So in this, in this lecture, we'll be looking at, uh, how the flow models differ by trip purpose by different types of land uses, how they can have two different regimes in their models. And in the models, one regime can be, uh, uh, for, uh, congested flow and one can be for uncongested flow. And what do we say when, uh, a model fits the data? Well, so how do we determine the fit of a model we'll just quickly look at now? We have that, uh, the macroscopic models, uh, Greenshields macroscopic model. And, uh, this is kind of the big picture of those models, where the relationship between speed directional flow, uh, The speed and flow and density is given by a QP is equal to VP, uh, times KPI. So this is something that all of you should remember sometimes in case of pedestrians, we also say that we do not use a density KP. Instead we use something called pedestrian space, which is just the inverse of density. Right. So density is pedestrian per square meter. Whereas space is meter square per pedestrian. So just to, uh, make, uh, make it clear how many people are there in an area, but meter square is called pedestrian density. Whereas how much space does one person need is called? Pedestrian space. Right? So I have, I need about certain amount of area to move freely in a space. So that is called pedestrian space, meter square park. But it's true when we look at different types of trip makers, right? So everybody who is walking on a pedestrian facility may not be walking for the same purpose. Right. Some could be. Students who are working, who are walking to go to school college, or wherever they're going. Some could be commuters they're office, NGOs that are, uh, walking along the street. Whereas broadly, another group could be shoppers are leisure travelers who are walking along the street. So they, we, it has been noticed that there are some, uh, significant differences between these three kind of. Uh, trip makers along the street. So when you plot this, uh, graph again between speed and density, you will see that the students usually have a greater free flow speed. Right? So when the density is zero, the speed is called free flow speed. So that means that students usually tend to walk faster. Right. And that is faster when compared to commuters are. Shoppers. And that is intuitive also in another sense that students are usually, uh, younger in age. So, uh, they would also tend to walk faster and then you will see that, uh, their free flow speed is very different. Whereas, uh, when it comes to commuters and shoppers, although the free flow speeds are very close to each other, but also they're different because, uh, commuters tend to walk lot a little bit faster as compared to the shoppers. Right. Shoppers. Walk, very leisurely. They are shopping. So they're not in a hurry to go anywhere. Whereas commuters maybe going to their workplaces are going, uh, running an errand for their office. So there'll be walking faster. So that is what something people, uh, many research across countries have shown next. Also now, uh, we have seen, we have seen, uh, the macroscopic models for. Um, for, uh, vehicles where we have seen the relationship between flow and density. But now here, if we change the same, uh, if you change the graph, uh, or the relationship between to see the relationship between flow and space, because remember a pedestrian, uh, in pedestrian realm, we sometimes use space instead of density. So when we see the, uh, graph between flow and space, we notice that. As flow in as a space increases, right? So need for space. So I may need a little bit over, a little less than one square meter to walk freely. Whereas some may need a little bit more than one square meter. So as the space increases, the flow also increases, but the increase is only up to us certain point. Right. The increases, maybe just about 1.1 meter squared per person, after that, if you keep on giving more space for people, right? So space is nothing, but if there is a person and the distance between the next person, right, this is hypothetically, that is the space. Now, if this space keeps on increasing, that means what turn it means is that there are very fewer number of people walking. Right because the distance between two people is too much, so much. That means there are overall in the system or in the sidewalk. There are fewer number of people walking. So when there are fewer number of people walking the floor, which is the number of people per minute, per meter also reduces, right? So up a sidewalk may have a lot of open spaces. A person is walking very freely. Yes. They have their freedom to walk, but at the same time, there are not enough people to then justify such sidewalks. Right? So you will see many times there are wide footpaths, but hardly any people are walking on that. So the one person or two people that are walking, they have enough ample amount of space, but the floor, which is the total number of people. Moving across a certain point per minute is very less. So we optimally you see just about 1.1 meter, a meter squared per person is optimal enough to have a very high floor. It, it keep on giving more space to people. Actually the floor it starts to reduce or the flow starts to reduce. So when we are designing, uh, sidewalks, we have to be very careful as to how much. Uh, with the, the sidewalk should be there versus the number of people that are going to walk on that sidewalk. We don't want to give to wider sidewalk if the volume of people actually, who are going to use it is very less, because that was, that will mean that the flow is very low. Flow is very low. Usually means that you are not utilizing the capacity of that sidewalk to a optimal extent or maximum extent. Then similarly the third kind of a, um, a macroscopic model that we have seen is four vehicles. What we had seen it for speed and flow. Similarly, if you convert it, uh, for pedestrian case, uh, into speed and space, we see that again. Similarly, if we keep on increasing the space between, uh, for each person, right? Space, meaning meter squared per person. So if we keep on increasing the space beyond a certain point, the speed is no longer going to increase. Because again, what happens is people can walk only at a certain pace, right? A certain speed. They cannot just because that is too much free space available to them. They cannot walk any faster because that is beyond their physical ability to walk any faster. Right. So on an average, maybe people walk at around one meter per second. If you keep on eating me one may think that, well, let me build a cider, a wider sidewalk, uh, where, uh, I'm giving about four meters square per person. So I'm giving one person almost four square meters to walk comfortably. And you would, and maybe the thought is that if I give more space, the people can walk faster, but it has been noticed that any space. More than about 1.5 to around 1.75 square meter does not bring about any increasing speed. Hello? Yes. If the space is less than one, then there is a much larger reduction in speed. So when in the, in congested situations, when one person doesn't have enough space or less, the order space less than. One meter squared for himself or herself, then their speed does get restricted. Okay. So that is the idea of space, uh, developing a relationship between space and speed, uh, versus developing in between flow and, uh, speed. Okay. Which is more of a, uh, vehicular, uh, relationship. And this is more of a, this is more of a pedestrian relationship, or this is more of a vehicular relationship, right? Okay. Okay. So now the other thing to remember is in all of these relationships, we have looked at, uh, say the relationship between speed and density, uh, flow and density, speed and flow or flow and area, uh, space. We have looked at it saying that they are single regime models. So, what do we mean by single Reggie models? Is that yeah, that the, that the same speed and density relation is valid for the entire range of densities seen in the traffic strip. So if we say, go back to the speed and density relationship, so let us see the speed and density relationship. We are saying that one model. Right. So each of this is a model, right? We can develop a model for the students. This is a model for the commuters and the bottom line is a model for them. Shoppers. When I say model they're all statistical models, right? So what we are looking at simplistically is to say that even at this density, the slope of these lines is the same ad. As versus two at this density. So we are also seeing that the slope of this line is constant B whatever, be the density. However, people have found out that there is a cutoff point at certain density where the SA the model has a different regime. So for example, the slope, maybe this, whereas after that point, the slope maybe very steep. Right. When you have such a model, they are called to Reggie models. So if I can draw the same thing here, if it is a two Reggie model, what usually happens is that the slope here up to a certain density is this. Whereas beyond that density, the slope kind of changes. So then you can make good distinction that these two are both representing the students only, but beyond a certain density. They have different behavior. The student, the student population that are walking have different behaviors. So these are usually called two Reggie models. And why people are looking at as a, as, as they become more and more mature in this field. Why are they trying to, uh, look at these things up? They are saying that if it is an uncongested flow situation, so there is some density or some density, cutoff or threshold at which, uh, beyond which the. My foot part of the sidewalk becomes congested, right? There's the density increases. It becomes congested. So in congested flow, people walk or people's behavior towards walking changes, vis-a-vis their behavior or walking, uh, under uncongested situation. So the top part usually is the uncongested situation, right? Because here the density zero. And so you can have free flow speed from here. So as the density increases, your speed reduces, but beyond a certain density, now it isn't congested flow situation. So this, this half is a congested flow situation. Whereas this isn't uncongested flow situation and your behavior changes. So what people are now looking at more and more are. Speed density relation will also be different in different zones of densities. Right? They're saying that there are multiple regime densities to regime separated two Reggie models. They're separated for congested and for uncongested. Whereas for simplicity of understanding, we have looked so far at all of these models as a just single Reggie model. So we'd not, are not making any distinction between uncongested flow and congested flow. Mostly a single ready models are usually used in a regular application of these, uh, models. Whereas in theoretical studies and to improve models, people start looking at two regime models. So let us, uh, so this is an example again, right? So when you are looking at pedestrian flow and pedestrian density, What some of these newer, so in 1.0, so this is a simulation software. Great. Uh, within one point or the, uh, at different, uh, uh, uh, meters is what they have, uh, developed this for. And Fruin was the initial, uh, researcher who had developed, uh, this model, uh, back in the 1970s. And when he had developed it for bi-directional flow, The red line he had made that there is no, uh, no, he had made no distinction between congested and an uncongested flow. And he had said that this one model fits the behavior of all the pedestrians beat on an empty sidewalk or beaten up congested sidewalk. However, when he started looking at unidirectional flows, he saw that. So unidirectional flows, meaning, uh, people moving, uh, in, in the same footpath, people are moving. In, um, uh, only one direction, right? Otherwise bi-directional meaning on the same foot part, they're moving in both directions. When you start having that kind of a sidewalk, he saw that if you see the dotted red line, he saw that there are two different models in the uncongested flow. The model kind of goes like this, whereas in the congested flow, the model goes like this, right? So there are two different models. They're dotted. Uh, dash lines. Similarly, when we SIM, uh, simulations are done using wisdom, they also, we SIM could actually model the con uncongested flow pretty well. Right? All the black lines you see, but it was not able to model the congested flow pretty well. So there is essentially people have started to make this distinction as to how there are two different regimes of the same model. Similarly, this is a example of. Uh, pedestrian flow and pedestrian speed. Again, they have been able to model the uncongested flow pretty well, whereas when it comes to the simulation tools, as well as the, uh, unidirectional flow, um, it becomes very difficult to model the congested flow. Okay. No. Why are we telling you about all of these models? Right. Uh, w why should so much focus be paid, uh, towards these, uh, models and these relationship between the, uh, three basic parameters, speed density and flow, because these are utilized to design the capacity of pedestrian facilities, capacity, meaning what should be the width and for how long should you build the site pedestrian facility for. Right. If you don't know how people move, uh, if you don't know the relationship between the flow and the speed parameters or the flow and the space parameters, then you would not be able to say. You would need a two meter wide sidewalk for the next, um, one kilometer or do you need a five meter wide sidewalk for the next half a kilometer, right? You do not then can, cannot estimate the right capacity of the sidewalk. And then you make, uh, um, sidewalks that are underutilized or not utilized at all. And, um, that has policy and condition implications. So also that is one of the reasons. And then that eventually affects what is called the pedestrian level of service, right? The person level of service we will get into in the next few lectures, uh, classifies, uh, how many, uh, Uh, pedestrians flow, uh, for a given wit offer sidewalk, for example, uh, one of the, uh, uh, one of the ways of measuring pedestrian, uh, level of services to see, uh, how much Y uh, facility you have and how many people are flowing through that facility. So don't design it, uh, and, uh, design it accurately level of service, which is an indication of how well, uh, the pedestrian facilities. Uh, are serving the users will, uh, will be effected. So the basis for everything is to know this relationship between the flop, which in turn tells you what capacity you should have. You should construct your pedestrian facilities, which in turn affects the level of service that is being provided, uh, by the pedestrian facilities. Hence, it's very important that you know about these models. Now quickly again, let us, uh, take you through an example of this model and how to solve it. So given is it you and, uh, speed and, um, uh, density relationship along a commercial street with no sidewalk. Okay. So they have no sidewalk facility, but we have observed, uh, that the people that are walking are, can be modeled using. This relationship. Right? So, uh, speed, uh, has a relationship with a pedestrian density. We have shown you the curves. This is the relationship. Now the question is asked, what is the average speed of pedestrians? When the densities are 0.9, 1.13, uh, pedestrian per meter squared, the graphical representation of the model, right? The graphical representation of this model and drive the other flow parameter. Uh, models. So you know that the other flow parameters, other than, uh, uh, speed and density are speed and flow and flow and density. So how do you derive that? So now the first question is to, uh, the speed of pedestrians at different densities. So it is just the very simple you have the equation, uh, the P value or the K values, and it gives you that. Uh, speed values. However, the interesting thing to notice again, like you already know by the, uh, car that has been shown earlier, the relationship that has been shown earlier is that as density increases from 0.9, 1.1 to three, the speed decreases, right? So speed, 4.9 is five 51.5, then decrease to 48.5 and then finally is resistor, right? So this is how usually. Uh, the relationships are so the higher, the densities higher density with accommodate less average pedestrian speed. Now, how do you, how do you show it? Uh, how do you show it in a graphical manner? So to show it in a graphical manner, we know that it has an inverse relationship, right? We know, but where does this cut? The exact Y axis. And where does this cut? The x-axis you have to know. Correct. So we all know that the graph cuts the y-axis. That point is called her free flow speed. So at free flow speed happens, the density is zero. So if you therefore mean mean free, uh, pedestals speed at uncongested condition V P F should be equal to. So if the density, if you say KP is equal to zero, then the VP is equal to 65 meters per minute, which gives you the. Free flow speeds. So now, you know that your first point is here. Secondly, what you know is that the slope of the line is minus 15 times, um, a KP value, but also what you know is the point at which, uh, this curve cuts. The x-axis is called the jam density. So at jam density, what happens is the speed becomes zero. So if you. Put the value of zero for VP, this KP automatically becomes KP becomes equal to, okay, John. Right? So if this is Zillow, you put it as Cajun and you'll know that K jam now is for 4.33 bed per meter squared.So now, you know, your, uh, mean free flow speed, you know, your jam density. So you can. Draw this curve. So whenever you're told to graphically represent, uh, an equation, you have to know all the parameters, all such parameters, so that you can draw the graph, a graph. If you don't know this and just draw a graph like this, then we know that you understand that it has a negative, uh, it has a negative slope, but you have not actually calculated these two points accurately. All right. It is necessary for you to do that. Right. And then we know how to measure or how to estimate the other parameters of the, uh, of the, uh, relationship, uh, of the macroscopic relationship, because we know the formula. So if only VP is equal, this has given you convert, uh, you convert VP in terms of Q and K. Put UNK here, you get the parabolic equation between now the relationship is between Q and K. So flow and density, which is parabolic. We already know. Similarly, if you convert this into a relationship between a Q and VP, that is also a parabolic flow and speed is convert this into a flow and speed lesson. And now if you want to convert, don't have, don't want to have density, which in many cases, uh, pedestrian cases, we don't. See the density like that, but see it in terms of space, then this becomes an inverse parable, right? Why we are telling you all of this is because, uh, in the very recent Indo at seam, uh, that has come out, these, uh, relationships have been depicted for different types of. Uh, pedestrian facilities. So if it's a terminal pedestrian facility, terminal, meaning, uh, it's a, it's an end point. So it's like a last stop or a bus stop, a bus station or something like that. Then the floor density curves are, uh, the simplest cover letters. Look at the simplest covers their speed density curve. The speed density curve is given by VP equal to 81.49 minus. 21.16. Whereas if it is, uh, if the facility is in a commercial place, so it's a sidewalk we're in a commercial place, then you'll see that the speed density relationship is 64.62. So what immediately, what this should tell you is that, uh, at, at terminal places, right at terminal places, but it's still the free flow speed of pedestrians is higher than when compared to. Commercial places, right? In commercial places, people will usually walk slower. There. Maybe there are resident, uh, maybe there are shops where they are shopping. So the commercial areas and usually the, the number of people in the commercial is walking is also high. So the speeds may be low and hence the free flow speed is lower than in case of. Terminal spaces that are in terminal facilities, where people usually have got off a bus or got off a Metro late, and then they want to walk pretty fast, right? They want to get to their homes or offices or something. So that is the easiest, uh, uh, relationship to be understood. Uh, so here are, uh, SIM also develops different models for different types of land uses. Right. So you see that there are these one, two, three, four, five, there are five different land uses for which these models have been developed. Land use terminal land use institutional land dues, commercial residential. So when I say land users, so it is the land use that is, uh, butting this facility, right? Or the land use that is, uh, contained, uh, containing this facility. So if it's a recreational land use, meaning maybe it is a, there's a foot part that is along. Cedar park, for example, terminal, we just told you, uh, institutional meaning maybe there is a, a big office building or a big school or a big college, some kind of institution. Uh, there's a sidewalk along it, commercial, meaning again, uh, market area, residential, meaning near your homes. So it has been seen that people behave differently. Right. We showed you earlier. Also, if you are a, uh, student versus if you are a shopper versus if you are a commuter, your, um, behavior is different. It is furthermore, it has been founded. If the sidewalk is along different types of land uses people behave differently. Right? So you see that, uh, the green one, which is the recreational recreational model, sorry for the recreational model, you see the slope is very gentle. And the free flow speed is much less than for the others. That meaning that people usually want to walk very leisurely, very slowly along foot paths that are closer to recreational facilities or about take recreational facilities. Whereas if there are other ones, they walk usually faster than that, that is the relationship. Similarly, the other relationships are now whenever people want to know how good. Is the model that you have developed. What people usually try to mean is to calculate what is called an R-squared. So these are all simple linear regression equations, right? Not simple linear regression equations. So for linear regression equations, what usually they want to see is it trend line like this, right. Is there a trend line that explains all of your data sets? So these may be all of your data sets the relationship between X. And the relationship between X and Y. Right? So these may be all your datasets, but is there a common trend line that can explain the behavior of majority of them? So that trendline, usually the higher the R squared or the R-squared closer to one, uh, if the asker is closer to one, then you usually say that. That model fits your data very well. So R squared can be, the model can be a light and have a negative slope or a positive slope. Doesn't matter. It could be like that also, but if it is closer to one now in this case, it is saying it is equal to one, which also sometimes it's not very good because, uh, it might depict that you have, uh, done something with the dataset and removed outliers, which. Maybe are important, uh, for your situation, but, uh, you have removed them and you got a perfect fit, which is always not good. You don't have to always work towards art square should be tending towards one. Uh, that should be your goal, which should not be, uh, not necessarily be equal to one. And if it is something like where there is no relationship, then you have to look at other types of, uh, Uh, models, which can explain your data, right? If there is no relationship between X and Y, then it is, uh, no linear relationship between X and Y. Then you start looking at other kinds of models. Okay. So R squared is the goodness of fit index.
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