 So, today's class most of the time we will try to spend in looking over a case scenario and then followed with some practice problems. So, as a case let us take a look at this notion of say traffic congestion, this can even illustrate what kind of policy resistance it comes up and how we can iteratively build a model which try to encompass at least the key aspects towards traffic congestion. First some background, so there it the freight transport by road has risen from 6 billion tons. This is for statistics for India in 51 to 1100 billion ton per kilometer in 2000. So over 50 years and passenger traffic has also raised from 23 billion passenger kilometer to 2875 billion passengers per kilometer during the same period. So, the order of the growth has been tremendous, annual growth rate of traffic expected to be 10 to 11 percent, whenever there is a boom in automobile sector that leads to increase in road traffic in the future. So, that is the part of the traffic, but when you look at the capacity that is the road network, national highways carrying 40 percent of traffic, among national highways and state highways only 2 percent of the lengths are 4 lanes, 34 percent 2 lanes, 64 percent single lane. This is as of 2004, it is not that updated, but both would have increased since then. So, we can assume a parallel or similar kind of increases that increase in the cars on the road versus road network, the main idea is the trend, the growth percentages that has been tremendous both in the capacity as well as in the traffic. So now, this is, so we would like to now model this concept of traffic congestion as well as how we are going to say increase the capacity. A simple example we saw last class as an example for delays itself captured this, where you try to model traffic congestion as well as how do you add capacity, how do you add capacity to road network, increasing amount of paved roads so that you can reach further places or widening the existing roads, either way it is going to involve construction. So, only way we are going to increase capacity is by road construction, either way maybe we will build better bridges, do some shortcuts for different locations, tunnels or expand the road network by widening, lane widening is very common as well as expanding the reach of paved roads into more remote areas or new locations and things like that. So models for traffic congestion, let us look at an open loop view, pretty much what we want to try is that whenever there seems to be some congestion, how do you measure congestion? Congestion in the road, after a delay we find that road construction happens. Most of the time it is not the other way, initially there is a lot of congestion, then government invests in expanding the road facilities there and as soon as road construction is finished traffic seems lighter and going smoother, so how do you measure this congestion? How can you measure this congestion? What do you mean by congestion? Density of vehicles on the road, what does it affect? Speed, eventually what? Travel time. Travel time, so what we are interested in is congestion or delay or let us make a better note called as travel time. So we can expect that as travel time increases then we want after some delay it will prompt action to expand the road capacity by involving road construction. This is what we want to do because the density in one hand can be a little misleading because it is taking it from point in time or the speed can be regulated but as long as traffic seems to be moving and you are getting from one point to other within your preferred travel time then people are happy. But this kind of open loop is not sufficient because even in your own narration in our head what we want to do is move now towards the, let us start with the basic closed loop. So in closed loop view let us use a couple of variables, let us say call it travel time and what we are trying to model is this entire scenario of how traffic congestion as well as movement in the road traffic, how it is affecting our road construction and highway capacity expansion and the interrelated system within that. That is a very broad scope and we will just hydrate it and move ahead. So let us define congestion by using what we call as travel time. What affects travel time? Some of the things is already given, what affects travel time? Density or you have the number of vehicles on the road as well as the capacity of the road as highway capacity affects it as well as number of vehicles or the volume of traffic. We will make it as part of the capacity. It is a highway capacity. We will determine the quality is bad, we will just transfer it into the. So travel time is depends on capacity of highway. Capacity of highway in turn depends on the number of lanes, it depends on the quality of the road, number of signals that is there, other bottlenecks that can appear at some distinct points in time, I mean distinct point in space depends on capacity of highway and volume of traffic. We have two variables which we say can affect our travel time. But road construction does not immediately happen. Suppose even if the road is narrow or there is only single or two-lane road, but as long as there is no new pressure to expand the road network, road construction is not affected or as there is no pressure to reduce congestion. So let us create, identify new variable called as pressure to reduce congestion. So more the pressure, we can expect that this is going to result in as there is more pressure to reduce congestion, it will prompt their government to take up road construction activities where road construction activities once it is completed will result in increased capacity for the highways. So this is what we kind of can expect. But this pressure to reduce congestion itself will depend on not just the travel time, it will be a function of both the travel time as well as a desired travel time. If it takes 10 minutes to go somewhere and people are happy with the 10 minutes, then there is not real pressure to reduce it. But if you want to do the same thing in 5 minutes, then that is when the pressure comes. So we want to define those goals explicitly. So let us try to draw this preliminary loop right now as our first closed loop. So let us just define by our highway capacity as well as traffic volume. Traffic volume is large, we expect the travel time to be large. Highway capacity is large, we expect the traffic volume time to be lower. And based on the travel time, let us have a variable called as pressure to reduce congestion. And as pressure to reduce congestion is large, we take up road construction and as road construction after a delay results in increased highway capacity. So as pressure to reduce congestion increases, road construction activities are going to increase as road construction activity increases. After some delay, the highway capacity will increase. It takes time to expand roads, time to you know, you have seen all that. And as highway capacity is more, then the travel time goes down. Now as travel time increases, the pressure to reduce congestion also needs to increase. But that increase is related to what we can call as desired travel time. So if you see here, what it just did is apply some other guidelines that we gave, we had. So we here, if you look at this loop, it is a negative feedback loop for road construction. So as travel time is more, there is pressure to reduce congestion. As pressure to reduce congestion increases, road construction activities increase. As road construction increases, after some delay, the capacity increases. And as capacity increases, the travel time falls. So that is a negative feedback that is happening. But what is the goal here? The goal will be, has to be done explicitly. It cannot keep on happening like this. So the goal is the desired travel time. Based on that, you decide whether it is to be a one lane to two lane or one to four lanes or one to six lanes. It depends on the desired travel time. So this defines your goal of the system. Another point is, while we are actually doing this kind of modeling, it is good to actually start thinking in some sort of units that we want to try. Because it will also help in identifying some new variables etc. that we may want in the system later. Like we have, say for example, these three variables, travel time, traffic volume, highway capacity or desired travel time. We can think of what kind of units that we can have for these. For example, we have variables like, let us say traffic volume. Good to consider units while modeling. Suggested units could be that if you want a traffic volume. It could be say, vehicle miles per day, then highway capacity. See eventually we want to compare highway capacity as well as our traffic volume. So it is good to then have both in the same units. Eventually that is how the model has to come to. Highway capacity can also be defined as the vehicle miles, say per time unit. Say per day or per minute or whatever it is. The same time units can be considered. So this means that okay, now we can compare it. Suppose you have defined highway capacity in terms of width of the road. Then comparing width of the road to vehicle miles per day, then how will we compare it? We cannot subtract it or we cannot divide it or something because the units are may not necessarily match. Again these are not all the variables. I am just giving illustration, traffic volume, highway capacity. Then travel time. Travel time itself could be say minutes per trip or something like that. So the idea here is we want to come up with some metric for what kind of units that we are looking at. This is how we are going to start measuring these values and this is how we are going to define some highway capacity and travel time it is going to be minutes per trip. Then we need to get some say example like average trip length or something. So then we can use it to figure out what is how we are going to measure the time units and things like that. This is just a kind of illustrative idea of how we are going to think about units. Now as this travel time increases as I told we are going to have congestion. So we define that as travel time is more then congestion is more. So and this is a negative feedback loop. Then we are in when we start building the model this is kind of small steps that we can take to figure out okay let us draw some broad loops to close the system and then let us look at currently what are the variables that are outside the loop. We have two variables one is a desired travel time other is the traffic volume. Desired travel time sounds like exogenous variable in the outside system because it is what people expect. Then we have traffic volume. Can traffic volume be made endogenous? What do you think affects the traffic volume? What all affects the traffic volume? Number of vehicles and then population affects the number of vehicles fine. Other number of vehicles what else? Just because there is lot of vehicles what do they need to do? The final units we just wrote as vehicle miles per day. Number of cars just goes vehicles. Where will I get miles and per day? Distance. The distance traveled. Quality of road affects the highway capacity. So the traffic volume itself it depends on the day of the week is fine. Let us what else? You have got two things. One is you have got the number of vehicles this time or number of cars and number of vehicles. Second is we need the vehicle miles per day. So for miles we need what is average distance or average trip distance. Average time is what you already got as travel time. Travel time you already got it but what you are looking at traffic volume. So one is how many cars are there and what is average distance per trip that they are making. And we can also have one more variable called average number of trips per day. If for example let us say if we have a nice highway or let us say even inside campus when there are shops right here the traffic is very low. So even if we forget some things we can make multiple trips to keep buying things. But if it is going to take you half an hour to reach let us say 2 kilometers to buy things then you might say even if you forget something you are not going to make the second trip. So if there is capacity is large then you may be able to make more discretionary trips or more number of trips you can make. There is one and two you will make longer trips define traffic volume as these three. Traffic volume can be affected by the trips per day, average trip length and vehicles in region. When people start taking more trips then my traffic volume is going to be more as average trip length increases traffic volume is going to be more there is more. Traffic congestion outside IIT is caused by people not from power, people not in power because they are all travelling large distances. People in western suburb want to work in eastern suburb, people in eastern suburb want to work in western suburb. So it causes traffic jam in power. And more cars in the region we can expect higher traffic volume and as was also suggested vehicle in the region depends on the population and to some extent on the economic activity of region. That is more the population more vehicles can be in the region and or just that does not affect we can also have let us say average vehicles per person as more average vehicle per person is there vehicle region is going to increase is going to contribute to traffic volume. Now let us try to close this loop by looking at suppose my travel time became easy if the travel time is large and my desired travel time was 15 minutes but my current travel time is say half an hour then it causes pressure to reduce congestion. Suppose the capacity is so big that you are now able to go there really fast so that will attract you to driving imagine nice well laid smooth roads we would love to drive we do not want to walk we do not want to take public transport we may just want to drive correct. So let us introduce that variable called attractiveness to driving. Here I am reversing the signs right just observe this travel time desired travel time here travel time is plus desired travel time minus some actual travel time is here but desired is here so there is huge gap then that causes pressure to reduce congestion but now if my travel time is only 5 minutes but my desired travel time is like I am willing to spend say 10 minutes to go there so that means you are more attracted to driving because roads are free and you can go at higher speed and it is much more pleasure driving and more you are attracted to drive you will do more trips per day you will just go out so that you can take a spin in your motorbike your new motorbike or new car you will go longer distance you will not go to the nearest shop you will say okay RIR what is this nice road why the hell am I stopping in nearest shop let me travel little longer distance and go to the other bigger shop or some other fancy place which has near mall etc but you get the relation right like these are just opposing to each other you may not even have two variables but this is just to illustrate the point that travel time is too high and people you know for short distance themselves spend long time to go that causes pressure to reduce congestion but your travel time is quite smooth and your then that will attract you to do more driving you can have both ways you can have this one also will that will make people happy as more you are attracted to driving then your desired travel time also increases so that is not the only one which attracts us to driving we cannot consider okay let us kind of finish this loop first so here if you see as attract as let us say we are more at we are more prone to say prefer driving then we do more trips per day which increases in traffic volume which increases in travel time which then reduces your attractiveness to driving each time you are getting caught in traffic jam you are you do not like it so this becomes a negative feedback loop same thing goes here as more attractive to driving you do more longer trips but then as traffic volume increases travel time increases then your attractiveness driving comes down it takes much longer time so we try to avoid or come up with another route things like that again becomes a balancing loop attractiveness to driving to by average vehicles per person please not enough we just go through this only looking at private vehicles ownership many times the attractiveness to private vehicles of course there is a simple pleasure there is also the utility of it because most of the commute is for work so we would like to bring in the adequacy of public transit itself on how attractive it is to drive so let us introduce those two variables called as adequacy of public transit as a negative relation the sense more adequate the public transit is it reduces your attractiveness to driving if you have really good public transit it works the other way public transit is not good enough then attractiveness to driving increases and as more people start wanting cycles or preferring their own means of transport maybe you will get a better pair of shoes that is also part of private ownership so that can also reduce the what can you say public transit ridership now let us look at what I have done I have introduced a new external variable called adequacy of public transit and attractiveness to driving and as attractiveness to driving becomes larger the public transit ridership falls down and it also has as it falls down the number of vehicles average vehicle per person also increase I could have put up direct link also but let just go diagram for now so people start buying more vehicles and as they start buying more vehicles more vehicle regional increase which will increase the traffic volume which reduces the travel time which reduces the attractiveness to driving so this again I have one two three negative feedback links so this also becomes a larger negative feedback within the system