1 00:00:01,000 --> 00:00:03,340 The following content is provided under a Creative 2 00:00:03,340 --> 00:00:04,760 Commons license. 3 00:00:04,760 --> 00:00:06,970 Your support will help MIT OpenCourseWare 4 00:00:06,970 --> 00:00:11,060 continue to offer high-quality educational resources for free. 5 00:00:11,060 --> 00:00:13,600 To make a donation or to view additional materials 6 00:00:13,600 --> 00:00:17,560 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,560 --> 00:00:18,922 at ocw.mit.edu. 8 00:00:21,954 --> 00:00:24,890 GABRIEL SANCHEZ-MARTINEZ: Let's get started with Lecture 9. 9 00:00:24,890 --> 00:00:29,170 Today's lecture is very relevant to the homework. 10 00:00:29,170 --> 00:00:32,590 We'll run through some examples of performance models 11 00:00:32,590 --> 00:00:35,610 like the ones you have to work on or specify 12 00:00:35,610 --> 00:00:37,840 an estimate on your homework. 13 00:00:37,840 --> 00:00:40,730 So we talk about performance, performance models. 14 00:00:40,730 --> 00:00:41,680 What is performance? 15 00:00:41,680 --> 00:00:43,510 What do we mean by performance? 16 00:00:43,510 --> 00:00:49,420 Typically, this is a key word used to describe output 17 00:00:49,420 --> 00:00:53,860 and some sense of how well the output service was, so 18 00:00:53,860 --> 00:00:57,620 both in terms of the operator and in terms of the passenger. 19 00:00:57,620 --> 00:01:02,320 So things like running times, waiting times, headways, things 20 00:01:02,320 --> 00:01:04,810 that you can measure from observations 21 00:01:04,810 --> 00:01:08,530 of service being delivered, we call that performance, 22 00:01:08,530 --> 00:01:09,690 generally. 23 00:01:09,690 --> 00:01:12,280 And the kinds of models that we're going to look at 24 00:01:12,280 --> 00:01:15,190 are wait time models, service variation 25 00:01:15,190 --> 00:01:17,960 along routes where you'll see how it's not always the same. 26 00:01:17,960 --> 00:01:20,830 You might start off at the terminal one way 27 00:01:20,830 --> 00:01:23,277 and end up a very different way. 28 00:01:23,277 --> 00:01:25,610 We'll look at running time models and dwell time models. 29 00:01:25,610 --> 00:01:29,510 So these are the components of running times. 30 00:01:29,510 --> 00:01:32,110 Let's start with waiting time. 31 00:01:32,110 --> 00:01:35,660 Well, before that, actually, let's think about-- 32 00:01:35,660 --> 00:01:37,840 so here are some kinds of models. 33 00:01:37,840 --> 00:01:41,280 Why would we be interested in modeling these things? 34 00:01:45,040 --> 00:01:50,526 What applications can you think of for these kinds of models? 35 00:01:50,526 --> 00:01:52,510 That's two ways of putting the same question. 36 00:01:55,690 --> 00:01:58,000 Why would we be interested in having a waiting time 37 00:01:58,000 --> 00:02:01,830 model or a running time model? 38 00:02:01,830 --> 00:02:02,955 Just give some examples. 39 00:02:06,190 --> 00:02:07,290 Yeah, over in the back? 40 00:02:07,290 --> 00:02:11,135 AUDIENCE: We want to understand how people are waiting exactly. 41 00:02:11,135 --> 00:02:12,510 GABRIEL SANCHEZ-MARTINEZ: But you 42 00:02:12,510 --> 00:02:13,884 can go out and measure it, right? 43 00:02:17,910 --> 00:02:21,990 You can go out and do a survey and observe 44 00:02:21,990 --> 00:02:23,310 how people are waiting. 45 00:02:23,310 --> 00:02:27,600 So why would a model be good? 46 00:02:27,600 --> 00:02:31,670 What would a model do that observations don't do? 47 00:02:31,670 --> 00:02:32,870 AUDIENCE: For a new service. 48 00:02:32,870 --> 00:02:33,690 GABRIEL SANCHEZ-MARTINEZ: For a new service, 49 00:02:33,690 --> 00:02:35,310 OK, we're starting to get some ideas, yeah. 50 00:02:35,310 --> 00:02:36,610 AUDIENCE: Or changes in service. 51 00:02:36,610 --> 00:02:37,500 GABRIEL SANCHEZ-MARTINEZ: Changes in service-- 52 00:02:37,500 --> 00:02:39,830 OK, so can you give me an example of that? 53 00:02:39,830 --> 00:02:42,900 AUDIENCE: Like [INAUDIBLE] making the Victoria line 20% 54 00:02:42,900 --> 00:02:43,874 more frequent. 55 00:02:43,874 --> 00:02:46,290 GABRIEL SANCHEZ-MARTINEZ: OK, so if you increase frequency 56 00:02:46,290 --> 00:02:51,679 on the Victoria line, what does that do to demand and to dwell 57 00:02:51,679 --> 00:02:52,220 time and to-- 58 00:02:52,220 --> 00:02:54,845 AUDIENCE: The Oxford Circuit is now closed for 30 minutes a day 59 00:02:54,845 --> 00:02:56,520 because can only interchange, not enter. 60 00:02:56,520 --> 00:02:58,020 GABRIEL SANCHEZ-MARTINEZ: OK, so you 61 00:02:58,020 --> 00:03:01,790 can try to use them to predict service changes 62 00:03:01,790 --> 00:03:05,460 or changes in performance due to changes in service. 63 00:03:05,460 --> 00:03:07,300 What else, any other ideas? 64 00:03:10,478 --> 00:03:11,926 [INAUDIBLE]? 65 00:03:11,926 --> 00:03:14,490 AUDIENCE: [INAUDIBLE] service variation, 66 00:03:14,490 --> 00:03:18,770 that's [INAUDIBLE] to crowded and to crowded on the bus. 67 00:03:18,770 --> 00:03:22,940 It's depending on how the planned headway 68 00:03:22,940 --> 00:03:28,100 sort of variance over time will impact how many people end up 69 00:03:28,100 --> 00:03:29,300 boarding on a bus. 70 00:03:29,300 --> 00:03:32,420 Let's say, a bus that came substantially 71 00:03:32,420 --> 00:03:35,730 after the previous bus might be crowded. 72 00:03:35,730 --> 00:03:38,522 And I won't be able board. 73 00:03:38,522 --> 00:03:40,800 And you might even have-- 74 00:03:40,800 --> 00:03:42,565 so denied boardings for-- 75 00:03:42,565 --> 00:03:44,106 GABRIEL SANCHEZ-MARTINEZ: [INAUDIBLE] 76 00:03:44,106 --> 00:03:45,170 will you summarize that? 77 00:03:45,170 --> 00:03:48,480 Maybe I'll try summarizing. 78 00:03:48,480 --> 00:03:49,970 You could use one of these models 79 00:03:49,970 --> 00:03:54,150 to fill in things about performance 80 00:03:54,150 --> 00:03:56,310 that you can't measure directly from data. 81 00:03:56,310 --> 00:04:00,900 So you gave an example of measuring headways and then 82 00:04:00,900 --> 00:04:03,870 estimating crowding on the bus based on headways. 83 00:04:03,870 --> 00:04:05,065 Was that more or less-- 84 00:04:05,065 --> 00:04:05,690 AUDIENCE: Yeah. 85 00:04:05,690 --> 00:04:06,260 GABRIEL SANCHEZ-MARTINEZ: --getting at-- 86 00:04:06,260 --> 00:04:06,420 AUDIENCE: [INAUDIBLE] 87 00:04:06,420 --> 00:04:08,878 GABRIEL SANCHEZ-MARTINEZ: OK, trying to generalize a little 88 00:04:08,878 --> 00:04:10,080 bit-- 89 00:04:10,080 --> 00:04:11,295 any other ideas? 90 00:04:11,295 --> 00:04:12,690 [INAUDIBLE]? 91 00:04:12,690 --> 00:04:16,980 AUDIENCE: I mean, you could use this as a [INAUDIBLE] model 92 00:04:16,980 --> 00:04:18,660 in any other simulation. 93 00:04:18,660 --> 00:04:20,493 GABRIEL SANCHEZ-MARTINEZ: Simulation models, 94 00:04:20,493 --> 00:04:22,830 so if you want to do a simulation model of a system 95 00:04:22,830 --> 00:04:27,270 that has transit in it, you need all these things to-- 96 00:04:27,270 --> 00:04:28,050 right? 97 00:04:28,050 --> 00:04:31,260 Because you need to have your [INAUDIBLE] which, 98 00:04:31,260 --> 00:04:34,620 it could be a bus, or a passenger waiting, or dwelling, 99 00:04:34,620 --> 00:04:35,860 or moving between stops. 100 00:04:35,860 --> 00:04:38,190 So you need these things. 101 00:04:38,190 --> 00:04:40,530 Any other applications, practical, very 102 00:04:40,530 --> 00:04:42,153 practical applications? 103 00:04:45,394 --> 00:04:50,840 So for dwell time and running time models, 104 00:04:50,840 --> 00:04:53,090 we all have our prediction of when 105 00:04:53,090 --> 00:04:55,490 the bus is coming to the stop. 106 00:04:55,490 --> 00:04:58,370 So you need some model to take the bus from where it is right 107 00:04:58,370 --> 00:05:01,160 now, which you can measure, to how long 108 00:05:01,160 --> 00:05:02,920 will it take me to reach this stop. 109 00:05:02,920 --> 00:05:06,680 And so the passenger can look up on the phone, how long will 110 00:05:06,680 --> 00:05:07,640 I have to wait. 111 00:05:07,640 --> 00:05:10,250 So these are all examples. 112 00:05:10,250 --> 00:05:11,240 And there are more. 113 00:05:11,240 --> 00:05:12,890 Let's start with waiting time. 114 00:05:12,890 --> 00:05:18,620 So we've already seen this first simple model of waiting time. 115 00:05:18,620 --> 00:05:23,370 We say that the expected waiting time is half the headway. 116 00:05:23,370 --> 00:05:26,630 And here, we are considering both waiting time and headway 117 00:05:26,630 --> 00:05:29,610 to be stochastic quantities. 118 00:05:29,610 --> 00:05:35,060 So if we observe many headways over a long period of time, 119 00:05:35,060 --> 00:05:37,550 we see a probability distribution of headways. 120 00:05:37,550 --> 00:05:40,130 And we're saying, the average headway divided by 2 121 00:05:40,130 --> 00:05:42,320 should equal the average waiting time. 122 00:05:42,320 --> 00:05:44,420 Now, this is a very simple model. 123 00:05:44,420 --> 00:05:47,000 And we know there are some problems with it. 124 00:05:47,000 --> 00:05:52,460 So we are assuming that passengers arrive independent 125 00:05:52,460 --> 00:05:54,020 of vehicle departure times. 126 00:05:54,020 --> 00:05:56,840 We are assuming that vehicles are departing 127 00:05:56,840 --> 00:05:58,830 at equal intervals deterministically, 128 00:05:58,830 --> 00:06:01,880 so they're sort of evenly spaced, 129 00:06:01,880 --> 00:06:03,456 and that every passenger can board 130 00:06:03,456 --> 00:06:04,580 the first vehicle they see. 131 00:06:04,580 --> 00:06:09,050 So nobody is left behind by a bus that is too full. 132 00:06:09,050 --> 00:06:12,450 So obviously, these things don't always hold. 133 00:06:12,450 --> 00:06:16,010 And particularly, the part of vehicles departing exactly 134 00:06:16,010 --> 00:06:19,110 at equal intervals doesn't hold. 135 00:06:19,110 --> 00:06:20,040 It rarely holds. 136 00:06:23,470 --> 00:06:25,250 AUDIENCE: Even if they depart, there 137 00:06:25,250 --> 00:06:28,510 are a lot of things along the way that impact the variation. 138 00:06:28,510 --> 00:06:29,360 GABRIEL SANCHEZ-MARTINEZ: Yeah, so but we're 139 00:06:29,360 --> 00:06:31,880 saying departing from each stop deterministically, not just 140 00:06:31,880 --> 00:06:33,500 the terminal. 141 00:06:33,500 --> 00:06:36,140 So obviously, there are some problems with this. 142 00:06:36,140 --> 00:06:39,020 And so how do we need to-- 143 00:06:39,020 --> 00:06:44,490 well, if we start taking care of some of these things, 144 00:06:44,490 --> 00:06:45,950 what will happen to waiting time? 145 00:06:45,950 --> 00:06:47,390 Will it decrease or increase? 146 00:06:51,230 --> 00:06:53,534 So if we take this particular one. 147 00:06:53,534 --> 00:06:55,200 This is the strongest assumption, right? 148 00:06:55,200 --> 00:06:58,235 So that vehicles depart deterministically 149 00:06:58,235 --> 00:07:00,510 at equal intervals-- 150 00:07:00,510 --> 00:07:03,210 and we say, no, they don't, actually. 151 00:07:03,210 --> 00:07:04,710 Some of them depart late. 152 00:07:04,710 --> 00:07:06,390 Some of them depart early. 153 00:07:06,390 --> 00:07:07,687 There is bunching. 154 00:07:07,687 --> 00:07:09,770 So what happens to waiting time when that happens? 155 00:07:09,770 --> 00:07:10,500 AUDIENCE: It goes up. 156 00:07:10,500 --> 00:07:12,041 GABRIEL SANCHEZ-MARTINEZ: It goes up. 157 00:07:12,041 --> 00:07:15,479 So how do we adjust this model to account for that? 158 00:07:15,479 --> 00:07:16,270 Let's look at that. 159 00:07:16,270 --> 00:07:18,066 AUDIENCE: Doesn't it depend though 160 00:07:18,066 --> 00:07:19,440 when the passengers are arriving? 161 00:07:19,440 --> 00:07:21,981 GABRIEL SANCHEZ-MARTINEZ: If you look at an average or a long 162 00:07:21,981 --> 00:07:26,760 period of time, and you'd have the same number of vehicles 163 00:07:26,760 --> 00:07:30,960 and drivers, so you either-- the best thing you can do if people 164 00:07:30,960 --> 00:07:32,314 are arriving randomly is to-- 165 00:07:32,314 --> 00:07:33,980 AUDIENCE: OK, they're arriving randomly. 166 00:07:33,980 --> 00:07:35,646 GABRIEL SANCHEZ-MARTINEZ: Yeah, so we're 167 00:07:35,646 --> 00:07:38,380 only tackling that assumption. 168 00:07:38,380 --> 00:07:42,180 OK, so there are some issues, as we said. 169 00:07:42,180 --> 00:07:44,190 There is bulk arrivals. 170 00:07:44,190 --> 00:07:47,730 So you have a bus stop right outside of a train station. 171 00:07:47,730 --> 00:07:49,020 And the train has just left. 172 00:07:49,020 --> 00:07:50,700 And a bunch of people get off the train 173 00:07:50,700 --> 00:07:52,980 and they all want to board the bus. 174 00:07:52,980 --> 00:07:54,560 That's not being captured by this. 175 00:07:57,360 --> 00:08:01,490 And we can think of the passenger arrival process 176 00:08:01,490 --> 00:08:04,350 in steps, from having no information 177 00:08:04,350 --> 00:08:06,270 to having a lot of information. 178 00:08:06,270 --> 00:08:09,420 So random arrivals is what we typically 179 00:08:09,420 --> 00:08:13,200 assume, certainly for high-frequency service, 180 00:08:13,200 --> 00:08:16,350 and in some cases, for all kinds of service. 181 00:08:16,350 --> 00:08:17,869 And that's more problematic. 182 00:08:17,869 --> 00:08:19,410 If you know how long-headway service, 183 00:08:19,410 --> 00:08:22,410 people are going to try to time their arrival 184 00:08:22,410 --> 00:08:24,080 at the stop to the schedule. 185 00:08:24,080 --> 00:08:26,660 But you will see some models that-- 186 00:08:26,660 --> 00:08:29,280 in the literature, that, for simplicity, 187 00:08:29,280 --> 00:08:31,520 especially service planning models, for simplicity, 188 00:08:31,520 --> 00:08:32,740 might assume this anyway. 189 00:08:32,740 --> 00:08:34,049 You have a question. 190 00:08:34,049 --> 00:08:36,035 AUDIENCE: Like random, but Poisson distributed, 191 00:08:36,035 --> 00:08:37,200 or just, like, random? 192 00:08:37,200 --> 00:08:39,330 GABRIEL SANCHEZ-MARTINEZ: Yeah, Poisson distributed-- sometimes 193 00:08:39,330 --> 00:08:40,470 it could just be random. 194 00:08:40,470 --> 00:08:43,650 Typically the assumption made is that it's a Poisson process. 195 00:08:43,650 --> 00:08:46,020 So inter-arrival times are negative exponential 196 00:08:46,020 --> 00:08:48,300 distribution. 197 00:08:48,300 --> 00:08:51,240 OK, then some passengers will time their arrivals 198 00:08:51,240 --> 00:08:52,500 to minimum waiting. 199 00:08:52,500 --> 00:08:58,770 So this could be that you have a phone app or a schedule. 200 00:08:58,770 --> 00:09:01,200 And you show up a couple of minutes before. 201 00:09:01,200 --> 00:09:04,530 You're trying to minimize your waiting time. 202 00:09:04,530 --> 00:09:07,560 You're trying to arrive shortly before the vehicle departs. 203 00:09:07,560 --> 00:09:09,930 And then there is the running to the vehicle 204 00:09:09,930 --> 00:09:14,080 before it leaves the stop, and therefore, I have no waiting. 205 00:09:14,080 --> 00:09:15,870 That's everyone's favorite, right? 206 00:09:15,870 --> 00:09:19,650 So if we look at a graph of expected headway 207 00:09:19,650 --> 00:09:23,790 on the horizontal axis and expected waiting 208 00:09:23,790 --> 00:09:25,620 time on the vertical axis, you will 209 00:09:25,620 --> 00:09:27,650 see that around 10 minutes-- 210 00:09:27,650 --> 00:09:30,150 some people say 15, some people say 10, but somewhere around 211 00:09:30,150 --> 00:09:31,260 there, 12-- 212 00:09:31,260 --> 00:09:37,310 so you will see that the actual waiting 213 00:09:37,310 --> 00:09:40,670 time that you would observe is much lower than what 214 00:09:40,670 --> 00:09:42,710 the simple model says it is, which 215 00:09:42,710 --> 00:09:46,610 is half the headway for headways longer than 10, 12, 15 minutes. 216 00:09:46,610 --> 00:09:49,430 And I think those people will have some strategy 217 00:09:49,430 --> 00:09:51,770 to minimize their waiting time. 218 00:09:51,770 --> 00:09:56,330 So below that amount, people tend to not time their arrivals 219 00:09:56,330 --> 00:09:56,960 at stops. 220 00:09:56,960 --> 00:10:02,240 And they tend to arrive randomly, essentially. 221 00:10:02,240 --> 00:10:04,960 So if you try to take the Red line here in Boston, 222 00:10:04,960 --> 00:10:06,920 you just show up at whatever time. 223 00:10:06,920 --> 00:10:09,820 And typically, most people won't be looking at their phone 224 00:10:09,820 --> 00:10:12,460 and trying to time their arrival, although that 225 00:10:12,460 --> 00:10:14,230 is happening increasingly. 226 00:10:14,230 --> 00:10:17,230 So for those people, the model, the simple model 227 00:10:17,230 --> 00:10:20,940 of headway divided by 2 tends to underestimate 228 00:10:20,940 --> 00:10:21,940 the actual waiting time. 229 00:10:21,940 --> 00:10:24,090 And that's due to headway variability, mostly. 230 00:10:27,850 --> 00:10:33,420 So we have to take care of that assumption. 231 00:10:33,420 --> 00:10:36,090 Let's look at a formulation that relaxes that. 232 00:10:36,090 --> 00:10:37,950 Let's say that vehicle departures are not 233 00:10:37,950 --> 00:10:39,930 regular and deterministic. 234 00:10:39,930 --> 00:10:44,290 So let's refine the model to take care of that. 235 00:10:44,290 --> 00:10:47,580 So let's define n as a function of headway 236 00:10:47,580 --> 00:10:49,500 to be the number of passengers arriving 237 00:10:49,500 --> 00:10:54,690 at some headway and the mean waiting time of headway, h, 238 00:10:54,690 --> 00:10:59,360 to be, well, that mean headway of some specific headway 239 00:10:59,360 --> 00:11:03,260 that we observe, and then g to be the probability density 240 00:11:03,260 --> 00:11:04,290 function of headway. 241 00:11:04,290 --> 00:11:06,300 So we observe many of these highways. 242 00:11:06,300 --> 00:11:09,150 And we have a histogram of headways. 243 00:11:09,150 --> 00:11:13,740 So if you want to compute the expected waiting 244 00:11:13,740 --> 00:11:16,050 time across all passengers, what we want 245 00:11:16,050 --> 00:11:18,300 is the expected total passenger waiting 246 00:11:18,300 --> 00:11:21,710 time divided by the expected number of passengers 247 00:11:21,710 --> 00:11:25,110 over many, many observations of vehicles leaving. 248 00:11:25,110 --> 00:11:29,670 And that is expressed mathematically in the equation 249 00:11:29,670 --> 00:11:30,540 below. 250 00:11:30,540 --> 00:11:33,300 We have integrals, in this case, from 0 to infinity, 251 00:11:33,300 --> 00:11:35,410 because headways can't be negative, 252 00:11:35,410 --> 00:11:38,460 so we're not going from negative infinity to positive infinity. 253 00:11:38,460 --> 00:11:43,080 And we're saying, yeah, for any given headway, 254 00:11:43,080 --> 00:11:44,910 we'll multiply the number of people times 255 00:11:44,910 --> 00:11:46,920 how much they wait on average times 256 00:11:46,920 --> 00:11:49,540 the probability that I see that headway. 257 00:11:49,540 --> 00:11:52,140 And let's integrate over all possible headways 258 00:11:52,140 --> 00:11:54,930 and divide that by the number of people on each headway. 259 00:11:54,930 --> 00:11:56,466 Does that make sense conceptually? 260 00:11:59,660 --> 00:12:02,690 OK, so now let's make some assumptions. 261 00:12:02,690 --> 00:12:05,900 Let's say that people do arrive uniformly 262 00:12:05,900 --> 00:12:08,990 with some rate, lambda. 263 00:12:08,990 --> 00:12:12,560 And so the number of people arriving in a given headway 264 00:12:12,560 --> 00:12:15,110 is going to be some arrival rate, which we've said 265 00:12:15,110 --> 00:12:17,520 is constant, times the headway. 266 00:12:17,520 --> 00:12:20,370 So if we say that 10 people arrive per minute, 267 00:12:20,370 --> 00:12:23,780 and lambda therefore is 10 passengers per minute, 268 00:12:23,780 --> 00:12:28,080 then if the headway is 2 minutes, we have 20 people. 269 00:12:28,080 --> 00:12:29,520 Does that make sense? 270 00:12:29,520 --> 00:12:32,130 Any questions so far? 271 00:12:32,130 --> 00:12:33,932 OK, and then-- 272 00:12:33,932 --> 00:12:36,015 AUDIENCE: Can you repeat just what you said there? 273 00:12:36,015 --> 00:12:38,120 GABRIEL SANCHEZ-MARTINEZ: Yeah, so lambda 274 00:12:38,120 --> 00:12:40,140 is the passenger arrival rate. 275 00:12:40,140 --> 00:12:43,536 And I gave an example of it being 10 passengers per minute. 276 00:12:43,536 --> 00:12:44,910 So is that something you measure. 277 00:12:44,910 --> 00:12:46,460 And we're assuming that that amount 278 00:12:46,460 --> 00:12:52,160 is constant for a specific time of day at a specific place. 279 00:12:52,160 --> 00:12:54,440 And then we're saying, if you-- once you 280 00:12:54,440 --> 00:12:57,410 assume that, then you observe a headway, which could be, say, 281 00:12:57,410 --> 00:12:58,130 2 minutes long. 282 00:12:58,130 --> 00:13:00,310 Then you multiply by the arrival rate. 283 00:13:00,310 --> 00:13:04,084 Now we say it's 20 passengers for that particular headway. 284 00:13:04,084 --> 00:13:06,500 And there might be another headway that is 5 minutes long. 285 00:13:06,500 --> 00:13:07,730 That's 50 people-- 286 00:13:07,730 --> 00:13:09,980 1 minute long, 10 people, and so forth. 287 00:13:09,980 --> 00:13:11,690 And then for each of those headways, 288 00:13:11,690 --> 00:13:14,210 the average waiting time is going 289 00:13:14,210 --> 00:13:18,514 to be half of that headway, correct? 290 00:13:18,514 --> 00:13:19,430 AUDIENCE: Correct. 291 00:13:19,430 --> 00:13:21,096 GABRIEL SANCHEZ-MARTINEZ: We're assuming 292 00:13:21,096 --> 00:13:23,690 people arrive randomly, so within a particular headway, 293 00:13:23,690 --> 00:13:26,756 this is true. 294 00:13:26,756 --> 00:13:28,380 There's no problem with this assumption 295 00:13:28,380 --> 00:13:30,213 as long as you are comfortable with assuming 296 00:13:30,213 --> 00:13:32,840 that people arrive randomly. 297 00:13:32,840 --> 00:13:37,430 So now we have this equation for expected waiting time. 298 00:13:42,140 --> 00:13:47,490 Let me write on the board how we get to that from the integrals. 299 00:14:05,620 --> 00:14:08,770 Can somebody say why-- 300 00:14:08,770 --> 00:14:15,100 so why is it the case that, when we have headway variability, 301 00:14:15,100 --> 00:14:17,050 the waiting time goes up? 302 00:14:17,050 --> 00:14:20,651 What's the concept that leads to that? 303 00:14:20,651 --> 00:14:21,150 Eli? 304 00:14:21,150 --> 00:14:22,525 AUDIENCE: There is unreliability, 305 00:14:22,525 --> 00:14:26,360 and so people need to build in more [INAUDIBLE] time instead 306 00:14:26,360 --> 00:14:26,860 of-- 307 00:14:26,860 --> 00:14:28,276 GABRIEL SANCHEZ-MARTINEZ: There is 308 00:14:28,276 --> 00:14:31,400 something else that is sort of more fundamental happening. 309 00:14:31,400 --> 00:14:34,906 AUDIENCE: A bus that comes at a larger headway 310 00:14:34,906 --> 00:14:36,280 than it was planned will actually 311 00:14:36,280 --> 00:14:37,610 be collecting more people. 312 00:14:37,610 --> 00:14:41,890 So more people will be waiting more time. 313 00:14:41,890 --> 00:14:44,440 GABRIEL SANCHEZ-MARTINEZ: So from a passenger's perspective, 314 00:14:44,440 --> 00:14:46,481 the probability of arriving during a long headway 315 00:14:46,481 --> 00:14:48,677 is greater than arriving during a short headway. 316 00:14:48,677 --> 00:14:49,510 What is that called? 317 00:14:49,510 --> 00:14:52,974 There is a name for that that's important in transportation. 318 00:14:55,640 --> 00:15:02,200 Random-- random incidence, so this is something you, if you 319 00:15:02,200 --> 00:15:05,030 took 200, you should know that. 320 00:15:05,030 --> 00:15:08,350 But it's good to know, random incidence. 321 00:15:08,350 --> 00:15:11,860 It's called the random incidence paradox, 322 00:15:11,860 --> 00:15:14,770 because people without-- if you don't think about it too much, 323 00:15:14,770 --> 00:15:16,020 you say it's half the headway. 324 00:15:16,020 --> 00:15:19,870 And then somebody says, no, actually the average waiting 325 00:15:19,870 --> 00:15:21,140 time is much longer. 326 00:15:21,140 --> 00:15:21,980 Why is that? 327 00:15:21,980 --> 00:15:24,070 And you have to sort of solve the paradox. 328 00:15:24,070 --> 00:15:26,840 It isn't really a paradox, but here we are. 329 00:15:30,895 --> 00:15:33,860 So the expected waiting time-- 330 00:15:33,860 --> 00:15:38,560 we'll start from the equation, the last equation 331 00:15:38,560 --> 00:15:40,330 on slide four. 332 00:15:40,330 --> 00:15:43,180 And we're just going to plug in what our assumption is. 333 00:15:43,180 --> 00:15:48,500 So we have integral from 0 to infinity. 334 00:15:48,500 --> 00:15:52,240 And we said we had n of h. 335 00:15:52,240 --> 00:15:54,520 And we said that that's going to be lambda h. 336 00:15:54,520 --> 00:16:01,900 So let's just put that in, lambda h. 337 00:16:01,900 --> 00:16:05,470 And then we said, for waiting time, 338 00:16:05,470 --> 00:16:12,550 we said it was going to be 1/2, so 1/2 h. 339 00:16:12,550 --> 00:16:15,900 And then we're going to multiply times the PVF 340 00:16:15,900 --> 00:16:19,930 the probability of that headway times dh. 341 00:16:32,990 --> 00:16:36,392 All right, and here we have the same, 342 00:16:36,392 --> 00:16:37,600 but without the waiting time. 343 00:16:41,980 --> 00:16:43,600 So that's just substitution, shouldn't 344 00:16:43,600 --> 00:16:44,805 be anything wrong with that. 345 00:16:44,805 --> 00:16:50,410 So now, lambda and 1/2 come out. 346 00:16:50,410 --> 00:16:52,900 They're not a function of h, so they are constants. 347 00:16:52,900 --> 00:16:54,390 We can take them out. 348 00:16:54,390 --> 00:17:02,290 And we are left with, let's see, lambda over 2 integral from 0 349 00:17:02,290 --> 00:17:07,390 to infinity of h squared-- 350 00:17:07,390 --> 00:17:10,960 because we have two h's here-- 351 00:17:10,960 --> 00:17:15,819 times dh dh. 352 00:17:15,819 --> 00:17:19,140 And here we have-- lambda comes out. 353 00:17:19,140 --> 00:17:26,589 And we integrate from 0 to infinity of h dh dh. 354 00:17:26,589 --> 00:17:30,140 OK, now the first observation is that lambda 355 00:17:30,140 --> 00:17:33,770 has come out on both sides. 356 00:17:33,770 --> 00:17:36,500 So there's just one lambda on the top, one on the bottom, 357 00:17:36,500 --> 00:17:38,690 so we can cancel them out. 358 00:17:38,690 --> 00:17:40,670 So that's very convenient. 359 00:17:40,670 --> 00:17:42,230 What does that mean? 360 00:17:42,230 --> 00:17:44,030 This happens because we are assuming 361 00:17:44,030 --> 00:17:45,870 that lambda is constant. 362 00:17:45,870 --> 00:17:47,720 That wouldn't happen if it weren't constant. 363 00:17:47,720 --> 00:17:50,390 But that's convenient, right? 364 00:17:50,390 --> 00:17:52,760 So now we have some quantity that does not 365 00:17:52,760 --> 00:17:55,580 depend on the arrival rate. 366 00:17:55,580 --> 00:17:58,820 So the same equation is not affected by that. 367 00:17:58,820 --> 00:18:03,070 And let's see if we can recognize 368 00:18:03,070 --> 00:18:04,720 what these quantities are. 369 00:18:04,720 --> 00:18:08,000 So let's start with the one on the bottom. 370 00:18:08,000 --> 00:18:11,680 This is the definition of the expectancy. 371 00:18:11,680 --> 00:18:14,570 So this is just the average. 372 00:18:14,570 --> 00:18:19,330 So we're essentially saying, take every-- 373 00:18:19,330 --> 00:18:21,550 take the average headway, so take every headway, 374 00:18:21,550 --> 00:18:23,920 and multiply times the probability of observing it. 375 00:18:23,920 --> 00:18:25,610 Add them all up. 376 00:18:25,610 --> 00:18:28,520 That's what the average is. 377 00:18:28,520 --> 00:18:36,290 So we have 2 times the expected headway. 378 00:18:36,290 --> 00:18:40,570 And at the top, we have the expectants of h squared. 379 00:18:44,180 --> 00:18:47,700 So I think that takes us to where we are 380 00:18:47,700 --> 00:18:51,450 on this slide, which is great. 381 00:18:51,450 --> 00:18:55,620 So part of it was just recognizing 382 00:18:55,620 --> 00:18:58,410 that that was the definition of expectants, so that's great. 383 00:18:58,410 --> 00:19:05,000 And now, somehow, we go from here to this equation. 384 00:19:05,000 --> 00:19:12,730 It's not entirely clear, but it's convenient 385 00:19:12,730 --> 00:19:17,260 that variance is a function of the expectants of h squared. 386 00:19:17,260 --> 00:19:20,020 So I'll just remind you, if you haven't seen it, 387 00:19:20,020 --> 00:19:23,179 then introduce you to the definition-- or not quite 388 00:19:23,179 --> 00:19:24,220 the definition, actually. 389 00:19:24,220 --> 00:19:27,400 This is you take the definition and you expand it 390 00:19:27,400 --> 00:19:29,080 and collect like terms. 391 00:19:29,080 --> 00:19:33,590 And then you get that the variance of h 392 00:19:33,590 --> 00:19:46,450 is the expectation of h squared minus the expectation of h 393 00:19:46,450 --> 00:19:51,700 quantity squared, so means of square minus squares mean. 394 00:19:51,700 --> 00:19:53,280 You might have seen that before. 395 00:19:53,280 --> 00:19:59,520 So essentially, if we solve that equation for expectation of h 396 00:19:59,520 --> 00:20:02,930 squared and plug that in here, we can collect like terms 397 00:20:02,930 --> 00:20:05,580 and get this last equation. 398 00:20:05,580 --> 00:20:09,170 So now we have that the expected waiting 399 00:20:09,170 --> 00:20:13,680 time is half the headway times some adjustment factor. 400 00:20:13,680 --> 00:20:18,170 And that adjustment factor is 1 plus some amount-- 401 00:20:18,170 --> 00:20:22,340 that amount is what we call the coefficient of variation-- 402 00:20:22,340 --> 00:20:22,879 squared. 403 00:20:22,879 --> 00:20:24,920 So coefficient of variation is standard deviation 404 00:20:24,920 --> 00:20:26,010 divided by mean. 405 00:20:26,010 --> 00:20:28,360 So we have normalizing the standard deviation. 406 00:20:28,360 --> 00:20:29,450 You square that quantity. 407 00:20:29,450 --> 00:20:30,940 You add it to 1. 408 00:20:30,940 --> 00:20:35,310 And that'll increase your waiting time by some amount. 409 00:20:38,510 --> 00:20:40,160 Let's stop here for a second. 410 00:20:40,160 --> 00:20:43,550 Any questions with the derivation and what 411 00:20:43,550 --> 00:20:44,570 the meaning of this is? 412 00:20:47,200 --> 00:20:47,790 Yes, Henry? 413 00:20:47,790 --> 00:20:49,410 AUDIENCE: Can you go back to that step 414 00:20:49,410 --> 00:20:52,939 where you went from the integral to 2 e of h. 415 00:20:52,939 --> 00:20:54,230 GABRIEL SANCHEZ-MARTINEZ: Here? 416 00:20:54,230 --> 00:20:54,890 AUDIENCE: Yeah. 417 00:20:54,890 --> 00:20:57,670 GABRIEL SANCHEZ-MARTINEZ: So this is a matter of definition. 418 00:20:57,670 --> 00:21:00,010 So this here, the integral from 0-- 419 00:21:00,010 --> 00:21:03,000 from negative infinity to positive infinity of some 420 00:21:03,000 --> 00:21:07,540 amount times the probability of observing that amount over all 421 00:21:07,540 --> 00:21:09,490 possibilities of the amount-- 422 00:21:09,490 --> 00:21:12,220 that's why we go from negative infinity to positive infinity-- 423 00:21:12,220 --> 00:21:15,020 equals the average of that amount, 424 00:21:15,020 --> 00:21:17,080 or the expectation of that amount. 425 00:21:17,080 --> 00:21:20,150 So that part-- do you understand? 426 00:21:20,150 --> 00:21:21,160 AUDIENCE: [INAUDIBLE] 427 00:21:21,160 --> 00:21:22,520 GABRIEL SANCHEZ-MARTINEZ: Yes? 428 00:21:22,520 --> 00:21:24,890 And then here, we're just saying, well, 429 00:21:24,890 --> 00:21:27,800 this is the same thing, except of the amount instead 430 00:21:27,800 --> 00:21:30,090 of being h is h squared. 431 00:21:30,090 --> 00:21:34,070 So we're taking the average of h squared. 432 00:21:34,070 --> 00:21:36,230 This comes from here to here from the definition 433 00:21:36,230 --> 00:21:39,120 of expectation. 434 00:21:39,120 --> 00:21:41,540 And the reason we're not going from negative infinity 435 00:21:41,540 --> 00:21:44,270 to positive infinity is that h can only 436 00:21:44,270 --> 00:21:46,610 take non-negative values. 437 00:21:46,610 --> 00:21:48,230 It can be 0 or higher. 438 00:21:48,230 --> 00:21:51,890 So it's the same thing to integrate from 0 to infinity, 439 00:21:51,890 --> 00:21:54,310 in that case. 440 00:21:54,310 --> 00:21:57,030 Does that clear up the question? 441 00:21:57,030 --> 00:21:58,285 Any other questions? 442 00:22:01,170 --> 00:22:09,030 So now we have this modified model of waiting time, 443 00:22:09,030 --> 00:22:10,560 mainly for high-frequency service, 444 00:22:10,560 --> 00:22:12,310 where people are arriving randomly 445 00:22:12,310 --> 00:22:15,070 but vehicles are not necessarily departing stops 446 00:22:15,070 --> 00:22:17,760 deterministically on equal intervals anymore. 447 00:22:17,760 --> 00:22:20,730 Now these vehicles can come-- 448 00:22:20,730 --> 00:22:23,300 as long as they come independent of the passenger arrival 449 00:22:23,300 --> 00:22:26,780 process, it doesn't really matter how they come. 450 00:22:26,780 --> 00:22:31,260 This model calculates the expected waiting time 451 00:22:31,260 --> 00:22:32,650 across all passengers. 452 00:22:32,650 --> 00:22:34,740 So we have some cases that we can look at. 453 00:22:34,740 --> 00:22:36,360 First case is simple. 454 00:22:36,360 --> 00:22:38,430 Let's say that the variance is 0. 455 00:22:38,430 --> 00:22:40,980 That means we go back to the assumption of, 456 00:22:40,980 --> 00:22:43,740 well, if the variance of headway is 0, 457 00:22:43,740 --> 00:22:47,120 vehicles are arriving exactly every five minutes, say. 458 00:22:47,120 --> 00:22:50,890 They're deterministically at equally spaced intervals. 459 00:22:50,890 --> 00:22:55,610 So we then have that the coefficients of variation is 0. 460 00:22:55,610 --> 00:22:57,610 And we end up with the same thing we had before. 461 00:22:57,610 --> 00:23:00,180 So that really checks out with our previous model 462 00:23:00,180 --> 00:23:03,570 in that special case, but we generalized the model. 463 00:23:03,570 --> 00:23:06,580 So we can try it with different vehicle arrival or departure 464 00:23:06,580 --> 00:23:08,150 processes. 465 00:23:08,150 --> 00:23:11,002 One that we have here is, let's say that vehicle departures are 466 00:23:11,002 --> 00:23:12,820 a Poisson process. 467 00:23:12,820 --> 00:23:16,770 So this would mean that they are arriving randomly, essentially, 468 00:23:16,770 --> 00:23:21,690 and that the time of arrival of a vehicle at a stop 469 00:23:21,690 --> 00:23:24,270 is independent of any other previous arrival. 470 00:23:24,270 --> 00:23:27,480 So that would be a good model for a service that 471 00:23:27,480 --> 00:23:29,610 is controlled very loosely. 472 00:23:29,610 --> 00:23:32,220 They are hardly controlled. 473 00:23:32,220 --> 00:23:34,290 So you essentially sometimes see bunching. 474 00:23:34,290 --> 00:23:35,130 Sometimes you don't. 475 00:23:35,130 --> 00:23:36,546 Sometimes you have a long headway. 476 00:23:36,546 --> 00:23:38,080 Sometimes you don't. 477 00:23:38,080 --> 00:23:43,230 So the definition-- if you go through the Poisson process 478 00:23:43,230 --> 00:23:46,710 and look at the definitions of the probability distributions 479 00:23:46,710 --> 00:23:48,720 of inter-arrival times, et cetera, 480 00:23:48,720 --> 00:23:51,630 you will see that the variance in such a process 481 00:23:51,630 --> 00:23:53,700 is the square of the expectation, so 482 00:23:53,700 --> 00:23:55,300 the square of the mean. 483 00:23:55,300 --> 00:23:58,600 And if we plug that into our equation, 484 00:23:58,600 --> 00:24:03,120 we get that the expected waiting time is the expected headway. 485 00:24:03,120 --> 00:24:07,620 So people are essentially waiting about as long 486 00:24:07,620 --> 00:24:11,210 as your mean headway, which is a bad situation. 487 00:24:11,210 --> 00:24:13,470 You've doubled their waiting time 488 00:24:13,470 --> 00:24:17,970 from half the headway to the full headway. 489 00:24:17,970 --> 00:24:21,150 If you have a system where your vehicles are all bunched 490 00:24:21,150 --> 00:24:23,460 and you're running pairs of bunches, 491 00:24:23,460 --> 00:24:28,650 that's more or less what you get, because you have-- 492 00:24:28,650 --> 00:24:30,410 if you had vehicles every 5 minutes, now 493 00:24:30,410 --> 00:24:33,672 you have two vehicles every 10. 494 00:24:33,672 --> 00:24:37,470 Your average headway is five, but your waiting time 495 00:24:37,470 --> 00:24:40,700 is now five as well. 496 00:24:40,700 --> 00:24:43,290 OK, what about the third case? 497 00:24:43,290 --> 00:24:47,070 The headway sequence is 5, 15, 5, 15, 5, 15-- 498 00:24:47,070 --> 00:24:49,710 thank you, keep going like that, so not quite bunched 499 00:24:49,710 --> 00:24:52,240 all the way, but there is a lot of variability. 500 00:24:52,240 --> 00:24:55,260 So I'm saying that the expected headway is 10. 501 00:24:55,260 --> 00:24:56,836 Does that make sense? 502 00:24:56,836 --> 00:24:58,120 It's 5, 15, 5, 15. 503 00:24:58,120 --> 00:25:01,520 The average of that process is 10. 504 00:25:01,520 --> 00:25:04,840 So we have an average headway of 10. 505 00:25:04,840 --> 00:25:07,084 And now what's the expected waiting time? 506 00:25:10,720 --> 00:25:19,630 So it helps to draw a timeline and to divide that timeline 507 00:25:19,630 --> 00:25:36,910 into pieces, so 5, 15, 5, 15. 508 00:25:36,910 --> 00:25:47,670 So we have vehicles arriving or departing here, here, here, 509 00:25:47,670 --> 00:25:49,690 and here. 510 00:25:49,690 --> 00:25:53,154 And there is a five-minute headway here, 511 00:25:53,154 --> 00:25:55,570 and a 15-minute headway here, a five-minute headway there, 512 00:25:55,570 --> 00:25:57,040 and a 15-minute headway here. 513 00:25:57,040 --> 00:25:58,860 And people are arriving randomly. 514 00:25:58,860 --> 00:26:01,450 People arrive independent of this process. 515 00:26:01,450 --> 00:26:05,260 So what you'll see is that, the first thing 516 00:26:05,260 --> 00:26:09,130 is, if you were to arrive on a five-minute headway, 517 00:26:09,130 --> 00:26:11,080 the average waiting time for you would 518 00:26:11,080 --> 00:26:15,470 be half of that five-minute headway, 2 and 1/2 minutes. 519 00:26:15,470 --> 00:26:18,470 And if you arrive on 15-minute headway, 520 00:26:18,470 --> 00:26:20,530 then your expected waiting time, given 521 00:26:20,530 --> 00:26:24,580 that you arrive on a 15-minute headway is 7 and 1/2 minutes. 522 00:26:24,580 --> 00:26:28,360 So that's the 2.5 and the 7.5 on this equation. 523 00:26:28,360 --> 00:26:29,980 And then we have the probabilities 524 00:26:29,980 --> 00:26:32,350 of having arrived on any of those headways. 525 00:26:32,350 --> 00:26:35,380 Because the 15-minute headway is 3 times longer 526 00:26:35,380 --> 00:26:38,110 than the 5-minute headway, you are 3 times more 527 00:26:38,110 --> 00:26:40,930 likely to arrive during a 15-minute headway 528 00:26:40,930 --> 00:26:42,310 than during a 5-minute headway. 529 00:26:42,310 --> 00:26:46,600 So therefore, 25% of passengers will arrive 530 00:26:46,600 --> 00:26:48,280 on the 15-minute headways. 531 00:26:48,280 --> 00:26:53,320 And you essentially computing a weighted average, 532 00:26:53,320 --> 00:26:56,560 so 6 and 1/4 minutes. 533 00:26:59,482 --> 00:27:00,456 Questions on this? 534 00:27:13,130 --> 00:27:15,380 All right, another assumption that we 535 00:27:15,380 --> 00:27:17,060 were looking at in the first model 536 00:27:17,060 --> 00:27:18,770 that we are still having in this model 537 00:27:18,770 --> 00:27:20,730 is that people can board the first vehicle 538 00:27:20,730 --> 00:27:22,730 that they see arrive. 539 00:27:22,730 --> 00:27:26,870 So no vehicle is too full to take passengers. 540 00:27:26,870 --> 00:27:28,850 No vehicle is leaving people behind. 541 00:27:28,850 --> 00:27:29,975 That's not true. 542 00:27:29,975 --> 00:27:34,310 If we look at any process, and we say here 543 00:27:34,310 --> 00:27:37,880 that w0 is the expected waiting time 544 00:27:37,880 --> 00:27:40,580 when people can board their first vehicle, 545 00:27:40,580 --> 00:27:45,290 so when no vehicle is full, and we then 546 00:27:45,290 --> 00:27:48,630 look at all the expected waiting times for passengers including 547 00:27:48,630 --> 00:27:52,490 passengers that are left behind, when that capacity is 548 00:27:52,490 --> 00:27:53,540 being reached-- 549 00:27:53,540 --> 00:27:55,640 so here on the horizontal, we have rho, 550 00:27:55,640 --> 00:27:57,820 which is flow over capacity. 551 00:28:00,209 --> 00:28:02,000 You may have seen this kind of relationship 552 00:28:02,000 --> 00:28:02,940 from queuing theory. 553 00:28:02,940 --> 00:28:06,470 So when rho is 0, it means that you will have some supply 554 00:28:06,470 --> 00:28:08,090 and nobody is using it. 555 00:28:08,090 --> 00:28:11,270 If rho is 0.5, then your capacity 556 00:28:11,270 --> 00:28:14,780 is double of the demand. 557 00:28:14,780 --> 00:28:16,250 That's OK usually. 558 00:28:16,250 --> 00:28:18,590 If rho is 1, you are-- 559 00:28:18,590 --> 00:28:20,600 your capacity equals your demand. 560 00:28:20,600 --> 00:28:23,780 And what we see is that the expected waiting 561 00:28:23,780 --> 00:28:27,735 time for service will start to approach infinity. 562 00:28:27,735 --> 00:28:31,130 There is a term that we use in, say, queuing theory, 563 00:28:31,130 --> 00:28:32,466 for such a system. 564 00:28:32,466 --> 00:28:33,340 Does anybody know it? 565 00:28:38,370 --> 00:28:42,930 We say that the system blows up when this happens. 566 00:28:42,930 --> 00:28:45,000 If you hear that, you'll see that. 567 00:28:45,000 --> 00:28:49,190 If you take any sort of queuing theory, you'll hear that. 568 00:28:49,190 --> 00:28:52,980 So as rho equals 1, let's say that you have-- you're 569 00:28:52,980 --> 00:28:55,620 providing a bus service with capacity 570 00:28:55,620 --> 00:28:59,550 for 100 people per minute. 571 00:28:59,550 --> 00:29:03,560 That's pretty high. 572 00:29:03,560 --> 00:29:07,060 And then your demand equals 100 people per minute. 573 00:29:07,060 --> 00:29:09,540 So if people are arriving uniformly, 574 00:29:09,540 --> 00:29:12,450 you might just provide a capacity for everyone. 575 00:29:12,450 --> 00:29:14,160 And nobody is being left behind. 576 00:29:14,160 --> 00:29:16,870 But that doesn't happen, because people arrive randomly. 577 00:29:20,940 --> 00:29:23,810 Some people will leave with some remaining capacity. 578 00:29:23,810 --> 00:29:26,670 And the next vehicle won't have enough space for that person. 579 00:29:26,670 --> 00:29:28,890 And then the vehicles themselves will not 580 00:29:28,890 --> 00:29:31,650 arrive perfectly evenly, so now you 581 00:29:31,650 --> 00:29:34,710 have a chain reaction of people waiting and not 582 00:29:34,710 --> 00:29:36,000 being able to board. 583 00:29:36,000 --> 00:29:38,640 And unless you supply more capacity, 584 00:29:38,640 --> 00:29:40,410 unless you increase the capacity, 585 00:29:40,410 --> 00:29:42,600 you're going to have more and more people waiting 586 00:29:42,600 --> 00:29:43,960 and being left behind. 587 00:29:43,960 --> 00:29:46,200 So the actual expected waiting time 588 00:29:46,200 --> 00:29:51,160 would increase as your ratio of flow to capacity approaches 1. 589 00:29:51,160 --> 00:29:53,220 That's the takeaway here. 590 00:29:53,220 --> 00:29:58,050 And you could think of two different lines, one 591 00:29:58,050 --> 00:30:01,260 for low reliability when there is a lot of bunching, 592 00:30:01,260 --> 00:30:03,540 and therefore, a lot of the capacity that you provide 593 00:30:03,540 --> 00:30:07,440 is wasted on small headways that are not 594 00:30:07,440 --> 00:30:11,400 serving as many passengers as you assign them for, and then 595 00:30:11,400 --> 00:30:13,560 high reliability, where, yes, you still 596 00:30:13,560 --> 00:30:17,250 have some variability that you can't control. 597 00:30:17,250 --> 00:30:18,320 You've done what you can. 598 00:30:18,320 --> 00:30:20,940 And this is relatively high reliability, 599 00:30:20,940 --> 00:30:23,550 but still you have some variability, 600 00:30:23,550 --> 00:30:28,020 and therefore, you can't expect to reach that rho equals 1. 601 00:30:28,020 --> 00:30:29,310 It'll still blow up when you-- 602 00:30:29,310 --> 00:30:30,770 as you approach 1. 603 00:30:30,770 --> 00:30:33,870 But it's closer to 1, so that's good. 604 00:30:33,870 --> 00:30:36,670 Questions on this relationship? 605 00:30:36,670 --> 00:30:37,170 [INAUDIBLE]? 606 00:30:37,170 --> 00:30:40,032 AUDIENCE: What do most bus services operate 607 00:30:40,032 --> 00:30:41,902 at [INAUDIBLE]? 608 00:30:41,902 --> 00:30:43,860 GABRIEL SANCHEZ-MARTINEZ: I don't have a number 609 00:30:43,860 --> 00:30:45,370 off the top of my head. 610 00:30:45,370 --> 00:30:46,370 It varies a lot. 611 00:30:46,370 --> 00:30:48,390 And it varies a lot of by time period. 612 00:30:48,390 --> 00:30:54,290 So some services here in Boston operate during the peak leaving 613 00:30:54,290 --> 00:30:57,230 people behind every day. 614 00:30:57,230 --> 00:31:00,230 If you look at some of the most crowded stations 615 00:31:00,230 --> 00:31:03,260 in the tube in London, certainly they have to meter people 616 00:31:03,260 --> 00:31:04,630 going into stations. 617 00:31:04,630 --> 00:31:09,970 So they may have approached rho greater than 1, in this case. 618 00:31:09,970 --> 00:31:11,270 But it's non-stationary. 619 00:31:11,270 --> 00:31:13,400 So that's during the peak of the peak. 620 00:31:13,400 --> 00:31:17,120 And then the demand falls, so eventually, those people 621 00:31:17,120 --> 00:31:21,410 are served, of course, because the supply is kept up 622 00:31:21,410 --> 00:31:24,350 at some-- at the highest rate that they can keep it. 623 00:31:24,350 --> 00:31:26,660 And demand it comes down, so it stabilizes. 624 00:31:30,380 --> 00:31:31,340 Any other questions? 625 00:31:31,340 --> 00:31:32,298 That's a good question. 626 00:31:35,880 --> 00:31:37,750 All right, so service variation along 627 00:31:37,750 --> 00:31:41,060 routes-- so we've been talking about vehicles 628 00:31:41,060 --> 00:31:44,270 leaving on time or not on time, and bunching. 629 00:31:44,270 --> 00:31:46,910 So here's what happens. 630 00:31:46,910 --> 00:31:49,040 This is a space-time diagram. 631 00:31:49,040 --> 00:31:52,370 Have we seen these diagrams on this course yet? 632 00:31:52,370 --> 00:31:53,090 AUDIENCE: No. 633 00:31:53,090 --> 00:31:54,923 GABRIEL SANCHEZ-MARTINEZ: No-- OK, well this 634 00:31:54,923 --> 00:31:57,200 is a very important diagram in public transportation, 635 00:31:57,200 --> 00:31:57,380 certainly. 636 00:31:57,380 --> 00:31:58,540 AUDIENCE: And the homework says to make one. 637 00:31:58,540 --> 00:31:58,876 So we should probably-- 638 00:31:58,876 --> 00:31:58,960 GABRIEL SANCHEZ-MARTINEZ: Great. 639 00:31:58,960 --> 00:32:00,234 AUDIENCE: --learn it. 640 00:32:00,234 --> 00:32:01,400 I was wondering what it was. 641 00:32:01,400 --> 00:32:03,691 GABRIEL SANCHEZ-MARTINEZ: This is a space-time diagram. 642 00:32:03,691 --> 00:32:09,180 You have space on the y-axis and time on the x-axis. 643 00:32:09,180 --> 00:32:13,040 And what you see are some lines that 644 00:32:13,040 --> 00:32:16,550 show how a vehicle is moving across space and time. 645 00:32:16,550 --> 00:32:20,380 So there are variations of this. 646 00:32:20,380 --> 00:32:23,510 Sometimes people put time on the vertical and space 647 00:32:23,510 --> 00:32:24,460 on the horizontal. 648 00:32:24,460 --> 00:32:25,400 That's fine. 649 00:32:25,400 --> 00:32:29,390 And typically, you see little steps for each stop, 650 00:32:29,390 --> 00:32:31,490 because if a vehicle is stopped, time is running 651 00:32:31,490 --> 00:32:35,910 and the vehicle is not moving, so you see little steps. 652 00:32:35,910 --> 00:32:37,220 So you can see holding. 653 00:32:37,220 --> 00:32:38,300 You can see dwell times. 654 00:32:38,300 --> 00:32:40,790 You can see speeds. 655 00:32:40,790 --> 00:32:42,980 You can see recovery time at a terminal, everything 656 00:32:42,980 --> 00:32:43,880 on this diagram. 657 00:32:43,880 --> 00:32:47,001 So it's a great diagram for analyzing service. 658 00:32:47,001 --> 00:32:49,250 AUDIENCE: When we make, should we make them like this? 659 00:32:49,250 --> 00:32:49,890 Or like-- 660 00:32:49,890 --> 00:32:51,890 GABRIEL SANCHEZ-MARTINEZ: The little steps are-- 661 00:32:51,890 --> 00:32:52,540 AUDIENCE: Little steps-- 662 00:32:52,540 --> 00:32:54,990 GABRIEL SANCHEZ-MARTINEZ: Yeah, I mean, they're helpful. 663 00:32:54,990 --> 00:32:56,910 You'll see dwell times, so yeah. 664 00:32:56,910 --> 00:32:59,760 So this is a conceptual very simplified diagram, 665 00:32:59,760 --> 00:33:01,850 but here is the idea. 666 00:33:01,850 --> 00:33:06,230 So you have some scheduled trajectories in dashed lines. 667 00:33:06,230 --> 00:33:10,130 So we're essentially planning to run service every 10 minutes, 668 00:33:10,130 --> 00:33:15,540 departing from Stop 1 at 9:00, 9:10, 9:20, et cetera. 669 00:33:15,540 --> 00:33:17,990 And if things run according to plan, 670 00:33:17,990 --> 00:33:20,664 you will keep a 10-minute headway throughout the route. 671 00:33:20,664 --> 00:33:21,830 But that's not what happens. 672 00:33:21,830 --> 00:33:25,610 If we observe-- let's say that the driver that 673 00:33:25,610 --> 00:33:27,990 was driving the bus that departed at 9:10 674 00:33:27,990 --> 00:33:32,780 is a slow driver, so that driver drives slowly. 675 00:33:32,780 --> 00:33:36,960 So therefore, more time passes covering the same space. 676 00:33:36,960 --> 00:33:40,130 And then let's say that the 9:20 driver is a fast driver. 677 00:33:40,130 --> 00:33:42,890 So that driver covers more distance in less time. 678 00:33:42,890 --> 00:33:44,210 And they bunch. 679 00:33:44,210 --> 00:33:46,950 They meet somewhere before they reach the end of the route. 680 00:33:46,950 --> 00:33:49,580 They've platooned. 681 00:33:49,580 --> 00:33:53,210 And they're running together as a bunch. 682 00:33:53,210 --> 00:33:54,830 So what does that do to headway? 683 00:33:59,056 --> 00:34:00,680 AUDIENCE: What does what do to headway? 684 00:34:00,680 --> 00:34:01,990 GABRIEL SANCHEZ-MARTINEZ: What does this process of bunching 685 00:34:01,990 --> 00:34:03,172 do to headways? 686 00:34:03,172 --> 00:34:04,130 AUDIENCE: It increases. 687 00:34:04,130 --> 00:34:06,060 GABRIEL SANCHEZ-MARTINEZ: Right? 688 00:34:06,060 --> 00:34:08,800 Headways are increasing, because now you've-- 689 00:34:08,800 --> 00:34:10,690 well, the average headway remains the same, 690 00:34:10,690 --> 00:34:12,760 because you have a 0 and a big number. 691 00:34:12,760 --> 00:34:15,730 But what people actually see are the long headways, 692 00:34:15,730 --> 00:34:18,580 because the chance of arriving and just catching 693 00:34:18,580 --> 00:34:22,480 the 30-second headway is much smaller. 694 00:34:22,480 --> 00:34:24,699 So people are waiting more because these headways are 695 00:34:24,699 --> 00:34:25,810 longer. 696 00:34:25,810 --> 00:34:26,310 Yeah? 697 00:34:26,310 --> 00:34:27,940 AUDIENCE: Well, you probably have more people waiting. 698 00:34:27,940 --> 00:34:29,108 So there is [INAUDIBLE]-- 699 00:34:29,108 --> 00:34:30,858 GABRIEL SANCHEZ-MARTINEZ: Yeah, so there's 700 00:34:30,858 --> 00:34:32,659 a vicious cycle here, right? 701 00:34:32,659 --> 00:34:34,876 Yeah, we'll get to that on the next slide. 702 00:34:34,876 --> 00:34:37,001 AUDIENCE: It would be better if the fast driver got 703 00:34:37,001 --> 00:34:39,030 in front of the slow driver. 704 00:34:39,030 --> 00:34:41,710 GABRIEL SANCHEZ-MARTINEZ: Yeah, we'll talk about that too. 705 00:34:41,710 --> 00:34:42,793 We'll talk about that too. 706 00:34:45,690 --> 00:34:50,436 OK, so do we understand this process of bunching? 707 00:34:50,436 --> 00:34:52,760 Here is what you were talking about, Ari? 708 00:34:52,760 --> 00:34:54,909 So there is an issue when that happens. 709 00:34:54,909 --> 00:34:56,860 So now we have the little steps here. 710 00:34:56,860 --> 00:34:59,160 So this is Step 2, Step 3. 711 00:34:59,160 --> 00:35:01,180 And these are the dwell times. 712 00:35:01,180 --> 00:35:03,055 So nothing happened on bus one. 713 00:35:03,055 --> 00:35:06,340 It ran exactly as planned in this hypothetical example. 714 00:35:06,340 --> 00:35:09,990 But because this driver was slow, the headway was bigger. 715 00:35:09,990 --> 00:35:11,930 And more people arrived during the headway. 716 00:35:11,930 --> 00:35:14,410 So now the dwell times at Stop 2 are 717 00:35:14,410 --> 00:35:17,290 going to be longer for that driver that was driving slow. 718 00:35:17,290 --> 00:35:19,062 Now that bus has more passengers. 719 00:35:19,062 --> 00:35:21,520 And therefore, the probability of stopping at the next stop 720 00:35:21,520 --> 00:35:23,200 is higher, because you have more people, 721 00:35:23,200 --> 00:35:26,620 so the chance that at least one person on that bus 722 00:35:26,620 --> 00:35:29,980 wants to get off at the next one is higher. 723 00:35:29,980 --> 00:35:31,779 And if the bus is really crowded, 724 00:35:31,779 --> 00:35:33,820 the dwell time process is going to be slowed down 725 00:35:33,820 --> 00:35:38,610 by just friction between passengers. 726 00:35:38,610 --> 00:35:41,830 So this bus continues to be delayed, not just 727 00:35:41,830 --> 00:35:46,840 by the driver driving slow, but by the dwell time effect. 728 00:35:46,840 --> 00:35:49,270 And because that bus is being delayed, 729 00:35:49,270 --> 00:35:54,460 fewer people are arriving to see the next bus, the third bus. 730 00:35:57,090 --> 00:36:02,080 The dwell times of that third bus are shorter than planned. 731 00:36:02,080 --> 00:36:03,700 Fewer people are waiting for that bus. 732 00:36:03,700 --> 00:36:06,520 And therefore, that bus has a lower probability 733 00:36:06,520 --> 00:36:07,710 of having to stop. 734 00:36:07,710 --> 00:36:10,300 And if it stops, fewer people will board it. 735 00:36:10,300 --> 00:36:12,510 So that bus is going to run fast. 736 00:36:12,510 --> 00:36:16,374 Even if the drivers are now driving at the same speed, 737 00:36:16,374 --> 00:36:17,540 there is nothing you can do. 738 00:36:17,540 --> 00:36:18,837 They will bunch. 739 00:36:18,837 --> 00:36:21,170 Well, there is something you can-- you can control them. 740 00:36:21,170 --> 00:36:22,810 And that's a later topic in the course. 741 00:36:22,810 --> 00:36:27,310 So they pair or they bunch, that's what we're saying. 742 00:36:27,310 --> 00:36:30,790 OK, so if we think about how if you 743 00:36:30,790 --> 00:36:33,280 were to survey headways of different points 744 00:36:33,280 --> 00:36:35,320 along the route, you would see that, 745 00:36:35,320 --> 00:36:38,450 if you start at a terminal-- 746 00:36:38,450 --> 00:36:40,670 this is like a probability density function, 747 00:36:40,670 --> 00:36:41,900 but they're not normalized. 748 00:36:41,900 --> 00:36:44,800 So don't think about scale. 749 00:36:44,800 --> 00:36:46,600 So you have headway on the horizontal. 750 00:36:46,600 --> 00:36:50,600 And the probability density on the vertical. 751 00:36:50,600 --> 00:36:53,770 And so at the terminal or close to the start of the route, 752 00:36:53,770 --> 00:36:56,800 you will see something bell-shaped usually. 753 00:36:59,350 --> 00:37:02,620 It'll be around the scheduled departure time, 754 00:37:02,620 --> 00:37:05,320 or around the headway. 755 00:37:05,320 --> 00:37:09,190 And there will be some variability due to drivers 756 00:37:09,190 --> 00:37:13,210 not being exactly on time when they leave, and supervision 757 00:37:13,210 --> 00:37:15,220 issues, or whatever it is. 758 00:37:15,220 --> 00:37:18,650 So maybe the boarding process at the first stop 759 00:37:18,650 --> 00:37:20,260 introduces some perturbations. 760 00:37:20,260 --> 00:37:21,990 But it's essentially bell-shaped. 761 00:37:21,990 --> 00:37:23,930 And it has less variability. 762 00:37:23,930 --> 00:37:26,290 As you move to the middle of the route, 763 00:37:26,290 --> 00:37:30,200 you'll see that the bell is being-- 764 00:37:30,200 --> 00:37:34,930 it's getting fatter and fatter tails, so more variability 765 00:37:34,930 --> 00:37:37,740 in the headway because of the bunching problem 766 00:37:37,740 --> 00:37:40,780 that we just described. 767 00:37:40,780 --> 00:37:43,240 Usually by the time you see bunching, your-- 768 00:37:43,240 --> 00:37:46,780 which could be at the end, or it could be before the end, 769 00:37:46,780 --> 00:37:52,750 you'll have this distribution with two spots, 770 00:37:52,750 --> 00:37:57,190 a lot of vehicles having headway of 0 or very close to 0, 771 00:37:57,190 --> 00:37:59,290 and a lot of headways longer than that 772 00:37:59,290 --> 00:38:02,090 by some amount with a lot of variability. 773 00:38:02,090 --> 00:38:05,470 So these are pairs that are arriving. 774 00:38:05,470 --> 00:38:09,590 So every time you see a pair arriving, one of them is 0. 775 00:38:09,590 --> 00:38:11,620 And the other one is some longer headway. 776 00:38:11,620 --> 00:38:17,524 And they are affecting these two parts of the probability 777 00:38:17,524 --> 00:38:18,856 distribution. 778 00:38:18,856 --> 00:38:22,346 Is that understood? 779 00:38:22,346 --> 00:38:24,292 Any questions? 780 00:38:24,292 --> 00:38:25,224 [INAUDIBLE] 781 00:38:25,224 --> 00:38:27,750 AUDIENCE: So the big point you're talking about 782 00:38:27,750 --> 00:38:29,796 is the lower-- 783 00:38:29,796 --> 00:38:31,545 GABRIEL SANCHEZ-MARTINEZ: So essentially-- 784 00:38:31,545 --> 00:38:32,900 AUDIENCE: --the lower curve? 785 00:38:32,900 --> 00:38:36,200 GABRIEL SANCHEZ-MARTINEZ: --it's just, yes, this one right here. 786 00:38:36,200 --> 00:38:40,940 So essentially, it's just higher variability of headways. 787 00:38:40,940 --> 00:38:45,880 So variability of headways tends to be minimized 788 00:38:45,880 --> 00:38:47,450 at the start of a run. 789 00:38:47,450 --> 00:38:49,005 And then that degrades. 790 00:38:49,005 --> 00:38:52,340 It increases as you run the route until you have bunching. 791 00:38:52,340 --> 00:38:54,560 And when you have bunching, you reach this point 792 00:38:54,560 --> 00:38:58,430 of having some vehicles that are close to 0 and others 793 00:38:58,430 --> 00:38:59,810 that are anywhere else. 794 00:39:02,420 --> 00:39:04,460 So that means the waiting time is 795 00:39:04,460 --> 00:39:08,610 a function of headway as scheduled, 796 00:39:08,610 --> 00:39:10,730 also of headway reliability. 797 00:39:10,730 --> 00:39:13,990 So if you're running a service that is well controlled 798 00:39:13,990 --> 00:39:17,090 and you have good headways, then you may never reach this point. 799 00:39:17,090 --> 00:39:18,590 And you may actually have everything 800 00:39:18,590 --> 00:39:21,530 running somewhat bell-shaped, not too much variance. 801 00:39:21,530 --> 00:39:24,450 That's your ideal situation. 802 00:39:24,450 --> 00:39:27,950 And then it's also a function of where you are along the route. 803 00:39:27,950 --> 00:39:30,554 So if you're closer to a controlled point, which 804 00:39:30,554 --> 00:39:31,970 could be the terminal, or it could 805 00:39:31,970 --> 00:39:35,990 be anywhere along the route, if there is control en route, 806 00:39:35,990 --> 00:39:38,780 then you will have less variability of headways. 807 00:39:41,620 --> 00:39:45,050 As you run downstream of the last control point, 808 00:39:45,050 --> 00:39:48,000 then you will see greater variability. 809 00:39:48,000 --> 00:39:53,510 Great, so what factors affect the headway deterioration? 810 00:39:53,510 --> 00:39:54,830 Length of route is one. 811 00:39:54,830 --> 00:39:59,180 So if he no longer route, then it takes an hour and a half 812 00:39:59,180 --> 00:40:04,340 to cover the whole one-direction run, the half cycle, 813 00:40:04,340 --> 00:40:07,460 then this deterioration has-- 814 00:40:07,460 --> 00:40:11,030 that process has more time to act on the route. 815 00:40:11,030 --> 00:40:16,160 So you will see more headway unreliability, more bunching. 816 00:40:16,160 --> 00:40:19,440 The marginal dwell time per passengers is another factor. 817 00:40:19,440 --> 00:40:21,190 If you think of-- 818 00:40:21,190 --> 00:40:23,270 well, we just saw that the bunching process 819 00:40:23,270 --> 00:40:27,560 is largely affected by dwell time, 820 00:40:27,560 --> 00:40:29,150 because if you have a longer headway, 821 00:40:29,150 --> 00:40:30,191 more people are boarding. 822 00:40:30,191 --> 00:40:32,030 So if you think of an extreme case 823 00:40:32,030 --> 00:40:34,650 where everybody boards and alights instantly, 824 00:40:34,650 --> 00:40:37,010 then you don't have that effect anymore. 825 00:40:37,010 --> 00:40:40,670 So to the extent that people can board and alight 826 00:40:40,670 --> 00:40:45,500 fast relative to the time it takes to run the service, 827 00:40:45,500 --> 00:40:48,350 then this effect diminishes to the extent 828 00:40:48,350 --> 00:40:50,390 that dwell time is a larger percentage 829 00:40:50,390 --> 00:40:51,740 of the total runtime. 830 00:40:51,740 --> 00:40:53,120 Then it increases. 831 00:40:53,120 --> 00:40:54,650 You have more bunching. 832 00:40:54,650 --> 00:40:56,942 The stopping probability-- again, 833 00:40:56,942 --> 00:40:58,400 this has to do with how many people 834 00:40:58,400 --> 00:41:00,825 are on the bus and the stop spacing. 835 00:41:00,825 --> 00:41:04,260 If stops are spaced very close to each other, 836 00:41:04,260 --> 00:41:07,650 then you have a higher variance of where you stop. 837 00:41:13,880 --> 00:41:16,910 if you have a long distance between stations, 838 00:41:16,910 --> 00:41:18,890 if you think of a commuter rail line, 839 00:41:18,890 --> 00:41:20,660 then you're going to stop at every stop. 840 00:41:20,660 --> 00:41:22,640 And therefore, you decrease the variability 841 00:41:22,640 --> 00:41:25,280 of this effect of stopping. 842 00:41:25,280 --> 00:41:29,150 So you have the urban bus route in one hand 843 00:41:29,150 --> 00:41:32,420 and the commuter rail with long distances between stops, 844 00:41:32,420 --> 00:41:35,720 and therefore stopping at every stop, on the other hand. 845 00:41:35,720 --> 00:41:38,030 The schedule headway has an impact. 846 00:41:38,030 --> 00:41:40,820 If you schedule the headways every 30 minutes, 847 00:41:40,820 --> 00:41:42,920 it's unlikely you'll see bunching. 848 00:41:42,920 --> 00:41:46,216 But if you schedule the headways every two minutes, then 849 00:41:46,216 --> 00:41:47,840 it's very easy for you to see bunching. 850 00:41:47,840 --> 00:41:49,460 You have to control this very well 851 00:41:49,460 --> 00:41:52,480 to avoid bunching in that case. 852 00:41:52,480 --> 00:41:55,590 And driver behavior-- if your vehicles 853 00:41:55,590 --> 00:42:01,390 are driven by very good drivers or computers 854 00:42:01,390 --> 00:42:05,770 on the extreme right, driven with driverless trains, 855 00:42:05,770 --> 00:42:08,170 then you can really sort of try to make up 856 00:42:08,170 --> 00:42:10,410 for all of these things in driving 857 00:42:10,410 --> 00:42:12,250 and try to keep things even. 858 00:42:12,250 --> 00:42:16,150 If they are drivers that are not particularly well trained 859 00:42:16,150 --> 00:42:19,630 and have not received feedback, for example, on their driving 860 00:42:19,630 --> 00:42:23,560 speeds and other behavior, and not just that, 861 00:42:23,560 --> 00:42:26,560 but also being on time at the terminal 862 00:42:26,560 --> 00:42:29,830 and leaving exactly on time, things like that, 863 00:42:29,830 --> 00:42:31,360 would all affect this. 864 00:42:31,360 --> 00:42:35,350 So here is a simple model of how headway will deteriorate. 865 00:42:35,350 --> 00:42:37,870 So there is also a mistake on the integration. 866 00:42:37,870 --> 00:42:41,230 That should be p i, not p i minus 1, sorry about that. 867 00:42:44,330 --> 00:42:48,340 So we have that the headway deviation at some stop-- 868 00:42:48,340 --> 00:42:50,410 here, we're thinking about a scheduled headway 869 00:42:50,410 --> 00:42:52,040 and the actual headway we see. 870 00:42:52,040 --> 00:42:56,530 So if service is running exactly on time, ei is 0. 871 00:42:56,530 --> 00:43:00,400 If that bus is delayed, then ei is positive. 872 00:43:00,400 --> 00:43:02,170 And if it's running early, ei is negative. 873 00:43:02,170 --> 00:43:05,200 So you start out-- 874 00:43:05,200 --> 00:43:06,790 you go from one stop to the next. 875 00:43:06,790 --> 00:43:09,440 And the deviation of headway you see at some stop 876 00:43:09,440 --> 00:43:12,070 is going to be a function of how-- 877 00:43:12,070 --> 00:43:15,510 what the deviation was at the previous stop. 878 00:43:15,510 --> 00:43:17,260 So if you're already delayed, your chances 879 00:43:17,260 --> 00:43:19,720 of being delayed at the next stop are higher. 880 00:43:19,720 --> 00:43:21,070 That should make sense. 881 00:43:21,070 --> 00:43:25,510 Then you add the effect of running from that previous stop 882 00:43:25,510 --> 00:43:26,770 to this stop. 883 00:43:26,770 --> 00:43:28,990 If you are particularly slow doing that, 884 00:43:28,990 --> 00:43:33,000 then your delay will increase. 885 00:43:33,000 --> 00:43:35,800 If you try to drive fast to make up for that, 886 00:43:35,800 --> 00:43:38,410 then ti will be negative. 887 00:43:38,410 --> 00:43:43,960 And it will decrease the impact on headway deterioration. 888 00:43:43,960 --> 00:43:45,084 Yeah? 889 00:43:45,084 --> 00:43:47,500 AUDIENCE: Do they ever just-- so this seems to imply that, 890 00:43:47,500 --> 00:43:49,240 like, maybe I could fix my problem 891 00:43:49,240 --> 00:43:52,080 by just dropping a bus from the schedule. 892 00:43:52,080 --> 00:43:54,610 Like, let's say I was really behind, 893 00:43:54,610 --> 00:43:57,400 I could just delay all the other buses 894 00:43:57,400 --> 00:43:59,276 and, like, take up the schedule of the next-- 895 00:43:59,276 --> 00:44:01,400 GABRIEL SANCHEZ-MARTINEZ: Yeah, but you will have-- 896 00:44:01,400 --> 00:44:03,570 you still have a headway, a longer headway, right? 897 00:44:03,570 --> 00:44:04,030 AUDIENCE: Yeah, [INAUDIBLE]-- 898 00:44:04,030 --> 00:44:04,750 GABRIEL SANCHEZ-MARTINEZ: So you could make up 899 00:44:04,750 --> 00:44:06,190 your schedule that way. 900 00:44:06,190 --> 00:44:11,110 But you're essentially doing an accounting trick by doing that. 901 00:44:11,110 --> 00:44:13,990 You can make your vehicles look like they're all 902 00:44:13,990 --> 00:44:16,450 running on time by doing that. 903 00:44:16,450 --> 00:44:17,840 And you dropped one trip. 904 00:44:17,840 --> 00:44:20,160 But you still have a longer headway here 905 00:44:20,160 --> 00:44:23,350 and shorter headways there, and passengers boarding 906 00:44:23,350 --> 00:44:24,930 mostly that delayed vehicle. 907 00:44:24,930 --> 00:44:27,280 And so from a passenger's perspective, 908 00:44:27,280 --> 00:44:30,040 that doesn't do much. 909 00:44:30,040 --> 00:44:32,909 But then you have to consider the effect of, well, 910 00:44:32,909 --> 00:44:34,450 you might have-- you might want to do 911 00:44:34,450 --> 00:44:36,480 that if your drivers need to check out 912 00:44:36,480 --> 00:44:37,600 at some time in the day. 913 00:44:37,600 --> 00:44:40,960 And your options are either, you do this now, 914 00:44:40,960 --> 00:44:45,660 or at some time when they check out, you have to drop trips. 915 00:44:45,660 --> 00:44:47,680 And that could be during the peak. 916 00:44:47,680 --> 00:44:49,360 So that would be really bad, right? 917 00:44:49,360 --> 00:44:53,713 So then you would have to consider that strategy. 918 00:44:53,713 --> 00:44:55,855 AUDIENCE: And do buses typically post, like, 919 00:44:55,855 --> 00:44:58,152 a scheduled time that they're supposed 920 00:44:58,152 --> 00:44:59,110 to arrive at each stop? 921 00:44:59,110 --> 00:44:59,560 Because normally-- 922 00:44:59,560 --> 00:44:59,830 GABRIEL SANCHEZ-MARTINEZ: Yes. 923 00:44:59,830 --> 00:45:01,150 AUDIENCE: --I see, like-- 924 00:45:01,150 --> 00:45:01,950 GABRIEL SANCHEZ-MARTINEZ: --especially now, 925 00:45:01,950 --> 00:45:02,590 so typically-- 926 00:45:02,590 --> 00:45:04,077 AUDIENCE: Like up-to-date minutes. 927 00:45:04,077 --> 00:45:06,160 GABRIEL SANCHEZ-MARTINEZ: So London will say that. 928 00:45:06,160 --> 00:45:08,030 So if it's high frequency-- 929 00:45:08,030 --> 00:45:10,700 and London, the threshold is 12 minutes. 930 00:45:10,700 --> 00:45:15,760 So anything below, they'll say, AM peak between-- 931 00:45:15,760 --> 00:45:18,610 expect service every eight minutes, for example. 932 00:45:18,610 --> 00:45:22,120 And then as soon as the headway increases above that, 933 00:45:22,120 --> 00:45:23,560 they'll say the times. 934 00:45:23,560 --> 00:45:27,160 But if you look at service here, for example, here 935 00:45:27,160 --> 00:45:31,480 in Boston, every bus route here, if you look at the DTFS file, 936 00:45:31,480 --> 00:45:35,040 has specific times at each stop, even though DTFS 937 00:45:35,040 --> 00:45:38,260 has the option of giving a headway 938 00:45:38,260 --> 00:45:42,570 for high-frequency service, they don't do that here. 939 00:45:42,570 --> 00:45:45,110 AUDIENCE: So only service [INAUDIBLE] time appear? 940 00:45:45,110 --> 00:45:47,600 GABRIEL SANCHEZ-MARTINEZ: Yeah, but those DTFS file 941 00:45:47,600 --> 00:45:48,880 has times at every stop. 942 00:45:51,850 --> 00:45:53,500 So yeah, and that's another thing, 943 00:45:53,500 --> 00:45:55,250 that if you look at printed schedules that 944 00:45:55,250 --> 00:45:58,640 are posted somewhere and on paper, 945 00:45:58,640 --> 00:46:00,140 typically they'll have the terminals 946 00:46:00,140 --> 00:46:03,119 and some time points in between and not every single stop. 947 00:46:03,119 --> 00:46:05,410 AUDIENCE: And the time points [INAUDIBLE] approximately 948 00:46:05,410 --> 00:46:05,910 [INAUDIBLE]? 949 00:46:05,910 --> 00:46:07,409 GABRIEL SANCHEZ-MARTINEZ: Well, that 950 00:46:07,409 --> 00:46:09,710 depends on the control policy of that agency. 951 00:46:12,260 --> 00:46:16,850 So we'll talk more about control in a later lecture. 952 00:46:16,850 --> 00:46:19,040 So typically, you don't control at every stop. 953 00:46:19,040 --> 00:46:23,280 You control at, sometimes, just the terminal, and sometimes 954 00:46:23,280 --> 00:46:26,180 at the terminal and some key points in between. 955 00:46:26,180 --> 00:46:30,200 And those are timing points for control. 956 00:46:30,200 --> 00:46:37,760 OK, so back to this model, the headway deviation at some stop 957 00:46:37,760 --> 00:46:40,460 is, you start out from what it was in the previous stop. 958 00:46:40,460 --> 00:46:42,140 You add the effect of running time 959 00:46:42,140 --> 00:46:44,360 to that stop from the previous one. 960 00:46:44,360 --> 00:46:48,350 And then you add the effect of dwell time, essentially. 961 00:46:48,350 --> 00:46:51,870 So this pi, not pi minus 1, as I said earlier-- 962 00:46:51,870 --> 00:46:55,670 and so this is the arrival rate at the stop right now 963 00:46:55,670 --> 00:46:59,160 and multiplied by the boarding time per passenger. 964 00:46:59,160 --> 00:47:08,450 So this whole quantity is multiplied by this amount, 965 00:47:08,450 --> 00:47:10,890 which is the time it takes-- 966 00:47:10,890 --> 00:47:13,630 it's the deviation of the time it takes to arrive. 967 00:47:13,630 --> 00:47:16,090 So it's a deviation in headway, essentially. 968 00:47:16,090 --> 00:47:19,420 So if you're headway is now a minute longer than it used 969 00:47:19,420 --> 00:47:23,500 to be, then you will have a minute times the boarding 970 00:47:23,500 --> 00:47:26,860 rate per passenger times the arrival rate of passengers, 971 00:47:26,860 --> 00:47:30,810 extra people boarding, or extra time, extra time 972 00:47:30,810 --> 00:47:33,437 of dwell time affecting that bus, 973 00:47:33,437 --> 00:47:35,020 and therefore slowing it down further. 974 00:47:39,345 --> 00:47:40,220 Does that make sense? 975 00:47:40,220 --> 00:47:41,845 Or should I break it down a little bit? 976 00:47:44,710 --> 00:47:46,710 AUDIENCE: Did not answer that question, but just 977 00:47:46,710 --> 00:47:47,520 to ask a question? 978 00:47:47,520 --> 00:47:50,210 AUDIENCE: Yeah, [INAUDIBLE] interrupt anyone else. 979 00:47:50,210 --> 00:47:52,760 AUDIENCE: So is this accounting for the fact 980 00:47:52,760 --> 00:47:55,400 that, if there is no one getting on or off this stop, 981 00:47:55,400 --> 00:47:58,580 the first person to pull into a stop, 982 00:47:58,580 --> 00:48:01,770 decelerate, and open the door takes a given amount of time. 983 00:48:01,770 --> 00:48:03,950 And the marginal time to add an extra person 984 00:48:03,950 --> 00:48:05,832 is quite small, or relatively small. 985 00:48:05,832 --> 00:48:06,957 Is that accounted for here? 986 00:48:06,957 --> 00:48:08,623 GABRIEL SANCHEZ-MARTINEZ: No, this model 987 00:48:08,623 --> 00:48:10,700 is saying that it's a linear effect. 988 00:48:10,700 --> 00:48:13,100 You're not saying that the first person takes-- 989 00:48:13,100 --> 00:48:14,930 AUDIENCE: So two would take twice as long as one person. 990 00:48:14,930 --> 00:48:15,890 GABRIEL SANCHEZ-MARTINEZ: Exactly, yeah, 991 00:48:15,890 --> 00:48:17,330 so that is a simplification here. 992 00:48:17,330 --> 00:48:18,967 And it's a point that we will address 993 00:48:18,967 --> 00:48:20,300 towards the end of this lecture. 994 00:48:20,300 --> 00:48:21,980 AUDIENCE: OK, [INAUDIBLE] asking questions. 995 00:48:21,980 --> 00:48:23,270 GABRIEL SANCHEZ-MARTINEZ: Good segues, 996 00:48:23,270 --> 00:48:24,728 that's, like, the second one today. 997 00:48:24,728 --> 00:48:26,687 AUDIENCE: This is only extra [INAUDIBLE] right? 998 00:48:26,687 --> 00:48:28,478 GABRIEL SANCHEZ-MARTINEZ: So yeah, exactly, 999 00:48:28,478 --> 00:48:30,390 because this is a model of headway deviation. 1000 00:48:30,390 --> 00:48:33,230 So you have this passenger arrival rate and the boarding 1001 00:48:33,230 --> 00:48:35,647 time, but because you're only multiplying it-- 1002 00:48:35,647 --> 00:48:37,730 you're not multiplying it times the whole headway. 1003 00:48:37,730 --> 00:48:40,490 You're multiplying times the headway deviation, which 1004 00:48:40,490 --> 00:48:45,210 is the deviation at this stop when you arrive and this stop. 1005 00:48:45,210 --> 00:48:47,960 It's the deviation when you were arriving 1006 00:48:47,960 --> 00:48:51,380 at the previous stop plus the time it took you to reach 1007 00:48:51,380 --> 00:48:53,030 this stop from that time. 1008 00:48:53,030 --> 00:48:55,730 So now you have adjusted-- now you have the deviation arriving 1009 00:48:55,730 --> 00:48:57,680 at this stop. 1010 00:48:57,680 --> 00:48:59,150 If your headway is a minute longer, 1011 00:48:59,150 --> 00:49:02,660 than you need to add however much extra dwell time 1012 00:49:02,660 --> 00:49:05,155 you have to pick up that extra minute of passengers. 1013 00:49:05,155 --> 00:49:06,780 An that's what that last case is doing. 1014 00:49:11,589 --> 00:49:13,600 AUDIENCE: So at the beginning, the e 1015 00:49:13,600 --> 00:49:15,370 will be 0 at the first stop? 1016 00:49:15,370 --> 00:49:16,870 GABRIEL SANCHEZ-MARTINEZ: Hopefully. 1017 00:49:19,354 --> 00:49:20,770 It depends on your control policy. 1018 00:49:20,770 --> 00:49:25,480 So if you depart your terminal and your driver didn't show up 1019 00:49:25,480 --> 00:49:29,500 on time and they left a minute late from the even headway, 1020 00:49:29,500 --> 00:49:33,350 then you start out with one minute deviation. 1021 00:49:33,350 --> 00:49:36,620 So yeah, hopefully that is 0 at the beginning. 1022 00:49:36,620 --> 00:49:41,030 And then due to effect that we described here, 1023 00:49:41,030 --> 00:49:43,040 it starts deteriorating. 1024 00:49:43,040 --> 00:49:45,230 And this is a mathematical model to-- 1025 00:49:45,230 --> 00:49:49,220 a simple mathematical model to account for that. 1026 00:49:49,220 --> 00:49:50,750 So it's still a deterministic one. 1027 00:49:50,750 --> 00:49:52,580 Here is a probabilistic one. 1028 00:49:52,580 --> 00:49:56,569 The details for this formulation are on this paper. 1029 00:49:56,569 --> 00:49:57,985 If you're interested, let me know. 1030 00:49:57,985 --> 00:50:00,050 I will send it you. 1031 00:50:00,050 --> 00:50:04,260 What I want to highlight here are the quantities. 1032 00:50:04,260 --> 00:50:07,460 So this is a model of headway variance. 1033 00:50:07,460 --> 00:50:09,490 So we are looking at, now, headway 1034 00:50:09,490 --> 00:50:11,360 as a stochastic quantity. 1035 00:50:11,360 --> 00:50:14,870 And we're calculating the variance of headway 1036 00:50:14,870 --> 00:50:16,550 at some stop. 1037 00:50:16,550 --> 00:50:18,680 And again, we see the same pattern. 1038 00:50:18,680 --> 00:50:23,660 We see that it depends on what it was at the previous stop. 1039 00:50:23,660 --> 00:50:26,790 And then we add the effect of the running time, 1040 00:50:26,790 --> 00:50:30,170 so the variance of running times between consecutive stops, 1041 00:50:30,170 --> 00:50:33,920 to the extent that the drivers are different in driving, 1042 00:50:33,920 --> 00:50:36,350 then this variance increases. 1043 00:50:36,350 --> 00:50:38,990 And then you have these two terms-- 1044 00:50:38,990 --> 00:50:42,800 really, three terms, that have to do with dwell time. 1045 00:50:42,800 --> 00:50:44,180 All of these terms have-- 1046 00:50:44,180 --> 00:50:48,680 the q here is the mean number of passengers per bus served. 1047 00:50:48,680 --> 00:50:51,860 And c is the marginal dwell time per passenger-- so different 1048 00:50:51,860 --> 00:50:56,110 notation from the model that we just saw, but same quantities. 1049 00:50:56,110 --> 00:51:03,470 So we have something times that probability is-- 1050 00:51:03,470 --> 00:51:04,970 the bus will skip a stop, et cetera. 1051 00:51:04,970 --> 00:51:08,590 So all these things together are accounting for the dwell time 1052 00:51:08,590 --> 00:51:09,500 effect. 1053 00:51:09,500 --> 00:51:13,580 So the last term here is, again, looking 1054 00:51:13,580 --> 00:51:17,750 at c, which is the marginal dwell time per passenger. 1055 00:51:17,750 --> 00:51:19,700 So this also has to do with dwell time. 1056 00:51:19,700 --> 00:51:24,740 But this one is also multiplied by the covariance with headway. 1057 00:51:24,740 --> 00:51:29,200 So this is that effect of the relationship between headway 1058 00:51:29,200 --> 00:51:32,590 and sort of passenger arrivals, captured here 1059 00:51:32,590 --> 00:51:35,850 as the covariance between those two quantities. 1060 00:51:35,850 --> 00:51:38,720 So the key takeaway from this slide 1061 00:51:38,720 --> 00:51:41,930 is not exactly why this is the right equation. 1062 00:51:41,930 --> 00:51:46,120 If you're interested in that, read the paper. 1063 00:51:46,120 --> 00:51:49,340 I am interested in understanding what 1064 00:51:49,340 --> 00:51:53,810 goes into this equation in terms of what components 1065 00:51:53,810 --> 00:51:56,010 lead to a higher variance at some particular stop. 1066 00:51:59,600 --> 00:52:04,310 OK, we are ready to move to vehicle running time models 1067 00:52:04,310 --> 00:52:06,140 if no-- if there are no questions 1068 00:52:06,140 --> 00:52:09,280 on waiting time headway models. 1069 00:52:09,280 --> 00:52:11,540 OK, so let's do that. 1070 00:52:11,540 --> 00:52:13,100 So different levels of detail-- 1071 00:52:13,100 --> 00:52:15,890 we have some models that are very detailed. 1072 00:52:15,890 --> 00:52:17,390 There are microscopic model. 1073 00:52:20,250 --> 00:52:21,140 It's a simulation. 1074 00:52:21,140 --> 00:52:24,137 And they look at the vehicle motion, the interaction 1075 00:52:24,137 --> 00:52:24,970 with other vehicles. 1076 00:52:24,970 --> 00:52:27,590 So you might have private automobiles interacting 1077 00:52:27,590 --> 00:52:30,505 with a bus, for example, and traffic blocking the bus. 1078 00:52:30,505 --> 00:52:31,380 And you have signals. 1079 00:52:31,380 --> 00:52:32,796 So every little detail is modeled. 1080 00:52:32,796 --> 00:52:34,580 That's a microscopic model. 1081 00:52:34,580 --> 00:52:37,250 On the other hand, you have macroscopic models, 1082 00:52:37,250 --> 00:52:39,680 which are not looking at all those details. 1083 00:52:39,680 --> 00:52:42,680 Instead, they're saying, what are the running times that I 1084 00:52:42,680 --> 00:52:44,630 observe as a function of time of day, 1085 00:52:44,630 --> 00:52:49,100 and the driver, different components. 1086 00:52:49,100 --> 00:52:51,450 And macroscopically, we say, this is the running time. 1087 00:52:51,450 --> 00:52:55,479 So there is something in between called mesoscopic, when 1088 00:52:55,479 --> 00:52:57,020 you have some parts that are detailed 1089 00:52:57,020 --> 00:52:58,580 and others that are not-- 1090 00:52:58,580 --> 00:53:02,180 so different levels of modeling. 1091 00:53:05,240 --> 00:53:07,100 So running time, as we know, includes 1092 00:53:07,100 --> 00:53:11,990 dwell time, the movement time between stops, and any delays. 1093 00:53:11,990 --> 00:53:14,930 Delays could be because of signals, traffic signals, 1094 00:53:14,930 --> 00:53:15,920 for example. 1095 00:53:15,920 --> 00:53:18,470 So dwell time is a function of the number 1096 00:53:18,470 --> 00:53:20,390 of passengers boarding and arriving 1097 00:53:20,390 --> 00:53:23,030 as well as technology characteristics. 1098 00:53:23,030 --> 00:53:25,910 What are some examples of technology characteristics 1099 00:53:25,910 --> 00:53:28,679 that could affect dwell times? 1100 00:53:28,679 --> 00:53:31,730 AUDIENCE: [INAUDIBLE] cards, smart card. 1101 00:53:31,730 --> 00:53:34,230 GABRIEL SANCHEZ-MARTINEZ: So what your fare card technology, 1102 00:53:34,230 --> 00:53:37,260 so if you have smart cards, or coins, paying cash-- 1103 00:53:37,260 --> 00:53:38,531 very different, right? 1104 00:53:38,531 --> 00:53:39,030 What else? 1105 00:53:39,030 --> 00:53:40,610 AUDIENCE: Off-board fare collection. 1106 00:53:40,610 --> 00:53:41,820 GABRIEL SANCHEZ-MARTINEZ: Off-board fare collection-- 1107 00:53:41,820 --> 00:53:44,130 so if you remove the payment from the vehicle, 1108 00:53:44,130 --> 00:53:46,720 then that really helps with decreasing the variability. 1109 00:53:46,720 --> 00:53:47,270 What else? 1110 00:53:47,270 --> 00:53:49,020 AUDIENCE: All boarding. 1111 00:53:49,020 --> 00:53:50,570 GABRIEL SANCHEZ-MARTINEZ: Sorry? 1112 00:53:50,570 --> 00:53:51,590 AUDIENCE: All boarding? 1113 00:53:51,590 --> 00:53:52,470 GABRIEL SANCHEZ-MARTINEZ: All-door boarding? 1114 00:53:52,470 --> 00:53:53,060 AUDIENCE: All-door boarding. 1115 00:53:53,060 --> 00:53:53,680 GABRIEL SANCHEZ-MARTINEZ: So boarding 1116 00:53:53,680 --> 00:53:55,330 through all doors-- of course, if you 1117 00:53:55,330 --> 00:53:56,830 can board through multiple doors, 1118 00:53:56,830 --> 00:53:59,990 you have a faster dwell time process in this variable. 1119 00:53:59,990 --> 00:54:00,610 What else? 1120 00:54:00,610 --> 00:54:02,940 AUDIENCE: If the station level is the same as the-- 1121 00:54:02,940 --> 00:54:04,690 GABRIEL SANCHEZ-MARTINEZ: Level boarding-- 1122 00:54:04,690 --> 00:54:07,310 so if people don't have to climb steps to go from the curb 1123 00:54:07,310 --> 00:54:11,940 to the vehicle, that also helps. 1124 00:54:11,940 --> 00:54:12,830 Any other ideas? 1125 00:54:12,830 --> 00:54:13,370 Eli? 1126 00:54:13,370 --> 00:54:14,185 AUDIENCE: The type of the bus stop-- 1127 00:54:14,185 --> 00:54:16,460 am I pulling into a cutout and then I have to wait for traffic 1128 00:54:16,460 --> 00:54:17,090 to merge back in. 1129 00:54:17,090 --> 00:54:17,590 GABRIEL SANCHEZ-MARTINEZ: Yes. 1130 00:54:17,590 --> 00:54:18,740 AUDIENCE: [INAUDIBLE] stay in the lane [INAUDIBLE]---- 1131 00:54:18,740 --> 00:54:19,160 GABRIEL SANCHEZ-MARTINEZ: And that 1132 00:54:19,160 --> 00:54:21,110 depends on the definition of dwell time. 1133 00:54:21,110 --> 00:54:23,870 So if you say dwell time is only the amount of time needed 1134 00:54:23,870 --> 00:54:27,254 to serve passengers, then that shouldn't really be a factor. 1135 00:54:27,254 --> 00:54:29,420 But if you think of the dwell time as the whole time 1136 00:54:29,420 --> 00:54:32,120 it took me to stop and serve that stop, 1137 00:54:32,120 --> 00:54:33,650 then this would be included. 1138 00:54:33,650 --> 00:54:36,672 So you're looking at-- 1139 00:54:36,672 --> 00:54:46,520 let's see if I can get at this, clean a section here. 1140 00:54:53,540 --> 00:54:58,870 So one example is a bus bay here. 1141 00:54:58,870 --> 00:55:01,640 So this is a bus stop right here. 1142 00:55:01,640 --> 00:55:05,640 And maybe this is an intersection. 1143 00:55:05,640 --> 00:55:09,710 So the bus comes in here. 1144 00:55:09,710 --> 00:55:12,000 And it's waiting there. 1145 00:55:12,000 --> 00:55:16,440 But if there is traffic and there are cars right here, 1146 00:55:16,440 --> 00:55:19,996 after that bus is ready, it may have to-- 1147 00:55:19,996 --> 00:55:21,870 well, if it's right of the signal, it's fine. 1148 00:55:21,870 --> 00:55:28,050 But if this now merges and there is traffic here, 1149 00:55:28,050 --> 00:55:30,000 then that bus might be sort of stuck enough 1150 00:55:30,000 --> 00:55:34,050 to maneuver its way back into the traffic flow. 1151 00:55:34,050 --> 00:55:39,240 So a [INAUDIBLE] sign or stop sign have an impact. 1152 00:55:39,240 --> 00:55:41,660 What else? 1153 00:55:41,660 --> 00:55:42,684 Henry? 1154 00:55:42,684 --> 00:55:44,350 AUDIENCE: If you're serving a line where 1155 00:55:44,350 --> 00:55:46,016 there are a lot of people who are, like, 1156 00:55:46,016 --> 00:55:48,090 tourists and people, like, ask questions, 1157 00:55:48,090 --> 00:55:49,979 it could take forever to get on to the bus. 1158 00:55:49,979 --> 00:55:52,020 GABRIEL SANCHEZ-MARTINEZ: That's not a technology 1159 00:55:52,020 --> 00:55:55,290 characteristic, but it is a valid factor 1160 00:55:55,290 --> 00:55:56,827 that affects dwell times. 1161 00:55:56,827 --> 00:55:58,785 Or maybe if your technology is very complicated 1162 00:55:58,785 --> 00:56:00,790 and people have a lot of questions, 1163 00:56:00,790 --> 00:56:01,935 that could be a way to-- 1164 00:56:01,935 --> 00:56:03,560 AUDIENCE: Or maybe a lack of technology 1165 00:56:03,560 --> 00:56:05,100 where, like, you don't have-- 1166 00:56:05,100 --> 00:56:07,860 GABRIEL SANCHEZ-MARTINEZ: So OK, I think we gave good examples. 1167 00:56:07,860 --> 00:56:09,750 If we look at the typical bus running time 1168 00:56:09,750 --> 00:56:11,540 and how we break it down, this is 1169 00:56:11,540 --> 00:56:14,670 a typical bus in mixed traffic. 1170 00:56:14,670 --> 00:56:16,320 Somewhere between a 1/2 and 3/4 will 1171 00:56:16,320 --> 00:56:18,780 be spent moving between stops. 1172 00:56:18,780 --> 00:56:20,520 Between 10 and a 1/4-- 1173 00:56:20,520 --> 00:56:24,735 10% and 1/4 will be spent at stops, serving the stop. 1174 00:56:24,735 --> 00:56:27,930 And between 10% and 1/4 will be served 1175 00:56:27,930 --> 00:56:32,370 in traffic or waiting for a signal, so at 1176 00:56:32,370 --> 00:56:34,100 a red light, essentially. 1177 00:56:34,100 --> 00:56:34,600 Yeah? 1178 00:56:34,600 --> 00:56:37,835 AUDIENCE: [INAUDIBLE] about this a bit, but in movement time, 1179 00:56:37,835 --> 00:56:40,210 would that include all the time it takes for me to, like, 1180 00:56:40,210 --> 00:56:44,235 stop the bus? 1181 00:56:44,235 --> 00:56:46,860 GABRIEL SANCHEZ-MARTINEZ: Again, it depends on your definition. 1182 00:56:46,860 --> 00:56:51,300 But yes, I think in this break down, yes. 1183 00:56:51,300 --> 00:56:52,908 So the slowing down and accelerating 1184 00:56:52,908 --> 00:56:53,991 is sort of in there, yeah. 1185 00:56:57,360 --> 00:57:00,010 Any other questions? 1186 00:57:00,010 --> 00:57:05,387 OK, so let's look at some dwell time models. 1187 00:57:05,387 --> 00:57:07,470 Dwell times models are a component of running time 1188 00:57:07,470 --> 00:57:12,830 models, because dwell times are one of the key pieces 1189 00:57:12,830 --> 00:57:13,830 of running times. 1190 00:57:13,830 --> 00:57:17,550 And actually, another point about these, movement time, 1191 00:57:17,550 --> 00:57:20,550 dwell time, and delay, which of those three 1192 00:57:20,550 --> 00:57:25,435 do you think the agency has most control over? 1193 00:57:25,435 --> 00:57:26,310 AUDIENCE: Dwell time. 1194 00:57:26,310 --> 00:57:28,143 GABRIEL SANCHEZ-MARTINEZ: Dwell time, right? 1195 00:57:28,143 --> 00:57:29,245 Why? 1196 00:57:29,245 --> 00:57:32,830 Can an agency usually diminish traffic or do something 1197 00:57:32,830 --> 00:57:35,150 about traffic signals? 1198 00:57:35,150 --> 00:57:37,840 There are some things you can do, but usually not. 1199 00:57:37,840 --> 00:57:38,860 It's harder. 1200 00:57:38,860 --> 00:57:42,640 And movement time, that has to do with traffic and speed 1201 00:57:42,640 --> 00:57:46,970 limits, and the vehicle itself, and driver behavior. 1202 00:57:46,970 --> 00:57:49,270 So that's a little bit harder to change. 1203 00:57:49,270 --> 00:57:52,510 So dwell times are probably the one thing 1204 00:57:52,510 --> 00:57:55,054 here that an agency will target. 1205 00:57:55,054 --> 00:57:56,470 Some part of it, you can't change, 1206 00:57:56,470 --> 00:57:58,886 because it has to do with how many people are at the stop. 1207 00:57:58,886 --> 00:58:00,310 But you could increase frequency. 1208 00:58:00,310 --> 00:58:01,640 You could change the bus assigned. 1209 00:58:01,640 --> 00:58:02,764 We gave a bunch of factors. 1210 00:58:02,764 --> 00:58:04,780 And a lot of those are in control of the agency, 1211 00:58:04,780 --> 00:58:07,070 so under the control of the agency. 1212 00:58:07,070 --> 00:58:10,010 So let's go to the dwell time models for that reason. 1213 00:58:10,010 --> 00:58:17,050 There are some examples here, three papers, one on bus dwell 1214 00:58:17,050 --> 00:58:21,460 time, one on light rail dwell times for the Green line, 1215 00:58:21,460 --> 00:58:23,320 and one on heavy rail. 1216 00:58:23,320 --> 00:58:25,600 So we're going to look at the three of them, 1217 00:58:25,600 --> 00:58:29,450 just a high-level overview. 1218 00:58:29,450 --> 00:58:31,490 So let's look at some concepts first. 1219 00:58:31,490 --> 00:58:35,380 Vehicle dwell times affect system performance. 1220 00:58:35,380 --> 00:58:38,000 We've discussed why and how. 1221 00:58:38,000 --> 00:58:39,620 And they affect service quality. 1222 00:58:39,620 --> 00:58:42,240 We've also discussed why and how. 1223 00:58:42,240 --> 00:58:44,710 They are a critical element in vehicle bunching. 1224 00:58:44,710 --> 00:58:46,570 I think that has also been covered. 1225 00:58:46,570 --> 00:58:50,440 So they result in high headway variability, high passenger 1226 00:58:50,440 --> 00:58:52,630 waiting times, and uneven passenger loads. 1227 00:58:52,630 --> 00:58:54,640 That last point, we have mentioned, 1228 00:58:54,640 --> 00:58:57,670 but it was indirectly-- 1229 00:58:57,670 --> 00:58:59,530 the point has been made indirectly, right? 1230 00:58:59,530 --> 00:59:01,660 If more people are arriving at some stop, 1231 00:59:01,660 --> 00:59:03,400 at some bus that has a longer headway, 1232 00:59:03,400 --> 00:59:05,230 not only did those people wait longer, 1233 00:59:05,230 --> 00:59:07,780 but they are boarding a more crowded bus. 1234 00:59:07,780 --> 00:59:09,970 So their experience in the vehicle 1235 00:59:09,970 --> 00:59:11,690 is also going to be diminished. 1236 00:59:11,690 --> 00:59:14,870 It's going to be a lower-quality experience. 1237 00:59:14,870 --> 00:59:18,880 So we've covered that. 1238 00:59:18,880 --> 00:59:22,130 Dwell time impact on performance depends on substation spacing, 1239 00:59:22,130 --> 00:59:24,720 the mean dwell as a proportion of trip time, 1240 00:59:24,720 --> 00:59:27,830 the mean headway, and operations control procedures. 1241 00:59:27,830 --> 00:59:32,180 We have touched upon, not necessarily what can be done, 1242 00:59:32,180 --> 00:59:35,510 but we know that there are some things you can do to diminish 1243 00:59:35,510 --> 00:59:36,590 the headway variability. 1244 00:59:40,600 --> 00:59:41,960 Any examples of that, actually? 1245 00:59:41,960 --> 00:59:45,119 Anybody have suggestions on what can be done? 1246 00:59:45,119 --> 00:59:45,985 Eli? 1247 00:59:45,985 --> 00:59:49,630 AUDIENCE: You tell the bus that is trailing to, like, 1248 00:59:49,630 --> 00:59:50,650 wait longer. 1249 00:59:50,650 --> 00:59:52,108 GABRIEL SANCHEZ-MARTINEZ: Yeah, you 1250 00:59:52,108 --> 00:59:54,334 try to slow down the buses that are running fast. 1251 00:59:54,334 --> 00:59:56,000 So there's different ways of doing that. 1252 00:59:56,000 --> 00:59:58,120 You can tell the driver to drive slowly. 1253 00:59:58,120 --> 01:00:01,090 You can hold buses at stops for control points. 1254 01:00:01,090 --> 01:00:02,710 So you tell them, do not depart. 1255 01:00:02,710 --> 01:00:04,240 You have to wait a minute before you depart, 1256 01:00:04,240 --> 01:00:05,935 because you're running too fast, and you're 1257 01:00:05,935 --> 01:00:07,135 going to catch up to the previous bus. 1258 01:00:07,135 --> 01:00:08,290 So that's called holding. 1259 01:00:08,290 --> 01:00:10,060 So there has been a lot of research and holding 1260 01:00:10,060 --> 01:00:10,600 strategies. 1261 01:00:10,600 --> 01:00:12,740 AUDIENCE: Wouldn't the more typical thing 1262 01:00:12,740 --> 01:00:13,990 would be telling them to hold? 1263 01:00:13,990 --> 01:00:14,590 GABRIEL SANCHEZ-MARTINEZ: Yes. 1264 01:00:14,590 --> 01:00:15,730 AUDIENCE: It seems weird to ask someone 1265 01:00:15,730 --> 01:00:17,020 to slow down in traffic. 1266 01:00:17,020 --> 01:00:18,485 GABRIEL SANCHEZ-MARTINEZ: Eh, it's been done. 1267 01:00:18,485 --> 01:00:18,930 AUDIENCE: Really? 1268 01:00:18,930 --> 01:00:19,820 GABRIEL SANCHEZ-MARTINEZ: Yeah. 1269 01:00:19,820 --> 01:00:21,320 AUDIENCE: I wouldn't like to be that bus driver. 1270 01:00:21,320 --> 01:00:24,220 GABRIEL SANCHEZ-MARTINEZ: You would kind of surreptitiously 1271 01:00:24,220 --> 01:00:25,780 and only a little bit. 1272 01:00:28,910 --> 01:00:29,785 AUDIENCE: [INAUDIBLE] 1273 01:00:29,785 --> 01:00:32,243 GABRIEL SANCHEZ-MARTINEZ: And people don't like being held. 1274 01:00:32,243 --> 01:00:34,905 So driving a little bit slow is less obvious to the passengers. 1275 01:00:34,905 --> 01:00:36,223 AUDIENCE: [INAUDIBLE] 1276 01:00:38,342 --> 01:00:40,887 AUDIENCE: Yes, you have a bus hold just a few seconds 1277 01:00:40,887 --> 01:00:41,845 when the light changes. 1278 01:00:41,845 --> 01:00:42,345 [INAUDIBLE] 1279 01:00:42,345 --> 01:00:43,750 GABRIEL SANCHEZ-MARTINEZ: Yeah. 1280 01:00:43,750 --> 01:00:45,166 AUDIENCE: I could really get you-- 1281 01:00:45,166 --> 01:00:48,020 AUDIENCE: So that's sort of the best hold strategy. 1282 01:00:48,020 --> 01:00:50,327 Oh, we just missed it. 1283 01:00:50,327 --> 01:00:51,910 GABRIEL SANCHEZ-MARTINEZ: OK, so there 1284 01:00:51,910 --> 01:00:53,410 are operations controlled procedures 1285 01:00:53,410 --> 01:00:55,100 that can be used here. 1286 01:00:55,100 --> 01:00:57,790 So some examples based on these factors, on the one 1287 01:00:57,790 --> 01:00:58,960 hand you have commuter rail. 1288 01:00:58,960 --> 01:01:01,000 We gave that example earlier. 1289 01:01:01,000 --> 01:01:04,210 Little impact of dwell time on performance-- that makes sense. 1290 01:01:04,210 --> 01:01:06,160 Commuter rail has long distance between stops. 1291 01:01:06,160 --> 01:01:07,630 It stops at every stop. 1292 01:01:07,630 --> 01:01:11,380 And most of the time is spent in movement. 1293 01:01:11,380 --> 01:01:15,560 So the percentage of time spent dwelling is small. 1294 01:01:15,560 --> 01:01:18,050 On the other hand, you have a very long, high-frequency bus 1295 01:01:18,050 --> 01:01:18,850 route. 1296 01:01:18,850 --> 01:01:22,420 So the likelihood of bunching here is really high. 1297 01:01:22,420 --> 01:01:24,629 And it's hard to control this. 1298 01:01:24,629 --> 01:01:26,920 AUDIENCE: Especially, also the commuter rail, also it's 1299 01:01:26,920 --> 01:01:28,720 like a scheduled dwell. 1300 01:01:28,720 --> 01:01:30,220 GABRIEL SANCHEZ-MARTINEZ: Yeah, it's 1301 01:01:30,220 --> 01:01:31,360 a little longer than it has to be. 1302 01:01:31,360 --> 01:01:31,815 AUDIENCE: [INAUDIBLE] hold. 1303 01:01:31,815 --> 01:01:33,430 Then you don't actually-- you have 1304 01:01:33,430 --> 01:01:35,270 a few seconds at the end where no one is boarding or arriving. 1305 01:01:35,270 --> 01:01:36,160 GABRIEL SANCHEZ-MARTINEZ: Yeah, because you 1306 01:01:36,160 --> 01:01:37,990 might have printed the schedule it has 1307 01:01:37,990 --> 01:01:41,470 departure times for every stop. 1308 01:01:41,470 --> 01:01:44,020 AUDIENCE: I would argue that commuter rail has 1309 01:01:44,020 --> 01:01:46,750 sort of the most potential, especially in a transit system 1310 01:01:46,750 --> 01:01:48,835 like Boston where you have-- 1311 01:01:48,835 --> 01:01:51,460 well, the line acceleration, if you get the acceleration better 1312 01:01:51,460 --> 01:01:53,110 through different technology, and when 1313 01:01:53,110 --> 01:01:54,460 you don't have high-level platforms, 1314 01:01:54,460 --> 01:01:56,251 the dwell times get really long when you're 1315 01:01:56,251 --> 01:01:57,332 boarding a lot of people. 1316 01:01:57,332 --> 01:01:59,040 GABRIEL SANCHEZ-MARTINEZ: But you still-- 1317 01:01:59,040 --> 01:02:01,498 AUDIENCE: Plus then you could pull the schedule down if you 1318 01:02:01,498 --> 01:02:02,820 were [INAUDIBLE] dwell time. 1319 01:02:02,820 --> 01:02:05,486 GABRIEL SANCHEZ-MARTINEZ: So you could, if your schedule is off, 1320 01:02:05,486 --> 01:02:08,110 that could be an option. 1321 01:02:08,110 --> 01:02:11,140 Even if your schedule is good on a high-frequency long urban bus 1322 01:02:11,140 --> 01:02:13,600 tour line, it's not going to help. 1323 01:02:13,600 --> 01:02:14,920 You still have to control it. 1324 01:02:14,920 --> 01:02:15,503 And it's hard. 1325 01:02:15,503 --> 01:02:18,250 And you're going to get bunching. 1326 01:02:18,250 --> 01:02:20,202 So dwell time depends on many factors. 1327 01:02:20,202 --> 01:02:21,160 Some of them are human. 1328 01:02:21,160 --> 01:02:22,618 Some of them have to do with modes, 1329 01:02:22,618 --> 01:02:25,160 as we just saw, operating policies and practices, 1330 01:02:25,160 --> 01:02:26,920 weather, all these things. 1331 01:02:26,920 --> 01:02:35,410 So here is a list of models of dwell times for trains. 1332 01:02:35,410 --> 01:02:39,590 So if we look at a single door situation, 1333 01:02:39,590 --> 01:02:41,910 so people have to board and alight from the same door 1334 01:02:41,910 --> 01:02:44,540 and there is no congestion of passengers, 1335 01:02:44,540 --> 01:02:47,110 then this is the simplest model one could have. 1336 01:02:47,110 --> 01:02:50,782 Dwell time is some constant a, which 1337 01:02:50,782 --> 01:02:52,240 has to do with how long it takes me 1338 01:02:52,240 --> 01:02:54,880 to come to a full stop and open doors, 1339 01:02:54,880 --> 01:02:57,070 plus some amount times the number of people 1340 01:02:57,070 --> 01:02:59,980 getting on plus some amount times the number of people 1341 01:02:59,980 --> 01:03:01,490 getting off. 1342 01:03:01,490 --> 01:03:02,897 Does that make sense? 1343 01:03:02,897 --> 01:03:04,480 You sort of measure the average number 1344 01:03:04,480 --> 01:03:06,580 of seconds per passenger alighting, 1345 01:03:06,580 --> 01:03:09,600 the average number seconds per passenger boarding, 1346 01:03:09,600 --> 01:03:10,790 and some constant. 1347 01:03:10,790 --> 01:03:11,920 You run a regression model. 1348 01:03:11,920 --> 01:03:13,260 That's what you get. 1349 01:03:13,260 --> 01:03:17,890 OK, now what if you notice that, when the train is packed, 1350 01:03:17,890 --> 01:03:21,130 people take a lot longer to get off and on? 1351 01:03:21,130 --> 01:03:24,070 Then you want to add the effect of interference, 1352 01:03:24,070 --> 01:03:26,042 or congestion, or friction. 1353 01:03:26,042 --> 01:03:27,250 Some people call it friction. 1354 01:03:27,250 --> 01:03:32,270 So one way of doing that is to multiply-- 1355 01:03:32,270 --> 01:03:36,020 so we add a term, the passenger friction term. 1356 01:03:36,020 --> 01:03:37,990 We multiply the number of people that 1357 01:03:37,990 --> 01:03:41,230 were getting on and off times the number of standees 1358 01:03:41,230 --> 01:03:41,970 in the vehicle. 1359 01:03:41,970 --> 01:03:43,360 STD here stands for standees. 1360 01:03:47,460 --> 01:03:49,210 So the number of people getting on and off 1361 01:03:49,210 --> 01:03:53,830 are the people that would have moved faster if there 1362 01:03:53,830 --> 01:03:55,440 hadn't been any standees. 1363 01:03:55,440 --> 01:03:58,570 And to the extent that there are many standees, then 1364 01:03:58,570 --> 01:04:01,600 they encounter friction as they are getting on or off. 1365 01:04:01,600 --> 01:04:03,740 And that slows the vehicle down. 1366 01:04:03,740 --> 01:04:06,130 So that's one way of taking care of this. 1367 01:04:06,130 --> 01:04:09,400 If that car has m doors, then you 1368 01:04:09,400 --> 01:04:13,780 could run a model for every door and then pick 1369 01:04:13,780 --> 01:04:16,390 the door that was slowest. 1370 01:04:16,390 --> 01:04:18,220 So that's that model. 1371 01:04:18,220 --> 01:04:22,150 And if you say that, actually, people are more or less 1372 01:04:22,150 --> 01:04:24,340 evenly distributed inside the vehicle 1373 01:04:24,340 --> 01:04:26,810 and evenly distributed on the platform, 1374 01:04:26,810 --> 01:04:29,674 so if we are willing to assume that, 1375 01:04:29,674 --> 01:04:31,840 then we can just take the number of doors and divide 1376 01:04:31,840 --> 01:04:34,960 by number of doors, essentially. 1377 01:04:34,960 --> 01:04:38,020 So we're back to the previous model, 1378 01:04:38,020 --> 01:04:40,000 but now we're multiplying-- or dividing 1379 01:04:40,000 --> 01:04:44,656 by m, by the number of doors, because we have maybe 1,000 1380 01:04:44,656 --> 01:04:47,830 passengers or many hundreds of passengers, 1381 01:04:47,830 --> 01:04:50,220 but we have to divide by the number of doors 1382 01:04:50,220 --> 01:04:52,043 to bring them down to passengers per door. 1383 01:04:55,110 --> 01:05:03,020 If that train has several cars, then you 1384 01:05:03,020 --> 01:05:06,140 have a model for each car in the train. 1385 01:05:06,140 --> 01:05:11,010 And you take the maximum, so similar concept. 1386 01:05:11,010 --> 01:05:13,080 We were looking before at doors in one car. 1387 01:05:13,080 --> 01:05:15,580 Now we're looking at cars in one train. 1388 01:05:15,580 --> 01:05:18,170 And if that is our balance flows, 1389 01:05:18,170 --> 01:05:20,240 again, we can now divide by number 1390 01:05:20,240 --> 01:05:23,360 of doors per car and number of cars per train. 1391 01:05:23,360 --> 01:05:28,290 So we normalize the demand by the size of the train, 1392 01:05:28,290 --> 01:05:29,840 the number of doors. 1393 01:05:29,840 --> 01:05:31,070 And we include friction. 1394 01:05:31,070 --> 01:05:35,570 So here is some examples, some ideas for your problem set, 1395 01:05:35,570 --> 01:05:37,430 if you want. 1396 01:05:37,430 --> 01:05:38,480 Some of them may apply. 1397 01:05:38,480 --> 01:05:39,390 Others may not. 1398 01:05:39,390 --> 01:05:44,330 So this is the kind of thing you could think of. 1399 01:05:44,330 --> 01:05:47,810 So now we want to look at this study 1400 01:05:47,810 --> 01:05:50,930 by Milkovits published in 2008. 1401 01:05:50,930 --> 01:05:53,780 At the time, there had been some studies 1402 01:05:53,780 --> 01:05:55,190 with manually collected data. 1403 01:05:55,190 --> 01:05:59,870 So there was very limited data on infrequent events. 1404 01:05:59,870 --> 01:06:04,320 There was very limited data on crowding. 1405 01:06:04,320 --> 01:06:08,750 They were still using, I think, the token system. 1406 01:06:08,750 --> 01:06:12,560 So the previous study was based on the token system, 1407 01:06:12,560 --> 01:06:15,600 so it wasn't updated on new fare technology. 1408 01:06:15,600 --> 01:06:18,380 So there was another study that did have automatically 1409 01:06:18,380 --> 01:06:20,780 collected data, a little more recent, 1410 01:06:20,780 --> 01:06:23,300 but they hadn't taken into account 1411 01:06:23,300 --> 01:06:26,010 the effect of payment type. 1412 01:06:26,010 --> 01:06:29,080 So the AFC system does tell you, this is a pass, 1413 01:06:29,080 --> 01:06:34,040 or this is a ticket, but they hadn't-- they had ignored that 1414 01:06:34,040 --> 01:06:35,360 variable from the model. 1415 01:06:35,360 --> 01:06:38,220 And the fit of the model was poor. 1416 01:06:38,220 --> 01:06:40,220 And then we have a transit capacity and quality 1417 01:06:40,220 --> 01:06:42,800 of service manual, which says, assume half-second penalty 1418 01:06:42,800 --> 01:06:46,596 per passenger for crowding, so rule of thumb. 1419 01:06:46,596 --> 01:06:48,470 So there was interest in developing something 1420 01:06:48,470 --> 01:06:49,570 more sophisticated. 1421 01:06:49,570 --> 01:06:51,950 And they looked at buses and CTA for that. 1422 01:06:51,950 --> 01:07:00,780 So they looked at these factors, so boarding, alighting 1423 01:07:00,780 --> 01:07:04,930 passengers, counting them, onboard passengers, so load, 1424 01:07:04,930 --> 01:07:08,970 the fare media type, the alighting door selection, 1425 01:07:08,970 --> 01:07:10,640 so whether people boarded-- 1426 01:07:10,640 --> 01:07:13,560 alighted from the front door or from the back door, which has 1427 01:07:13,560 --> 01:07:15,900 an impact, and the bus type. 1428 01:07:15,900 --> 01:07:17,550 There were several bus designs. 1429 01:07:17,550 --> 01:07:21,420 And some of them were-- had wider doors and fare boxes 1430 01:07:21,420 --> 01:07:23,710 placed more optimally. 1431 01:07:23,710 --> 01:07:26,700 So all of that was taken into account. 1432 01:07:26,700 --> 01:07:33,030 And yeah, so data from the CTA in Chicago-- 1433 01:07:33,030 --> 01:07:36,600 they didn't consider timing points, 1434 01:07:36,600 --> 01:07:38,850 because at timing points, buses can be held. 1435 01:07:38,850 --> 01:07:42,230 So that could be erroneously included as dwell time, 1436 01:07:42,230 --> 01:07:43,680 so that was thrown out. 1437 01:07:43,680 --> 01:07:47,340 Only far-side stops-- so what's the difference 1438 01:07:47,340 --> 01:07:50,948 between a near-side stop and a far-side stop? 1439 01:07:50,948 --> 01:07:53,251 AUDIENCE: Far-side stop is after the intersection. 1440 01:07:53,251 --> 01:07:55,250 GABRIEL SANCHEZ-MARTINEZ: Right, after signals-- 1441 01:07:55,250 --> 01:07:57,830 so far-side stops are after signals. 1442 01:07:57,830 --> 01:07:59,670 They did not look at near-side stops, which 1443 01:07:59,670 --> 01:08:03,395 could be affected by the red light, the example 1444 01:08:03,395 --> 01:08:07,801 that Ari just gave of, as the bus is ready to leave, 1445 01:08:07,801 --> 01:08:08,550 a light turns red. 1446 01:08:08,550 --> 01:08:10,530 Now you have a longer dwell time. 1447 01:08:10,530 --> 01:08:14,050 The bus could leave its doors open just in case. 1448 01:08:14,050 --> 01:08:17,430 So that was thrown out. 1449 01:08:17,430 --> 01:08:18,979 OK, they looked-- 1450 01:08:18,979 --> 01:08:20,790 AUDIENCE: What do you mean by known stops? 1451 01:08:20,790 --> 01:08:23,373 GABRIEL SANCHEZ-MARTINEZ: So if there were any stops that were 1452 01:08:23,373 --> 01:08:25,586 not properly coded or-- 1453 01:08:25,586 --> 01:08:26,460 they were thrown out. 1454 01:08:29,340 --> 01:08:33,060 They had APC, so the Automatic Passenger Counting, all doors. 1455 01:08:33,060 --> 01:08:38,580 And they threw out data that was of bad APCs, 1456 01:08:38,580 --> 01:08:43,520 so buses that required zero boardings or zero alightings 1457 01:08:43,520 --> 01:08:46,250 were thrown out. 1458 01:08:46,250 --> 01:08:50,120 And they looked at each AFC transaction 1459 01:08:50,120 --> 01:08:54,471 and matched it to what-- 1460 01:08:54,471 --> 01:08:55,209 how they paid. 1461 01:08:55,209 --> 01:08:55,939 Was it a ticket? 1462 01:08:55,939 --> 01:08:57,529 Was it a smart card, et cetera? 1463 01:08:57,529 --> 01:09:01,700 So they looked at a whole month, November 2006. 1464 01:09:01,700 --> 01:09:04,170 So here's how the model works. 1465 01:09:04,170 --> 01:09:06,260 First, they realized that it was easier 1466 01:09:06,260 --> 01:09:11,029 to have one model for when the front door controls 1467 01:09:11,029 --> 01:09:15,439 the process and a separate model for when the rear door controls 1468 01:09:15,439 --> 01:09:15,950 the process. 1469 01:09:15,950 --> 01:09:19,490 So the first step was to predict which of the two processes 1470 01:09:19,490 --> 01:09:22,229 was going to dominate and control the dwell time, 1471 01:09:22,229 --> 01:09:25,220 and then from that, select that one model, or the, one 1472 01:09:25,220 --> 01:09:26,700 or the other. 1473 01:09:26,700 --> 01:09:30,020 And then they looked at including bus type 1474 01:09:30,020 --> 01:09:36,500 and traveling as a friction factor in each of these models. 1475 01:09:36,500 --> 01:09:38,600 So let's look at the high-level results. 1476 01:09:41,420 --> 01:09:43,790 This is the front door model. 1477 01:09:43,790 --> 01:09:46,939 So this is for when the front door dominates the process. 1478 01:09:46,939 --> 01:09:51,260 They had a pretty good adjusted r squared, 0.733. 1479 01:09:51,260 --> 01:09:55,820 And we see some dummy variables. 1480 01:09:55,820 --> 01:09:58,160 NABI is a type of bus here. 1481 01:09:58,160 --> 01:10:02,030 And NOVA and New Flyer are also types of buses. 1482 01:10:02,030 --> 01:10:09,380 So NABI, for some reason, tended to be a half second longer 1483 01:10:09,380 --> 01:10:10,680 dwell time overall. 1484 01:10:14,660 --> 01:10:19,450 This variable is the number of people getting on in the front. 1485 01:10:19,450 --> 01:10:22,660 They are excluding the first few passengers for the reason 1486 01:10:22,660 --> 01:10:25,620 that Ari brought up earlier in the lecture, 1487 01:10:25,620 --> 01:10:28,487 that the first few passengers take a little longer, 1488 01:10:28,487 --> 01:10:30,820 but once you have a [INAUDIBLE],, a stream of passengers 1489 01:10:30,820 --> 01:10:32,980 going in, then there's a more uniform rate. 1490 01:10:32,980 --> 01:10:38,230 So this is front on extra. 1491 01:10:38,230 --> 01:10:42,490 And we have 3.7, about. 1492 01:10:42,490 --> 01:10:46,060 But it was a little longer for NOVA buses and a little shorter 1493 01:10:46,060 --> 01:10:47,560 for NABI buses. 1494 01:10:47,560 --> 01:10:51,250 So you have to adjust that amount for the different buses. 1495 01:10:51,250 --> 01:10:54,400 So there was an interaction between that variable 1496 01:10:54,400 --> 01:10:57,640 and the dummies for bus type. 1497 01:10:57,640 --> 01:11:03,310 Then the effect of people getting off in the front 1498 01:11:03,310 --> 01:11:04,650 was accounted for. 1499 01:11:04,650 --> 01:11:07,350 So here you have-- 1500 01:11:07,350 --> 01:11:09,320 if you have three or more people, 1501 01:11:09,320 --> 01:11:12,570 so people in excess of three getting-- 1502 01:11:12,570 --> 01:11:14,280 people in excess of two, actually, 1503 01:11:14,280 --> 01:11:16,820 three or more people getting off in the front, 1504 01:11:16,820 --> 01:11:20,290 that had an impact as well, a positive impact, of course. 1505 01:11:20,290 --> 01:11:23,220 So these parts that I described were 1506 01:11:23,220 --> 01:11:27,390 included in the model for non-crowded 1507 01:11:27,390 --> 01:11:29,760 and in the model for crowded as well. 1508 01:11:29,760 --> 01:11:33,030 So they have separate parts of the model 1509 01:11:33,030 --> 01:11:35,420 that were for crowded and non-crowded conditions. 1510 01:11:35,420 --> 01:11:42,390 When it wasn't crowded, cards were about 2.6 second effect 1511 01:11:42,390 --> 01:11:45,900 for boarding per passenger. 1512 01:11:45,900 --> 01:11:49,800 Tickets were about 4.8, so almost 2 seconds slower 1513 01:11:49,800 --> 01:11:52,560 than smart cards. 1514 01:11:52,560 --> 01:11:59,700 And New Flyer tickets were not quite as longer as tickets 1515 01:11:59,700 --> 01:12:00,640 everywhere else. 1516 01:12:00,640 --> 01:12:02,815 So this is the bus that had pretty wide doors 1517 01:12:02,815 --> 01:12:07,440 and the fare box was placed in a better way 1518 01:12:07,440 --> 01:12:11,020 so that people could tap in as they went in. 1519 01:12:11,020 --> 01:12:13,710 You have a small advantage there. 1520 01:12:13,710 --> 01:12:16,290 Here you have the effect of people getting off 1521 01:12:16,290 --> 01:12:18,230 in the front for the first two passengers, 1522 01:12:18,230 --> 01:12:21,900 not the ones that are three and up, so 2.8 1523 01:12:21,900 --> 01:12:24,930 seconds per passenger extra. 1524 01:12:24,930 --> 01:12:27,390 And here is a dummy for if the sensor in the front 1525 01:12:27,390 --> 01:12:30,400 was blocked, which could have different effects. 1526 01:12:30,400 --> 01:12:33,970 But essentially, that could indicate crowding or something, 1527 01:12:33,970 --> 01:12:36,240 so that was included there. 1528 01:12:36,240 --> 01:12:39,360 What happens when it was crowded? 1529 01:12:39,360 --> 01:12:43,950 So when the load on the bus was high, 1530 01:12:43,950 --> 01:12:46,960 it didn't really matter if it was a card or a ticket. 1531 01:12:46,960 --> 01:12:50,630 So now we're just regressing on the number of AFC transactions. 1532 01:12:50,630 --> 01:12:54,330 And we're getting an average of 4.3 per passenger. 1533 01:12:54,330 --> 01:12:55,650 So it's slower. 1534 01:12:55,650 --> 01:12:58,140 It's much closer to the ticket quantity. 1535 01:12:58,140 --> 01:13:00,330 And therefore, the impact, the benefit 1536 01:13:00,330 --> 01:13:05,070 that you had of dwell time savings by smart card is lost. 1537 01:13:05,070 --> 01:13:07,890 So that savings, you have when it's not crowded, 1538 01:13:07,890 --> 01:13:09,370 but you lose when it's crowded. 1539 01:13:09,370 --> 01:13:09,870 Sonya? 1540 01:13:09,870 --> 01:13:13,950 AUDIENCE: Is that the same thing with off-board fare collection? 1541 01:13:13,950 --> 01:13:15,700 GABRIEL SANCHEZ-MARTINEZ: No, because when 1542 01:13:15,700 --> 01:13:17,158 you have off-board fare collection, 1543 01:13:17,158 --> 01:13:18,930 you can open all doors. 1544 01:13:18,930 --> 01:13:22,720 And so it becomes more like a train, where everybody gets off 1545 01:13:22,720 --> 01:13:24,320 first and then everybody gets on. 1546 01:13:24,320 --> 01:13:28,034 So this model is for more people boarding by the front 1547 01:13:28,034 --> 01:13:29,200 and alighting from the back. 1548 01:13:29,200 --> 01:13:31,199 But sometimes, some people get off in the front. 1549 01:13:35,770 --> 01:13:38,500 And here we have the friction factor, 1550 01:13:38,500 --> 01:13:41,560 so the number of passengers-- the number of standees 1551 01:13:41,560 --> 01:13:46,580 squared times passengers. 1552 01:13:46,580 --> 01:13:49,230 So this is going to-- 1553 01:13:49,230 --> 01:13:52,045 it's a small factor, a small coefficient, 1554 01:13:52,045 --> 01:13:54,440 but it's multiplied by some quantity squared, 1555 01:13:54,440 --> 01:13:57,300 so that kind of makes up the difference. 1556 01:13:57,300 --> 01:14:00,520 And because this is squared, this 1557 01:14:00,520 --> 01:14:02,650 is kind of a polynomial effect. 1558 01:14:02,650 --> 01:14:06,780 So the more people that are standing-- 1559 01:14:06,780 --> 01:14:08,189 it's not a linear increase. 1560 01:14:08,189 --> 01:14:10,480 If there many more people on the bus that are standing, 1561 01:14:10,480 --> 01:14:12,070 then it's a much slower dwell. 1562 01:14:12,070 --> 01:14:14,850 So this is a correction factor for a crowded bus. 1563 01:14:17,770 --> 01:14:19,600 OK, here is the rear door model. 1564 01:14:19,600 --> 01:14:23,620 So this is for when the rear door was predicted 1565 01:14:23,620 --> 01:14:25,554 to control the dwell time, so more 1566 01:14:25,554 --> 01:14:27,970 towards the end of the route where more people are getting 1567 01:14:27,970 --> 01:14:30,550 off than people are getting on. 1568 01:14:30,550 --> 01:14:33,730 We have, again, some impacts. 1569 01:14:33,730 --> 01:14:36,460 The design of vehicle was significant. 1570 01:14:36,460 --> 01:14:41,020 And the number of people getting off was, of course, a variable, 1571 01:14:41,020 --> 01:14:44,740 so about 1.7 seconds per passenger-- 1572 01:14:44,740 --> 01:14:48,250 more on NOVA buses, less on NABI buses. 1573 01:14:48,250 --> 01:14:50,920 And the friction factor, again-- so you 1574 01:14:50,920 --> 01:14:56,680 have the sort of general friction factor was 0.005. 1575 01:14:56,680 --> 01:15:01,780 It was 0.009 for normal buses and 0.002 for NABI buses. 1576 01:15:01,780 --> 01:15:03,980 So you have to sort of add these up 1577 01:15:03,980 --> 01:15:07,260 to get the effective friction factor on a particular bus. 1578 01:15:10,530 --> 01:15:12,960 All right, so the sort of key takeaways 1579 01:15:12,960 --> 01:15:16,110 are that the smart media loses the benefit 1580 01:15:16,110 --> 01:15:17,280 in crowded conditions. 1581 01:15:17,280 --> 01:15:19,230 We saw that. 1582 01:15:19,230 --> 01:15:21,360 The crowding impact increases exponentially. 1583 01:15:21,360 --> 01:15:24,750 These people tried linear standees. 1584 01:15:24,750 --> 01:15:28,390 And standees squared was a much better predictor. 1585 01:15:28,390 --> 01:15:34,520 So bus attributes impact dwell time. 1586 01:15:34,520 --> 01:15:37,770 The dummies for the bus design were significant. 1587 01:15:37,770 --> 01:15:38,940 And they had an impact. 1588 01:15:38,940 --> 01:15:42,000 So some of that had to do with the location of the ticket 1589 01:15:42,000 --> 01:15:42,820 reader. 1590 01:15:42,820 --> 01:15:47,700 And some of that had to do with wide doors that allowed people 1591 01:15:47,700 --> 01:15:49,920 to enter more comfortably and faster. 1592 01:15:52,770 --> 01:15:55,120 So that's good. 1593 01:15:55,120 --> 01:15:57,360 Let's move on to the Green line model, just 1594 01:15:57,360 --> 01:15:59,880 quickly, at a high level. 1595 01:15:59,880 --> 01:16:03,030 We're almost out of time, so I'm going to skip this slide. 1596 01:16:03,030 --> 01:16:06,300 You're, I think, all aware of what the Green line is, 1597 01:16:06,300 --> 01:16:09,850 so I don't have to cover this. 1598 01:16:09,850 --> 01:16:12,480 They looked at one-car train and two-car train models 1599 01:16:12,480 --> 01:16:13,440 separately. 1600 01:16:13,440 --> 01:16:15,060 So sometimes the Green line will run 1601 01:16:15,060 --> 01:16:17,700 single vehicles, single car, or sometimes 1602 01:16:17,700 --> 01:16:20,080 two cars paired together. 1603 01:16:20,080 --> 01:16:27,382 And the dwell time here was some constant times the number 1604 01:16:27,382 --> 01:16:29,340 of people who are getting on times the number-- 1605 01:16:29,340 --> 01:16:35,100 plus the number people getting off plus some friction factor. 1606 01:16:35,100 --> 01:16:38,400 And same here for two-car trains, but the coefficients 1607 01:16:38,400 --> 01:16:39,270 were different. 1608 01:16:39,270 --> 01:16:43,035 So that's the overall concept that I want to communicate. 1609 01:16:43,035 --> 01:16:47,640 Here is a table of the results from that model for the number 1610 01:16:47,640 --> 01:16:48,780 of people getting on. 1611 01:16:48,780 --> 01:16:52,260 So this is using the model to forecast. 1612 01:16:52,260 --> 01:16:54,100 These are not observations. 1613 01:16:54,100 --> 01:16:59,110 So if you feed 0 people getting on, you just get the constants. 1614 01:17:01,780 --> 01:17:04,240 If you feed the model 10 people getting on, 1615 01:17:04,240 --> 01:17:07,030 it depends on the load on the model. 1616 01:17:07,030 --> 01:17:10,470 If the load is, say, less than 53, 1617 01:17:10,470 --> 01:17:13,870 then you don't have the friction factor really 1618 01:17:13,870 --> 01:17:18,400 controlling anything, so you have 20.3 or 20.2, 1619 01:17:18,400 --> 01:17:21,130 so a very small effect of friction, 1620 01:17:21,130 --> 01:17:27,430 and therefore about the same time for both models. 1621 01:17:27,430 --> 01:17:28,900 If it's very crowded, then you do 1622 01:17:28,900 --> 01:17:33,220 have a more significant benefit for having a two-car train. 1623 01:17:33,220 --> 01:17:34,720 That makes sense, right? 1624 01:17:34,720 --> 01:17:35,806 You have more capacity. 1625 01:17:35,806 --> 01:17:37,180 And the same thing happens as you 1626 01:17:37,180 --> 01:17:40,900 move to 20 passengers getting on and 30 passengers getting on, 1627 01:17:40,900 --> 01:17:44,080 with those differences increasing when you have 1628 01:17:44,080 --> 01:17:49,400 a crowded train, and the differences 1629 01:17:49,400 --> 01:17:52,156 between one-car and two-car trains 1630 01:17:52,156 --> 01:17:54,280 not being that significant when you don't have much 1631 01:17:54,280 --> 01:17:56,820 [INAUDIBLE]. 1632 01:17:56,820 --> 01:17:59,750 So this should, more or less, make sense. 1633 01:17:59,750 --> 01:18:04,610 So all this is to show you that you have some ideas about what 1634 01:18:04,610 --> 01:18:05,590 controls dwell time. 1635 01:18:05,590 --> 01:18:08,630 You test your hypothesis on the data. 1636 01:18:08,630 --> 01:18:12,080 And you can try different things. 1637 01:18:12,080 --> 01:18:13,840 These are just examples. 1638 01:18:13,840 --> 01:18:17,620 So the findings from that research, dwell times 1639 01:18:17,620 --> 01:18:20,350 were quite sensitive to flows and loads. 1640 01:18:20,350 --> 01:18:22,750 The crowding effect might be non-linear. 1641 01:18:22,750 --> 01:18:25,870 They looked at non-linear effects . 1642 01:18:25,870 --> 01:18:28,240 Just like before. 1643 01:18:28,240 --> 01:18:31,030 The dwell times for multi-car trains, for two-car trains 1644 01:18:31,030 --> 01:18:33,165 were different than those for one-car trains. 1645 01:18:33,165 --> 01:18:36,370 The dwell time functions suggest high sensitivity of performance 1646 01:18:36,370 --> 01:18:37,600 to perturbations. 1647 01:18:37,600 --> 01:18:41,710 So we saw what those preparations 1648 01:18:41,710 --> 01:18:43,670 were earlier in this lecture. 1649 01:18:43,670 --> 01:18:47,950 And this model is sensitive to some of them. 1650 01:18:47,950 --> 01:18:50,860 Because of that, effect of real-time operations control 1651 01:18:50,860 --> 01:18:52,810 should be essential to operating the Green 1652 01:18:52,810 --> 01:18:54,280 line with even headways. 1653 01:18:54,280 --> 01:18:57,040 So this is more of a recommendation. 1654 01:18:57,040 --> 01:19:00,460 And another recommendation is that simulation models 1655 01:19:00,460 --> 01:19:03,460 of this kind of service should include 1656 01:19:03,460 --> 01:19:06,670 sophisticated dwell time models like the ones that 1657 01:19:06,670 --> 01:19:09,390 were estimated here to account for all those effects, 1658 01:19:09,390 --> 01:19:13,970 because otherwise, you would have a simple model that is 1659 01:19:13,970 --> 01:19:15,640 not very faithful to reality. 1660 01:19:20,660 --> 01:19:22,660 Running a mixed fleet, some with one-car trains, 1661 01:19:22,660 --> 01:19:24,530 some two-car trains is dangerous. 1662 01:19:24,530 --> 01:19:26,624 So what do they mean by dangerous? 1663 01:19:29,057 --> 01:19:31,390 So it wouldn't make much difference if it's not crowded. 1664 01:19:31,390 --> 01:19:34,130 But according to the model, on crowded conditions, 1665 01:19:34,130 --> 01:19:36,430 the two-car train will be faster. 1666 01:19:36,430 --> 01:19:41,020 So then you will have a bunching effect happening, 1667 01:19:41,020 --> 01:19:43,960 two-car trains catching up to one-car trains, 1668 01:19:43,960 --> 01:19:47,095 and therefore deterioration of service quality. 1669 01:19:47,095 --> 01:19:51,190 Here is a marginal boarding time on heavy rail 1670 01:19:51,190 --> 01:19:53,810 from the third paper. 1671 01:19:53,810 --> 01:19:55,580 I think this was looking at the Red line. 1672 01:19:55,580 --> 01:19:58,210 So this is the marginal boarding time 1673 01:19:58,210 --> 01:20:00,670 when everybody can sit down. 1674 01:20:03,910 --> 01:20:06,940 When you look at the number of through passengers-- 1675 01:20:06,940 --> 01:20:09,604 that is the number of passengers who are on the train when 1676 01:20:09,604 --> 01:20:11,770 the train arrives on the platform and don't get off, 1677 01:20:11,770 --> 01:20:14,530 so passengers riding through the station-- 1678 01:20:14,530 --> 01:20:18,910 when the number is 0, then this is how much boarding time you 1679 01:20:18,910 --> 01:20:22,240 have per passenger, more or less. 1680 01:20:22,240 --> 01:20:27,940 And then as you look at the effect of more and more people 1681 01:20:27,940 --> 01:20:30,420 are standing, than that kind of slows people down 1682 01:20:30,420 --> 01:20:31,780 as they board. 1683 01:20:31,780 --> 01:20:35,020 So this was a way of capturing that effect, 1684 01:20:35,020 --> 01:20:36,940 again, the fiction factor. 1685 01:20:36,940 --> 01:20:41,360 So you see a theme here, that this seems to be relevant. 1686 01:20:41,360 --> 01:20:44,620 And there have been research studies 1687 01:20:44,620 --> 01:20:48,490 looking at ways of including that effect in models. 1688 01:20:48,490 --> 01:20:50,470 These models could be used for improving 1689 01:20:50,470 --> 01:20:52,360 the accuracy of your waiting time estimate 1690 01:20:52,360 --> 01:20:54,610 that your smartphone gives you, or improving 1691 01:20:54,610 --> 01:20:59,490 how faithful a simulation model is, et cetera. 1692 01:20:59,490 --> 01:21:02,220 Here is the equation, very similar, 1693 01:21:02,220 --> 01:21:03,460 same structure we had before. 1694 01:21:03,460 --> 01:21:04,939 We have some constant. 1695 01:21:04,939 --> 01:21:06,480 We have the number of people boarding 1696 01:21:06,480 --> 01:21:08,727 per door, the number of people alighting per door. 1697 01:21:08,727 --> 01:21:10,560 In this case, we add them up, because people 1698 01:21:10,560 --> 01:21:12,890 alight first and then board. 1699 01:21:12,890 --> 01:21:15,870 And then we have some friction factor. 1700 01:21:15,870 --> 01:21:17,850 How that friction factor is calculated 1701 01:21:17,850 --> 01:21:19,730 has been different on every model, 1702 01:21:19,730 --> 01:21:21,860 but they all have a friction factor. 1703 01:21:21,860 --> 01:21:26,220 So these are all services that, at least at times, 1704 01:21:26,220 --> 01:21:27,420 are quite crowded. 1705 01:21:27,420 --> 01:21:31,500 So that was important and significant. 1706 01:21:31,500 --> 01:21:34,230 OK, if there are no questions, you may leave. 1707 01:21:34,230 --> 01:21:35,970 And if there are, I'll take them. 1708 01:21:35,970 --> 01:21:37,690 Sorry, don't feel that you have to wait, 1709 01:21:37,690 --> 01:21:40,440 because it's already 5:30.