready to move to vehicle running time models 00:52:04.310 --> 00:52:06.140 if no-- if there are no questions 00:52:06.140 --> 00:52:09.280 on waiting time headway models. 00:52:09.280 --> 00:52:11.540 OK, so let's do that. 00:52:11.540 --> 00:52:13.100 So different levels of detail-- 00:52:13.100 --> 00:52:15.890 we have some models that are very detailed. 00:52:15.890 --> 00:52:17.390 There are microscopic model. 00:52:20.250 --> 00:52:21.140 It's a simulation. 00:52:21.140 --> 00:52:24.137 And they look at the vehicle motion, the interaction 00:52:24.137 --> 00:52:24.970 with other vehicles. 00:52:24.970 --> 00:52:27.590 So you might have private automobiles interacting 00:52:27.590 --> 00:52:30.505 with a bus, for example, and traffic blocking the bus. 00:52:30.505 --> 00:52:31.380 And you have signals. 00:52:31.380 --> 00:52:32.796 So every little detail is modeled. 00:52:32.796 --> 00:52:34.580 That's a microscopic model. 00:52:34.580 --> 00:52:37.250 On the other hand, you have macroscopic models, 00:52:37.250 --> 00:52:39.680 which are not looking at all those details. 00:52:39.680 --> 00:52:42.680 Instead, they're saying, what are the running times that I 00:52:42.680 --> 00:52:44.630 observe as a function of time of day, 00:52:44.630 --> 00:52:49.100 and the driver, different components. 00:52:49.100 --> 00:52:51.450 And macroscopically, we say, this is the running time. 00:52:51.450 --> 00:52:55.479 So there is something in between called mesoscopic, when 00:52:55.479 --> 00:52:57.020 you have some parts that are detailed 00:52:57.020 --> 00:52:58.580 and others that are not-- 00:52:58.580 --> 00:53:02.180 so different levels of modeling. 00:53:05.240 --> 00:53:07.100 So running time, as we know, includes 00:53:07.100 --> 00:53:11.990 dwell time, the movement time between stops, and any delays. 00:53:11.990 --> 00:53:14.930 Delays could be because of signals, traffic signals, 00:53:14.930 --> 00:53:15.920 for example. 00:53:15.920 --> 00:53:18.470 So dwell time is a function of the number 00:53:18.470 --> 00:53:20.390 of passengers boarding and arriving 00:53:20.390 --> 00:53:23.030 as well as technology characteristics. 00:53:23.030 --> 00:53:25.910 What are some examples of technology characteristics 00:53:25.910 --> 00:53:28.679 that could affect dwell times? 00:53:28.679 --> 00:53:31.730 AUDIENCE: [INAUDIBLE] cards, smart card. 00:53:31.730 --> 00:53:34.230 GABRIEL SANCHEZ-MARTINEZ: So what your fare card technology, 00:53:34.230 --> 00:53:37.260 so if you have smart cards, or coins, paying cash-- 00:53:37.260 --> 00:53:38.531 very different, right? 00:53:38.531 --> 00:53:39.030 What else? 00:53:39.030 --> 00:53:40.610 AUDIENCE: Off-board fare collection. 00:53:40.610 --> 00:53:41.820 GABRIEL SANCHEZ-MARTINEZ: Off-board fare collection-- 00:53:41.820 --> 00:53:44.130 so if you remove the payment from the vehicle, 00:53:44.130 --> 00:53:46.720 then that really helps with decreasing the variability. 00:53:46.720 --> 00:53:47.270 What else? 00:53:47.270 --> 00:53:49.020 AUDIENCE: All boarding. 00:53:49.020 --> 00:53:50.570 GABRIEL SANCHEZ-MARTINEZ: Sorry? 00:53:50.570 --> 00:53:51.590 AUDIENCE: All boarding? 00:53:51.590 --> 00:53:52.470 GABRIEL SANCHEZ-MARTINEZ: All-door boarding? 00:53:52.470 --> 00:53:53.060 AUDIENCE: All-door boarding. 00:53:53.060 --> 00:53:53.680 GABRIEL SANCHEZ-MARTINEZ: So boarding 00:53:53.680 --> 00:53:55.330 through all doors-- of course, if you 00:53:55.330 --> 00:53:56.830 can board through multiple doors, 00:53:56.830 --> 00:53:59.990 you have a faster dwell time process in this variable. 00:53:59.990 --> 00:54:00.610 What else? 00:54:00.610 --> 00:54:02.940 AUDIENCE: If the station level is the same as the-- 00:54:02.940 --> 00:54:04.690 GABRIEL SANCHEZ-MARTINEZ: Level boarding-- 00:54:04.690 --> 00:54:07.310 so if people don't have to climb steps to go from the curb 00:54:07.310 --> 00:54:11.940 to the vehicle, that also helps. 00:54:11.940 --> 00:54:12.830 Any other ideas? 00:54:12.830 --> 00:54:13.370 Eli? 00:54:13.370 --> 00:54:14.185 AUDIENCE: The type of the bus stop-- 00:54:14.185 --> 00:54:16.460 am I pulling into a cutout and then I have to wait for traffic 00:54:16.460 --> 00:54:17.090 to merge back in. 00:54:17.090 --> 00:54:17.590 GABRIEL SANCHEZ-MARTINEZ: Yes. 00:54:17.590 --> 00:54:18.740 AUDIENCE: [INAUDIBLE] stay in the lane [INAUDIBLE]---- 00:54:18.740 --> 00:54:19.160 GABRIEL SANCHEZ-MARTINEZ: And that 00:54:19.160 --> 00:54:21.110 depends on the definition of dwell time. 00:54:21.110 --> 00:54:23.870 So if you say dwell time is only the amount of time needed 00:54:23.870 --> 00:54:27.254 to serve passengers, then that shouldn't really be a factor. 00:54:27.254 --> 00:54:29.420 But if you think of the dwell time as the whole time 00:54:29.420 --> 00:54:32.120 it took me to stop and serve that stop, 00:54:32.120 --> 00:54:33.650 then this would be included. 00:54:33.650 --> 00:54:36.672 So you're looking at-- 00:54:36.672 --> 00:54:46.520 let's see if I can get at this, clean a section here. 00:54:53.540 --> 00:54:58.870 So one example is a bus bay here. 00:54:58.870 --> 00:55:01.640 So this is a bus stop right here. 00:55:01.640 --> 00:55:05.640 And maybe this is an intersection. 00:55:05.640 --> 00:55:09.710 So the bus comes in here. 00:55:09.710 --> 00:55:12.000 And it's waiting there. 00:55:12.000 --> 00:55:16.440 But if there is traffic and there are cars right here, 00:55:16.440 --> 00:55:19.996 after that bus is ready, it may have to-- 00:55:19.996 --> 00:55:21.870 well, if it's right of the signal, it's fine. 00:55:21.870 --> 00:55:28.050 But if this now merges and there is traffic here, 00:55:28.050 --> 00:55:30.000 then that bus might be sort of stuck enough 00:55:30.000 --> 00:55:34.050 to maneuver its way back into the traffic flow. 00:55:34.050 --> 00:55:39.240 So a [INAUDIBLE] sign or stop sign have an impact. 00:55:39.240 --> 00:55:41.660 What else? 00:55:41.660 --> 00:55:42.684 Henry? 00:55:42.684 --> 00:55:44.350 AUDIENCE: If you're serving a line where 00:55:44.350 --> 00:55:46.016 there are a lot of people who are, like, 00:55:46.016 --> 00:55:48.090 tourists and people, like, ask questions, 00:55:48.090 --> 00:55:49.979 it could take forever to get on to the bus. 00:55:49.979 --> 00:55:52.020 GABRIEL SANCHEZ-MARTINEZ: That's not a technology 00:55:52.020 --> 00:55:55.290 characteristic, but it is a valid factor 00:55:55.290 --> 00:55:56.827 that affects dwell times. 00:55:56.827 --> 00:55:58.785 Or maybe if your technology is very complicated 00:55:58.785 --> 00:56:00.790 and people have a lot of questions, 00:56:00.790 --> 00:56:01.935 that could be a way to-- 00:56:01.935 --> 00:56:03.560 AUDIENCE: Or maybe a lack of technology 00:56:03.560 --> 00:56:05.100 where, like, you don't have-- 00:56:05.100 --> 00:56:07.860 GABRIEL SANCHEZ-MARTINEZ: So OK, I think we gave good examples. 00:56:07.860 --> 00:56:09.750 If we look at the typical bus running time 00:56:09.750 --> 00:56:11.540 and how we break it down, this is 00:56:11.540 --> 00:56:14.670 a typical bus in mixed traffic. 00:56:14.670 --> 00:56:16.320 Somewhere between a 1/2 and 3/4 will 00:56:16.320 --> 00:56:18.780 be spent moving between stops. 00:56:18.780 --> 00:56:20.520 Between 10 and a 1/4-- 00:56:20.520 --> 00:56:24.735 10% and 1/4 will be spent at stops, serving the stop. 00:56:24.735 --> 00:56:27.930 And between 10% and 1/4 will be served 00:56:27.930 --> 00:56:32.370 in traffic or waiting for a signal, so at 00:56:32.370 --> 00:56:34.100 a red light, essentially. 00:56:34.100 --> 00:56:34.600 Yeah? 00:56:34.600 --> 00:56:37.835 AUDIENCE: [INAUDIBLE] about this a bit, but in movement time, 00:56:37.835 --> 00:56:40.210 would that include all the time it takes for me to, like, 00:56:40.210 --> 00:56:44.235 stop the bus? 00:56:44.235 --> 00:56:46.860 GABRIEL SANCHEZ-MARTINEZ: Again, it depends on your definition. 00:56:46.860 --> 00:56:51.300 But yes, I think in this break down, yes. 00:56:51.300 --> 00:56:52.908 So the slowing down and accelerating 00:56:52.908 --> 00:56:53.991 is sort of in there, yeah. 00:56:57.360 --> 00:57:00.010 Any other questions? 00:57:00.010 --> 00:57:05.387 OK, so let's look at some dwell time models. 00:57:05.387 --> 00:57:07.470 Dwell times models are a component of running time 00:57:07.470 --> 00:57:12.830 models, because dwell times are one of the key pieces 00:57:12.830 --> 00:57:13.830 of running times. 00:57:13.830 --> 00:57:17.550 And actually, another point about these, movement time, 00:57:17.550 --> 00:57:20.550 dwell time, and delay, which of those three 00:57:20.550 --> 00:57:25.435 do you think the agency has most control over? 00:57:25.435 --> 00:57:26.310 AUDIENCE: Dwell time. 00:57:26.310 --> 00:57:28.143 GABRIEL SANCHEZ-MARTINEZ: Dwell time, right? 00:57:28.143 --> 00:57:29.245 Why? 00:57:29.245 --> 00:57:32.830 Can an agency usually diminish traffic or do something 00:57:32.830 --> 00:57:35.150 about traffic signals? 00:57:35.150 --> 00:57:37.840 There are some things you can do, but usually not. 00:57:37.840 --> 00:57:38.860 It's harder. 00:57:38.860 --> 00:57:42.640 And movement time, that has to do with traffic and speed 00:57:42.640 --> 00:57:46.970 limits, and the vehicle itself, and driver behavior. 00:57:46.970 --> 00:57:49.270 So that's a little bit harder to change. 00:57:49.270 --> 00:57:52.510 So dwell times are probably the one thing 00:57:52.510 --> 00:57:55.054 here that an agency will target. 00:57:55.054 --> 00:57:56.470 Some part of it, you can't change, 00:57:56.470 --> 00:57:58.886 because it has to do with how many people are at the stop. 00:57:58.886 --> 00:58:00.310 But you could increase frequency. 00:58:00.310 --> 00:58:01.640 You could change the bus assigned. 00:58:01.640 --> 00:58:02.764 We gave a bunch of factors. 00:58:02.764 --> 00:58:04.780 And a lot of those are in control of the agency, 00:58:04.780 --> 00:58:07.070 so under the control of the agency. 00:58:07.070 --> 00:58:10.010 So let's go to the dwell time models for that reason. 00:58:10.010 --> 00:58:17.050 There are some examples here, three papers, one on bus dwell 00:58:17.050 --> 00:58:21.460 time, one on light rail dwell times for the Green line, 00:58:21.460 --> 00:58:23.320 and one on heavy rail. 00:58:23.320 --> 00:58:25.600 So we're going to look at the three of them, 00:58:25.600 --> 00:58:29.450 just a high-level overview. 00:58:29.450 --> 00:58:31.490 So let's look at some concepts first. 00:58:31.490 --> 00:58:35.380 Vehicle dwell times affect system performance. 00:58:35.380 --> 00:58:38.000 We've discussed why and how. 00:58:38.000 --> 00:58:39.620 And they affect service quality. 00:58:39.620 --> 00:58:42.240 We've also discussed why and how. 00:58:42.240 --> 00:58:44.710 They are a critical element in vehicle bunching. 00:58:44.710 --> 00:58:46.570 I think that has also been covered. 00:58:46.570 --> 00:58:50.440 So they result in high headway variability, high passenger 00:58:50.440 --> 00:58:52.630 waiting times, and uneven passenger loads. 00:58:52.630 --> 00:58:54.640 That last point, we have mentioned, 00:58:54.640 --> 00:58:57.670 but it was indirectly-- 00:58:57.670 --> 00:58:59.530 the point has been made indirectly, right? 00:58:59.530 --> 00:59:01.660 If more people are arriving at some stop, 00:59:01.660 --> 00:59:03.400 at some bus that has a longer headway, 00:59:03.400 --> 00:59:05.230 not only did those people wait longer, 00:59:05.230 --> 00:59:07.780 but they are boarding a more crowded bus. 00:59:07.780 --> 00:59:09.970 So their experience in the vehicle 00:59:09.970 --> 00:59:11.690 is also going to be diminished. 00:59:11.690 --> 00:59:14.870 It's going to be a lower-quality experience. 00:59:14.870 --> 00:59:18.880 So we've covered that. 00:59:18.880 --> 00:59:22.130 Dwell time impact on performance depends on substation spacing, 00:59:22.130 --> 00:59:24.720 the mean dwell as a proportion of trip time, 00:59:24.720 --> 00:59:27.830 the mean headway, and operations control procedures. 00:59:27.830 --> 00:59:32.180 We have touched upon, not necessarily what can be done, 00:59:32.180 --> 00:59:35.510 but we know that there are some things you can do to diminish 00:59:35.510 --> 00:59:36.590 the headway variability. 00:59:40.600 --> 00:59:41.960 Any examples of that, actually? 00:59:41.960 --> 00:59:45.119 Anybody have suggestions on what can be done? 00:59:45.119 --> 00:59:45.985 Eli? 00:59:45.985 --> 00:59:49.630 AUDIENCE: You tell the bus that is trailing to, like, 00:59:49.630 --> 00:59:50.650 wait longer. 00:59:50.650 --> 00:59:52.108 GABRIEL SANCHEZ-MARTINEZ: Yeah, you 00:59:52.108 --> 00:59:54.334 try to slow down the buses that are running fast. 00:59:54.334 --> 00:59:56.000 So there's different ways of doing that. 00:59:56.000 --> 00:59:58.120 You can tell the driver to drive slowly. 00:59:58.120 --> 01:00:01.090 You can hold buses at stops for control points. 01:00:01.090 --> 01:00:02.710 So you tell them, do not depart. 01:00:02.710 --> 01:00:04.240 You have to wait a minute before you depart, 01:00:04.240 --> 01:00:05.935 because you're running too fast, and you're 01:00:05.935 --> 01:00:07.135 going to catch up to the previous bus. 01:00:07.135 --> 01:00:08.290 So that's called holding. 01:00:08.290 --> 01:00:10.060 So there has been a lot of research and holding 01:00:10.060 --> 01:00:10.600 strategies. 01:00:10.600 --> 01:00:12.740 AUDIENCE: Wouldn't the more typical thing 01:00:12.740 --> 01:00:13.990 would be telling them to hold? 01:00:13.990 --> 01:00:14.590 GABRIEL SANCHEZ-MARTINEZ: Yes. 01:00:14.590 --> 01:00:15.730 AUDIENCE: It seems weird to ask someone 01:00:15.730 --> 01:00:17.020 to slow down in traffic. 01:00:17.020 --> 01:00:18.485 GABRIEL SANCHEZ-MARTINEZ: Eh, it's been done. 01:00:18.485 --> 01:00:18.930 AUDIENCE: Really? 01:00:18.930 --> 01:00:19.820 GABRIEL SANCHEZ-MARTINEZ: Yeah. 01:00:19.820 --> 01:00:21.320 AUDIENCE: I wouldn't like to be that bus driver. 01:00:21.320 --> 01:00:24.220 GABRIEL SANCHEZ-MARTINEZ: You would kind of surreptitiously 01:00:24.220 --> 01:00:25.780 and only a little bit. 01:00:28.910 --> 01:00:29.785 AUDIENCE: [INAUDIBLE] 01:00:29.785 --> 01:00:32.243 GABRIEL SANCHEZ-MARTINEZ: And people don't like being held. 01:00:32.243 --> 01:00:34.905 So driving a little bit slow is less obvious to the passengers. 01:00:34.905 --> 01:00:36.223 AUDIENCE: [INAUDIBLE] 01:00:38.342 --> 01:00:40.887 AUDIENCE: Yes, you have a bus hold just a few seconds 01:00:40.887 --> 01:00:41.845 when the light changes. 01:00:41.845 --> 01:00:42.345 [INAUDIBLE] 01:00:42.345 --> 01:00:43.750 GABRIEL SANCHEZ-MARTINEZ: Yeah. 01:00:43.750 --> 01:00:45.166 AUDIENCE: I could really get you-- 01:00:45.166 --> 01:00:48.020 AUDIENCE: So that's sort of the best hold strategy. 01:00:48.020 --> 01:00:50.327 Oh, we just missed it. 01:00:50.327 --> 01:00:51.910 GABRIEL SANCHEZ-MARTINEZ: OK, so there 01:00:51.910 --> 01:00:53.410 are operations controlled procedures 01:00:53.410 --> 01:00:55.100 that can be used here. 01:00:55.100 --> 01:00:57.790 So some examples based on these factors, on the one 01:00:57.790 --> 01:00:58.960 hand you have commuter rail. 01:00:58.960 --> 01:01:01.000 We gave that example earlier. 01:01:01.000 --> 01:01:04.210 Little impact of dwell time on performance-- that makes sense. 01:01:04.210 --> 01:01:06.160 Commuter rail has long distance between stops. 01:01:06.160 --> 01:01:07.630 It stops at every stop. 01:01:07.630 --> 01:01:11.380 And most of the time is spent in movement. 01:01:11.380 --> 01:01:15.560 So the percentage of time spent dwelling is small. 01:01:15.560 --> 01:01:18.050 On the other hand, you have a very long, high-frequency bus 01:01:18.050 --> 01:01:18.850 route. 01:01:18.850 --> 01:01:22.420 So the likelihood of bunching here is really high. 01:01:22.420 --> 01:01:24.629 And it's hard to control this. 01:01:24.629 --> 01:01:26.920 AUDIENCE: Especially, also the commuter rail, also it's 01:01:26.920 --> 01:01:28.720 like a scheduled dwell. 01:01:28.720 --> 01:01:30.220 GABRIEL SANCHEZ-MARTINEZ: Yeah, it's 01:01:30.220 --> 01:01:31.360 a little longer than it has to be. 01:01:31.360 --> 01:01:31.815 AUDIENCE: [INAUDIBLE] hold. 01:01:31.815 --> 01:01:33.430 Then you don't actually-- you have 01:01:33.430 --> 01:01:35.270 a few seconds at the end where no one is boarding or arriving. 01:01:35.270 --> 01:01:36.160 GABRIEL SANCHEZ-MARTINEZ: Yeah, because you 01:01:36.160 --> 01:01:37.990 might have printed the schedule it has 01:01:37.990 --> 01:01:41.470 departure times for every stop. 01:01:41.470 --> 01:01:44.020 AUDIENCE: I would argue that commuter rail has 01:01:44.020 --> 01:01:46.750 sort of the most potential, especially in a transit system 01:01:46.750 --> 01:01:48.835 like Boston where you have-- 01:01:48.835 --> 01:01:51.460 well, the line acceleration, if you get the acceleration better 01:01:51.460 --> 01:01:53.110 through different technology, and when 01:01:53.110 --> 01:01:54.460 you don't have high-level platforms, 01:01:54.460 --> 01:01:56.251 the dwell times get really long when you're 01:01:56.251 --> 01:01:57.332 boarding a lot of people. 01:01:57.332 --> 01:01:59.040 GABRIEL SANCHEZ-MARTINEZ: But you still-- 01:01:59.040 --> 01:02:01.498 AUDIENCE: Plus then you could pull the schedule down if you 01:02:01.498 --> 01:02:02.820 were [INAUDIBLE] dwell time. 01:02:02.820 --> 01:02:05.486 GABRIEL SANCHEZ-MARTINEZ: So you could, if your schedule is off, 01:02:05.486 --> 01:02:08.110 that could be an option. 01:02:08.110 --> 01:02:11.140 Even if your schedule is good on a high-frequency long urban bus 01:02:11.140 --> 01:02:13.600 tour line, it's not going to help. 01:02:13.600 --> 01:02:14.920 You still have to control it. 01:02:14.920 --> 01:02:15.503 And it's hard. 01:02:15.503 --> 01:02:18.250 And you're going to get bunching. 01:02:18.250 --> 01:02:20.202 So dwell time depends on many factors. 01:02:20.202 --> 01:02:21.160 Some of them are human. 01:02:21.160 --> 01:02:22.618 Some of them have to do with modes, 01:02:22.618 --> 01:02:25.160 as we just saw, operating policies and practices, 01:02:25.160 --> 01:02:26.920 weather, all these things. 01:02:26.920 --> 01:02:35.410 So here is a list of models of dwell times for trains. 01:02:35.410 --> 01:02:39.590 So if we look at a single door situation, 01:02:39.590 --> 01:02:41.910 so people have to board and alight from the same door 01:02:41.910 --> 01:02:44.540 and there is no congestion of passengers, 01:02:44.540 --> 01:02:47.110 then this is the simplest model one could have. 01:02:47.110 --> 01:02:50.782 Dwell time is some constant a, which 01:02:50.782 --> 01:02:52.240 has to do with how long it takes me 01:02:52.240 --> 01:02:54.880 to come to a full stop and open doors, 01:02:54.880 --> 01:02:57.070 plus some amount times the number of people 01:02:57.070 --> 01:02:59.980 getting on plus some amount times the number of people 01:02:59.980 --> 01:03:01.490 getting off. 01:03:01.490 --> 01:03:02.897 Does that make sense? 01:03:02.897 --> 01:03:04.480 You sort of measure the average number 01:03:04.480 --> 01:03:06.580 of seconds per passenger alighting, 01:03:06.580 --> 01:03:09.600 the average number seconds per passenger boarding, 01:03:09.600 --> 01:03:10.790 and some constant. 01:03:10.790 --> 01:03:11.920 You run a regression model. 01:03:11.920 --> 01:03:13.260 That's what you get. 01:03:13.260 --> 01:03:17.890 OK, now what if you notice that, when the train is packed, 01:03:17.890 --> 01:03:21.130 people take a lot longer to get off and on? 01:03:21.130 --> 01:03:24.070 Then you want to add the effect of interference, 01:03:24.070 --> 01:03:26.042 or congestion, or friction. 01:03:26.042 --> 01:03:27.250 Some people call it friction. 01:03:27.250 --> 01:03:32.270 So one way of doing that is to multiply-- 01:03:32.270 --> 01:03:36.020 so we add a term, the passenger friction term. 01:03:36.020 --> 01:03:37.990 We multiply the number of people that 01:03:37.990 --> 01:03:41.230 were getting on and off times the number of standees 01:03:41.230 --> 01:03:41.970 in the vehicle. 01:03:41.970 --> 01:03:43.360 STD here stands for standees. 01:03:47.460 --> 01:03:49.210 So the number of people getting on and off 01:03:49.210 --> 01:03:53.830 are the people that would have moved faster if there 01:03:53.830 --> 01:03:55.440 hadn't been any standees. 01:03:55.440 --> 01:03:58.570 And to the extent that there are many standees, then 01:03:58.570 --> 01:04:01.600 they encounter friction as they are getting on or off. 01:04:01.600 --> 01:04:03.740 And that slows the vehicle down. 01:04:03.740 --> 01:04:06.130 So that's one way of taking care of this. 01:04:06.130 --> 01:04:09.400 If that car has m doors, then you 01:04:09.400 --> 01:04:13.780 could run a model for every door and then pick 01:04:13.780 --> 01:04:16.390 the door that was slowest. 01:04:16.390 --> 01:04:18.220 So that's that model. 01:04:18.220 --> 01:04:22.150 And if you say that, actually, people are more or less 01:04:22.150 --> 01:04:24.340 evenly distributed inside the vehicle 01:04:24.340 --> 01:04:26.810 and evenly distributed on the platform, 01:04:26.810 --> 01:04:29.674 so if we are willing to assume that, 01:04:29.674 --> 01:04:31.840 then we can just take the number of doors and divide 01:04:31.840 --> 01:04:34.960 by number of doors, essentially. 01:04:34.960 --> 01:04:38.020 So we're back to the previous model, 01:04:38.020 --> 01:04:40.000 but now we're multiplying-- or dividing 01:04:40.000 --> 01:04:44.656 by m, by the number of doors, because we have maybe 1,000 01:04:44.656 --> 01:04:47.830 passengers or many hundreds of passengers, 01:04:47.830 --> 01:04:50.220 but we have to divide by the number of doors 01:04:50.220 --> 01:04:52.043 to bring them down to passengers per door. 01:04:55.110 --> 01:05:03.020 If that train has several cars, then you 01:05:03.020 --> 01:05:06.140 have a model for each car in the train. 01:05:06.140 --> 01:05:11.010 And you take the maximum, so similar concept. 01:05:11.010 --> 01:05:13.080 We were looking before at doors in one car. 01:05:13.080 --> 01:05:15.580 Now we're looking at cars in one train. 01:05:15.580 --> 01:05:18.170 And if that is our balance flows, 01:05:18.170 --> 01:05:20.240 again, we can now divide by number 01:05:20.240 --> 01:05:23.360 of doors per car and number of cars per train. 01:05:23.360 --> 01:05:28.290 So we normalize the demand by the size of the train, 01:05:28.290 --> 01:05:29.840 the number of doors. 01:05:29.840 --> 01:05:31.070 And we include friction. 01:05:31.070 --> 01:05:35.570 So here is some examples, some ideas for your problem set, 01:05:35.570 --> 01:05:37.430 if you want. 01:05:37.430 --> 01:05:38.480 Some of them may apply. 01:05:38.480 --> 01:05:39.390 Others may not. 01:05:39.390 --> 01:05:44.330 So this is the kind of thing you could think of. 01:05:44.330 --> 01:05:47.810 So now we want to look at this study 01:05:47.810 --> 01:05:50.930 by Milkovits published in 2008. 01:05:50.930 --> 01:05:53.780 At the time, there had been some studies 01:05:53.780 --> 01:05:55.190 with manually collected data. 01:05:55.190 --> 01:05:59.870 So there was very limited data on infrequent events. 01:05:59.870 --> 01:06:04.320 There was very limited data on crowding. 01:06:04.320 --> 01:06:08.750 They were still using, I think, the token system. 01:06:08.750 --> 01:06:12.560 So the previous study was based on the token system, 01:06:12.560 --> 01:06:15.600 so it wasn't updated on new fare technology. 01:06:15.600 --> 01:06:18.380 So there was another study that did have automatically 01:06:18.380 --> 01:06:20.780 collected data, a little more recent, 01:06:20.780 --> 01:06:23.300 but they hadn't taken into account 01:06:23.300 --> 01:06:26.010 the effect of payment type. 01:06:26.010 --> 01:06:29.080 So the AFC system does tell you, this is a pass, 01:06:29.080 --> 01:06:34.040 or this is a ticket, but they hadn't-- they had ignored that 01:06:34.040 --> 01:06:35.360 variable from the model. 01:06:35.360 --> 01:06:38.220 And the fit of the model was poor. 01:06:38.220 --> 01:06:40.220 And then we have a transit capacity and quality 01:06:40.220 --> 01:06:42.800 of service manual, which says, assume half-second penalty 01:06:42.800 --> 01:06:46.596 per passenger for crowding, so rule of thumb. 01:06:46.596 --> 01:06:48.470 So there was interest in developing something 01:06:48.470 --> 01:06:49.570 more sophisticated. 01:06:49.570 --> 01:06:51.950 And they looked at buses and CTA for that. 01:06:51.950 --> 01:07:00.780 So they looked at these factors, so boarding, alighting 01:07:00.780 --> 01:07:04.930 passengers, counting them, onboard passengers, so load, 01:07:04.930 --> 01:07:08.970 the fare media type, the alighting door selection, 01:07:08.970 --> 01:07:10.640 so whether people boarded-- 01:07:10.640 --> 01:07:13.560 alighted from the front door or from the back door, which has 01:07:13.560 --> 01:07:15.900 an impact, and the bus type. 01:07:15.900 --> 01:07:17.550 There were several bus designs. 01:07:17.550 --> 01:07:21.420 And some of them were-- had wider doors and fare boxes 01:07:21.420 --> 01:07:23.710 placed more optimally. 01:07:23.710 --> 01:07:26.700 So all of that was taken into account. 01:07:26.700 --> 01:07:33.030 And yeah, so data from the CTA in Chicago-- 01:07:33.030 --> 01:07:36.600 they didn't consider timing points, 01:07:36.600 --> 01:07:38.850 because at timing points, buses can be held. 01:07:38.850 --> 01:07:42.230 So that could be erroneously included as dwell time, 01:07:42.230 --> 01:07:43.680 so that was thrown out. 01:07:43.680 --> 01:07:47.340 Only far-side stops-- so what's the difference 01:07:47.340 --> 01:07:50.948 between a near-side stop and a far-side stop? 01:07:50.948 --> 01:07:53.251 AUDIENCE: Far-side stop is after the intersection. 01:07:53.251 --> 01:07:55.250 GABRIEL SANCHEZ-MARTINEZ: Right, after signals-- 01:07:55.250 --> 01:07:57.830 so far-side stops are after signals. 01:07:57.830 --> 01:07:59.670 They did not look at near-side stops, which 01:07:59.670 --> 01:08:03.395 could be affected by the red light, the example 01:08:03.395 --> 01:08:07.801 that Ari just gave of, as the bus is ready to leave, 01:08:07.801 --> 01:08:08.550 a light turns red. 01:08:08.550 --> 01:08:10.530 Now you have a longer dwell time. 01:08:10.530 --> 01:08:14.050 The bus could leave its doors open just in case. 01:08:14.050 --> 01:08:17.430 So that was thrown out. 01:08:17.430 --> 01:08:18.979 OK, they looked-- 01:08:18.979 --> 01:08:20.790 AUDIENCE: What do you mean by known stops? 01:08:20.790 --> 01:08:23.373 GABRIEL SANCHEZ-MARTINEZ: So if there were any stops that were 01:08:23.373 --> 01:08:25.586 not properly coded or-- 01:08:25.586 --> 01:08:26.460 they were thrown out. 01:08:29.340 --> 01:08:33.060 They had APC, so the Automatic Passenger Counting, all doors. 01:08:33.060 --> 01:08:38.580 And they threw out data that was of bad APCs, 01:08:38.580 --> 01:08:43.520 so buses that required zero boardings or zero alightings 01:08:43.520 --> 01:08:46.250 were thrown out. 01:08:46.250 --> 01:08:50.120 And they looked at each AFC transaction 01:08:50.120 --> 01:08:54.471 and matched it to what-- 01:08:54.471 --> 01:08:55.209 how they paid. 01:08:55.209 --> 01:08:55.939 Was it a ticket? 01:08:55.939 --> 01:08:57.529 Was it a smart card, et cetera? 01:08:57.529 --> 01:09:01.700 So they looked at a whole month, November 2006. 01:09:01.700 --> 01:09:04.170 So here's how the model works. 01:09:04.170 --> 01:09:06.260 First, they realized that it was easier 01:09:06.260 --> 01:09:11.029 to have one model for when the front door controls 01:09:11.029 --> 01:09:15.439 the process and a separate model for when the rear door controls 01:09:15.439 --> 01:09:15.950 the process. 01:09:15.950 --> 01:09:19.490 So the first step was to predict which of the two processes 01:09:19.490 --> 01:09:22.229 was going to dominate and control the dwell time, 01:09:22.229 --> 01:09:25.220 and then from that, select that one model, or the, one 01:09:25.220 --> 01:09:26.700 or the other. 01:09:26.700 --> 01:09:30.020 And then they looked at including bus type 01:09:30.020 --> 01:09:36.500 and traveling as a friction factor in each of these models. 01:09:36.500 --> 01:09:38.600 So let's look at the high-level results. 01:09:41.420 --> 01:09:43.790 This is the front door model. 01:09:43.790 --> 01:09:46.939 So this is for when the front door dominates the process. 01:09:46.939 --> 01:09:51.260 They had a pretty good adjusted r squared, 0.733. 01:09:51.260 --> 01:09:55.820 And we see some dummy variables. 01:09:55.820 --> 01:09:58.160 NABI is a type of bus here. 01:09:58.160 --> 01:10:02.030 And NOVA and New Flyer are also types of buses. 01:10:02.030 --> 01:10:09.380 So NABI, for some reason, tended to be a half second longer 01:10:09.380 --> 01:10:10.680 dwell time overall. 01:10:14.660 --> 01:10:19.450 This variable is the number of people getting on in the front. 01:10:19.450 --> 01:10:22.660 They are excluding the first few passengers for the reason 01:10:22.660 --> 01:10:25.620 that Ari brought up earlier in the lecture, 01:10:25.620 --> 01:10:28.487 that the first few passengers take a little longer, 01:10:28.487 --> 01:10:30.820 but once you have a [INAUDIBLE],, a stream of passengers 01:10:30.820 --> 01:10:32.980 going in, then there's a more uniform rate. 01:10:32.980 --> 01:10:38.230 So this is front on extra. 01:10:38.230 --> 01:10:42.490 And we have 3.7, about. 01:10:42.490 --> 01:10:46.060 But it was a little longer for NOVA buses and a little shorter 01:10:46.060 --> 01:10:47.560 for NABI buses. 01:10:47.560 --> 01:10:51.250 So you have to adjust that amount for the different buses. 01:10:51.250 --> 01:10:54.400 So there was an interaction between that variable 01:10:54.400 --> 01:10:57.640 and the dummies for bus type. 01:10:57.640 --> 01:11:03.310 Then the effect of people getting off in the front 01:11:03.310 --> 01:11:04.650 was accounted for. 01:11:04.650 --> 01:11:07.350 So here you have-- 01:11:07.350 --> 01:11:09.320 if you have three or more people, 01:11:09.320 --> 01:11:12.570 so people in excess of three getting-- 01:11:12.570 --> 01:11:14.280 people in excess of two, actually, 01:11:14.280 --> 01:11:16.820 three or more people getting off in the front, 01:11:16.820 --> 01:11:20.290 that had an impact as well, a positive impact, of course. 01:11:20.290 --> 01:11:23.220 So these parts that I described were 01:11:23.220 --> 01:11:27.390 included in the model for non-crowded 01:11:27.390 --> 01:11:29.760 and in the model for crowded as well. 01:11:29.760 --> 01:11:33.030 So they have separate parts of the model 01:11:33.030 --> 01:11:35.420 that were for crowded and non-crowded conditions. 01:11:35.420 --> 01:11:42.390 When it wasn't crowded, cards were about 2.6 second effect 01:11:42.390 --> 01:11:45.900 for boarding per passenger. 01:11:45.900 --> 01:11:49.800 Tickets were about 4.8, so almost 2 seconds slower 01:11:49.800 --> 01:11:52.560 than smart cards. 01:11:52.560 --> 01:11:59.700 And New Flyer tickets were not quite as longer as tickets 01:11:59.700 --> 01:12:00.640 everywhere else. 01:12:00.640 --> 01:12:02.815 So this is the bus that had pretty wide doors 01:12:02.815 --> 01:12:07.440 and the fare box was placed in a better way 01:12:07.440 --> 01:12:11.020 so that people could tap in as they went in. 01:12:11.020 --> 01:12:13.710 You have a small advantage there. 01:12:13.710 --> 01:12:16.290 Here you have the effect of people getting off 01:12:16.290 --> 01:12:18.230 in the front for the first two passengers, 01:12:18.230 --> 01:12:21.900 not the ones that are three and up, so 2.8 01:12:21.900 --> 01:12:24.930 seconds per passenger extra. 01:12:24.930 --> 01:12:27.390 And here is a dummy for if the sensor in the front 01:12:27.390 --> 01:12:30.400 was blocked, which could have different effects. 01:12:30.400 --> 01:12:33.970 But essentially, that could indicate crowding or something, 01:12:33.970 --> 01:12:36.240 so that was included there. 01:12:36.240 --> 01:12:39.360 What happens when it was crowded? 01:12:39.360 --> 01:12:43.950 So when the load on the bus was high, 01:12:43.950 --> 01:12:46.960 it didn't really matter if it was a card or a ticket. 01:12:46.960 --> 01:12:50.630 So now we're just regressing on the number of AFC transactions. 01:12:50.630 --> 01:12:54.330 And we're getting an average of 4.3 per passenger. 01:12:54.330 --> 01:12:55.650 So it's slower. 01:12:55.650 --> 01:12:58.140 It's much closer to the ticket quantity. 01:12:58.140 --> 01:13:00.330 And therefore, the impact, the benefit 01:13:00.330 --> 01:13:05.070 that you had of dwell time savings by smart card is lost. 01:13:05.070 --> 01:13:07.890 So that savings, you have when it's not crowded, 01:13:07.890 --> 01:13:09.370 but you lose when it's crowded. 01:13:09.370 --> 01:13:09.870 Sonya? 01:13:09.870 --> 01:13:13.950 AUDIENCE: Is that the same thing with off-board fare collection? 01:13:13.950 --> 01:13:15.700 GABRIEL SANCHEZ-MARTINEZ: No, because when 01:13:15.700 --> 01:13:17.158 you have off-board fare collection, 01:13:17.158 --> 01:13:18.930 you can open all doors. 01:13:18.930 --> 01:13:22.720 And so it becomes more like a train, where everybody gets off 01:13:22.720 --> 01:13:24.320 first and then everybody gets on. 01:13:24.320 --> 01:13:28.034 So this model is for more people boarding by the front 01:13:28.034 --> 01:13:29.200 and alighting from the back. 01:13:29.200 --> 01:13:31.199 But sometimes, some people get off in the front. 01:13:35.770 --> 01:13:38.500 And here we have the friction factor, 01:13:38.500 --> 01:13:41.560 so the number of passengers-- the number of standees 01:13:41.560 --> 01:13:46.580 squared times passengers. 01:13:46.580 --> 01:13:49.230 So this is going to-- 01:13:49.230 --> 01:13:52.045 it's a small factor, a small coefficient, 01:13:52.045 --> 01:13:54.440 but it's multiplied by some quantity squared, 01:13:54.440 --> 01:13:57.300 so that kind of makes up the difference. 01:13:57.300 --> 01:14:00.520 And because this is squared, this 01:14:00.520 --> 01:14:02.650 is kind of a polynomial effect. 01:14:02.650 --> 01:14:06.780 So the more people that are standing-- 01:14:06.780 --> 01:14:08.189 it's not a linear increase. 01:14:08.189 --> 01:14:10.480 If there many more people on the bus that are standing, 01:14:10.480 --> 01:14:12.070 then it's a much slower dwell. 01:14:12.070 --> 01:14:14.850 So this is a correction factor for a crowded bus. 01:14:17.770 --> 01:14:19.600 OK, here is the rear door model. 01:14:19.600 --> 01:14:23.620 So this is for when the rear door was predicted 01:14:23.620 --> 01:14:25.554 to control the dwell time, so more 01:14:25.554 --> 01:14:27.970 towards the end of the route where more people are getting 01:14:27.970 --> 01:14:30.550 off than people are getting on. 01:14:30.550 --> 01:14:33.730 We have, again, some impacts. 01:14:33.730 --> 01:14:36.460 The design of vehicle was significant. 01:14:36.460 --> 01:14:41.020 And the number of people getting off was, of course, a variable, 01:14:41.020 --> 01:14:44.740 so about 1.7 seconds per passenger-- 01:14:44.740 --> 01:14:48.250 more on NOVA buses, less on NABI buses. 01:14:48.250 --> 01:14:50.920 And the friction factor, again-- so you 01:14:50.920 --> 01:14:56.680 have the sort of general friction factor was 0.005. 01:14:56.680 --> 01:15:01.780 It was 0.009 for normal buses and 0.002 for NABI buses. 01:15:01.780 --> 01:15:03.980 So you have to sort of add these up 01:15:03.980 --> 01:15:07.260 to get the effective friction factor on a particular bus. 01:15:10.530 --> 01:15:12.960 All right, so the sort of key takeaways 01:15:12.960 --> 01:15:16.110 are that the smart media loses the benefit 01:15:16.110 --> 01:15:17.280 in crowded conditions. 01:15:17.280 --> 01:15:19.230 We saw that. 01:15:19.230 --> 01:15:21.360 The crowding impact increases exponentially. 01:15:21.360 --> 01:15:24.750 These people tried linear standees. 01:15:24.750 --> 01:15:28.390 And standees squared was a much better predictor. 01:15:28.390 --> 01:15:34.520 So bus attributes impact dwell time. 01:15:34.520 --> 01:15:37.770 The dummies for the bus design were significant. 01:15:37.770 --> 01:15:38.940 And they had an impact. 01:15:38.940 --> 01:15:42.000 So some of that had to do with the location of the ticket 01:15:42.000 --> 01:15:42.820 reader. 01:15:42.820 --> 01:15:47.700 And some of that had to do with wide doors that allowed people 01:15:47.700 --> 01:15:49.920 to enter more comfortably and faster. 01:15:52.770 --> 01:15:55.120 So that's good. 01:15:55.120 --> 01:15:57.360 Let's move on to the Green line model, just 01:15:57.360 --> 01:15:59.880 quickly, at a high level. 01:15:59.880 --> 01:16:03.030 We're almost out of time, so I'm going to skip this slide. 01:16:03.030 --> 01:16:06.300 You're, I think, all aware of what the Green line is, 01:16:06.300 --> 01:16:09.850 so I don't have to cover this. 01:16:09.850 --> 01:16:12.480 They looked at one-car train and two-car train models 01:16:12.480 --> 01:16:13.440 separately. 01:16:13.440 --> 01:16:15.060 So sometimes the Green line will run 01:16:15.060 --> 01:16:17.700 single vehicles, single car, or sometimes 01:16:17.700 --> 01:16:20.080 two cars paired together. 01:16:20.080 --> 01:16:27.382 And the dwell time here was some constant times the number 01:16:27.382 --> 01:16:29.340 of people who are getting on times the number-- 01:16:29.340 --> 01:16:35.100 plus the number people getting off plus some friction factor. 01:16:35.100 --> 01:16:38.400 And same here for two-car trains, but the coefficients 01:16:38.400 --> 01:16:39.270 were different. 01:16:39.270 --> 01:16:43.035 So that's the overall concept that I want to communicate. 01:16:43.035 --> 01:16:47.640 Here is a table of the results from that model for the number 01:16:47.640 --> 01:16:48.780 of people getting on. 01:16:48.780 --> 01:16:52.260 So this is using the model to forecast. 01:16:52.260 --> 01:16:54.100 These are not observations. 01:16:54.100 --> 01:16:59.110 So if you feed 0 people getting on, you just get the constants. 01:17:01.780 --> 01:17:04.240 If you feed the model 10 people getting on, 01:17:04.240 --> 01:17:07.030 it depends on the load on the model. 01:17:07.030 --> 01:17:10.470 If the load is, say, less than 53, 01:17:10.470 --> 01:17:13.870 then you don't have the friction factor really 01:17:13.870 --> 01:17:18.400 controlling anything, so you have 20.3 or 20.2, 01:17:18.400 --> 01:17:21.130 so a very small effect of friction, 01:17:21.130 --> 01:17:27.430 and therefore about the same time for both models. 01:17:27.430 --> 01:17:28.900 If it's very crowded, then you do 01:17:28.900 --> 01:17:33.220 have a more significant benefit for having a two-car train. 01:17:33.220 --> 01:17:34.720 That makes sense, right? 01:17:34.720 --> 01:17:35.806 You have more capacity. 01:17:35.806 --> 01:17:37.180 And the same thing happens as you 01:17:37.180 --> 01:17:40.900 move to 20 passengers getting on and 30 passengers getting on, 01:17:40.900 --> 01:17:44.080 with those differences increasing when you have 01:17:44.080 --> 01:17:49.400 a crowded train, and the differences 01:17:49.400 --> 01:17:52.156 between one-car and two-car trains 01:17:52.156 --> 01:17:54.280 not being that significant when you don't have much 01:17:54.280 --> 01:17:56.820 [INAUDIBLE]. 01:17:56.820 --> 01:17:59.750 So this should, more or less, make sense. 01:17:59.750 --> 01:18:04.610 So all this is to show you that you have some ideas about what 01:18:04.610 --> 01:18:05.590 controls dwell time. 01:18:05.590 --> 01:18:08.630 You test your hypothesis on the data. 01:18:08.630 --> 01:18:12.080 And you can try different things. 01:18:12.080 --> 01:18:13.840 These are just examples. 01:18:13.840 --> 01:18:17.620 So the findings from that research, dwell times 01:18:17.620 --> 01:18:20.350 were quite sensitive to flows and loads. 01:18:20.350 --> 01:18:22.750 The crowding effect might be non-linear. 01:18:22.750 --> 01:18:25.870 They looked at non-linear effects . 01:18:25.870 --> 01:18:28.240 Just like before. 01:18:28.240 --> 01:18:31.030 The dwell times for multi-car trains, for two-car trains 01:18:31.030 --> 01:18:33.165 were different than those for one-car trains. 01:18:33.165 --> 01:18:36.370 The dwell time functions suggest high sensitivity of performance 01:18:36.370 --> 01:18:37.600 to perturbations. 01:18:37.600 --> 01:18:41.710 So we saw what those preparations 01:18:41.710 --> 01:18:43.670 were earlier in this lecture. 01:18:43.670 --> 01:18:47.950 And this model is sensitive to some of them. 01:18:47.950 --> 01:18:50.860 Because of that, effect of real-time operations control 01:18:50.860 --> 01:18:52.810 should be essential to operating the Green 01:18:52.810 --> 01:18:54.280 line with even headways. 01:18:54.280 --> 01:18:57.040 So this is more of a recommendation. 01:18:57.040 --> 01:19:00.460 And another recommendation is that simulation models 01:19:00.460 --> 01:19:03.460 of this kind of service should include 01:19:03.460 --> 01:19:06.670 sophisticated dwell time models like the ones that 01:19:06.670 --> 01:19:09.390 were estimated here to account for all those effects, 01:19:09.390 --> 01:19:13.970 because otherwise, you would have a simple model that is 01:19:13.970 --> 01:19:15.640 not very faithful to reality. 01:19:20.660 --> 01:19:22.660 Running a mixed fleet, some with one-car trains, 01:19:22.660 --> 01:19:24.530 some two-car trains is dangerous. 01:19:24.530 --> 01:19:26.624 So what do they mean by dangerous? 01:19:29.057 --> 01:19:31.390 So it wouldn't make much difference if it's not crowded. 01:19:31.390 --> 01:19:34.130 But according to the model, on crowded conditions, 01:19:34.130 --> 01:19:36.430 the two-car train will be faster. 01:19:36.430 --> 01:19:41.020 So then you will have a bunching effect happening, 01:19:41.020 --> 01:19:43.960 two-car trains catching up to one-car trains, 01:19:43.960 --> 01:19:47.095 and therefore deterioration of service quality. 01:19:47.095 --> 01:19:51.190 Here is a marginal boarding time on heavy rail 01:19:51.190 --> 01:19:53.810 from the third paper. 01:19:53.810 --> 01:19:55.580 I think this was looking at the Red line. 01:19:55.580 --> 01:19:58.210 So this is the marginal boarding time 01:19:58.210 --> 01:20:00.670 when everybody can sit down. 01:20:03.910 --> 01:20:06.940 When you look at the number of through passengers-- 01:20:06.940 --> 01:20:09.604 that is the number of passengers who are on the train when 01:20:09.604 --> 01:20:11.770 the train arrives on the platform and don't get off, 01:20:11.770 --> 01:20:14.530 so passengers riding through the station-- 01:20:14.530 --> 01:20:18.910 when the number is 0, then this is how much boarding time you 01:20:18.910 --> 01:20:22.240 have per passenger, more or less. 01:20:22.240 --> 01:20:27.940 And then as you look at the effect of more and more people 01:20:27.940 --> 01:20:30.420 are standing, than that kind of slows people down 01:20:30.420 --> 01:20:31.780 as they board. 01:20:31.780 --> 01:20:35.020 So this was a way of capturing that effect, 01:20:35.020 --> 01:20:36.940 again, the fiction factor. 01:20:36.940 --> 01:20:41.360 So you see a theme here, that this seems to be relevant. 01:20:41.360 --> 01:20:44.620 And there have been research studies 01:20:44.620 --> 01:20:48.490 looking at ways of including that effect in models. 01:20:48.490 --> 01:20:50.470 These models could be used for improving 01:20:50.470 --> 01:20:52.360 the accuracy of your waiting time estimate 01:20:52.360 --> 01:20:54.610 that your smartphone gives you, or improving 01:20:54.610 --> 01:20:59.490 how faithful a simulation model is, et cetera. 01:20:59.490 --> 01:21:02.220 Here is the equation, very similar, 01:21:02.220 --> 01:21:03.460 same structure we had before. 01:21:03.460 --> 01:21:04.939 We have some constant. 01:21:04.939 --> 01:21:06.480 We have the number of people boarding 01:21:06.480 --> 01:21:08.727 per door, the number of people alighting per door. 01:21:08.727 --> 01:21:10.560 In this case, we add them up, because people 01:21:10.560 --> 01:21:12.890 alight first and then board. 01:21:12.890 --> 01:21:15.870 And then we have some friction factor. 01:21:15.870 --> 01:21:17.850 How that friction factor is calculated 01:21:17.850 --> 01:21:19.730 has been different on every model, 01:21:19.730 --> 01:21:21.860 but they all have a friction factor. 01:21:21.860 --> 01:21:26.220 So these are all services that, at least at times, 01:21:26.220 --> 01:21:27.420 are quite crowded. 01:21:27.420 --> 01:21:31.500 So that was important and significant. 01:21:31.500 --> 01:21:34.230 OK, if there are no questions, you may leave. 01:21:34.230 --> 01:21:35.970 And if there are, I'll take them. 01:21:35.970 --> 01:21:37.690 Sorry, don't feel that you have to wait, 01:21:37.690 --> 01:21:40.440 because it's already 5:30.