 I welcome you all for the today's session on fundamentals of traffic flow. In this class, we are going to discuss about the density, occupancy, spacing, and headway. I am Ashok Kumar, Aston Professor, Department of Civil Engineering, Walchand, and Stiff Technology, Solapur. Learning outcomes of the today's session. At the end of the session, students will able to calculate traffic density, occupancy, spacing, and headway. First, we'll discuss about the density. As we know that it is defined as the number of vehicles occupying a given length of a highway or a lane. It is expressed in vehicles per kilometer or vehicles per kilometer per lane. So if I take the 1000 meter length or 1 kilometer length in that how many number of vehicles are occupied in a per 100 meter length, 1000 meter length. So it is number of vehicles per kilometer per lane. It depends upon the spacing and some other parameters. It is very difficult to measure directly on the field because in a 1000 meter you need some elevated observations. If you are standing at an elevated, any elevated point, from there if you are able to observe a 1000 meter length, in that you can count the number of vehicles are there in that 1000 meter. So until and unless if you are not getting any elevated observation, then it is very difficult to measure density directly on the field. So we need to have for computing this density, we are going to depend upon the speed and flow rate measurement. With this we can calculate what is the density of that particular lane. It is very important in terms of quality of a traffic flow because it directly gives the demand. What is the demand for a traffic engineer? So number of vehicles spaced closely, number of vehicles are spaced largely and even this will give the driver what kind of the speed he has to select depending upon the spacing available in front of him. So the spacing is available very less then he will go with very less speed. If the spacing is more then he will decide to go with the high speed. So this density and your speed will give the rate of flow and even density also decides whether the flow is more or flow is lesser. So this is one of the macroscopic parameter for any traffic steam flow. It is measured, we know that it is number of vehicles spaced in a length L. So these are the number of vehicles are spaced 1, 2, 3, 4 in a length L. So number of vehicles divided by the L, length L say 1000 meter or 1500 meter, that gives the density K. So where K is the density in vehicle per unit distance. So unit distance usually we take it as 1 kilometer and N is the number of vehicles occupied in some length of roadway in a specified time and L is the length of roadway. So other the microscopic parameter, now we are going to go about the microscopic parameter. In this we will try to discuss about the occupancy spacing and headway. So as I told you it is very difficult to measure the density on the field directly. So if you have any modern detectors on the field we can measure the occupancy. With the occupancy we can measure the density. So what is the occupancy? occupancy is defined as the proportion of a time that a detector is occupied or covered by vehicle in a defined time period. So if your detectors are placed this magnetic detectors are placed on the road surface. So with that we can how much proportion of time that is the vehicle is occupied and that detector depending upon the percentage or the proportion of the time it is there on the detector we can fund out the with the occupancy we can fund out the what is the density. So here let Lv is the average length of the vehicle in meter and Ld is the length of the detector this is the usually magnetic detector and O is the occupancy over a given detector. With that we can calculate the density k equal to 1000 into O is the occupancy over given detector Lv plus Ld. So in this case let us take a occupancy is 0.2 and a 15 minute analysis period is taken. So within the 15 minute analysis period what is the occupancy we are trying to find it out with that correlation we are finding out the what is the density. So length of the vehicle is 8.5 meter and detector is 0.91 meter. So we can calculate what is the density for 15 minute we can convert into take it into 1000 meter length this is 8.5 plus 0.91 and 1000 multiplied by 0.2 that gives you 21.25 vehicles number of vehicles in per kilometer per lane. So this is the with the occupancy we can also calculate the what is the density on that particular section. This occupancy this detectors are placed in each lane. Suppose you have another adjacent lane you have to put the another detectors on the another adjacent lane so that at the end of the day we can calculate we can add both the densities and we can come up with a each lane density on the particular lane. So for that we have to conduct only for each lane this occupancy is going to be calculated then we can later we can add this both the density for each lane and left lane and right lane separately. Another term is a spacing this is a distance between two vehicles measured from front bumper of the vehicle to that of another front of the vehicle. So front of the bumper of the vehicle this is the front vehicle and this is the following vehicle. So if I measured from the front vehicle to following vehicle from bumper to bumper that is called as a spacing it is the inverse of your density. So s equal to 1 by k. Now you pass a video for over here and think for a minute and try to give the answer for this. So here it is in a during heavy congestion it is observed that the average spacing of vehicles in a queue length is in a mostly in a 6.5 meter spacing is given for you and ask you to calculate what is the jump density or density for a stopped vehicle. With the spacing in a using previous relation can you able to calculate what is the density over here. I hope you are able to give the answer for this question the answer the answer for this one is you can put the spacing equal to that is I told you spacing is inverse of your density so spacing s equal to 1 by k. So in this one spacing is given you need to find out what is the jump density over here. So per kilometer usually I told you density is calculated per kilometer so per kilometer is 1000 meter length and spacing is given 6.5 meter you can calculate 153.85 vehicles per kilometer this is the density for this question. Going here to the another important microscopic parameter is the headway. This is headway is defined as a time interval or a distance between successive vehicles. So it is measured in terms of from a distance or in terms of time. Typically this is measured from bumper to bumper I told you either you measure in time or measure in a space. So these are classified into two ways time headway and space headway. So let us see the time headway it is difference between the time when the front of the vehicle arrives. Suppose if you are standing over here and measuring the front vehicle the first one you are recorded at 30 seconds and second vehicle recorded at 33 seconds. Let us run this animation to understand this time headway. Yes first vehicle is recorded as 30 seconds second vehicle is recorded at 33 seconds so deduct the this time headway is 33 minus 30 so 3 second is the time headway and how to calculate this time headway if all the headways are h in a time period t or which the flow has been measured. So I told you these are the all the headways hi is that measured headways in time so this is hi equal to t all the summation and number of vehicles are nt but the flow is defined as the number of vehicles per unit time. So say 15 minute interval or 10 minute interval usually 15 minute interval we take nt is the number of vehicles measured in time t and we know that the flow is number of vehicles per interval time t so put that hi hi equal to the t equal to hi and the question the q it becomes hi equal to h average so that h average the average headway. So in this case your nt will get cancelled over here and the final equation for a flow is q equal to 1 by h way so headway is inversely inverse of your flow in this case you can observe the average headway in terms of time is inverse of q suppose in example if you given 1200 vehicles per hour this is a flow is given you and ask you to calculate the what is the time headway. So here time headway is inverse of flow you can put the 1200 in your denominator to get the time headway that is 3 seconds so 3600 is to convert this hours into seconds. Another one space headway it is distance earlier it was a time now it was a distance again from the distance in the traffic stream from bumper to front of the bumper to the front of the rear vehicle so that is measured in time headway in meters again this is if all the vehicles headway in distance x so x is a distance over which the density has been measured so si equal to x so within the distance of x you have measured the how many headways in that particular length so this is usually the density we are going to measure say 1000 meter you have measured n number of headways that is si equal to x but the density k is the number of vehicles nx at a distance of x so we know that k equal to number of vehicles per unit length per unit length we take it as x so that is nothing but your total headway you are measured that is si equal to x so put the x equal to si in this case so si equal to s average that is your average headway distance in meter and it is inverse of the density so if I want to calculate the s average so 1 by k is nothing but your s average so in this case density is given 100 vehicles per kilometer ask you to calculate the space headway that case you can put the 100 as the density and hs is your headway in meters 1000 by 100 so 1000 is the total one kilometer length you are spaced at 100 meter so headway is 10 meter another the diagram with the this distance and time we can calculate the time space diagram we can draw the time space diagram so this inclined lines are a number of vehicles are spaced and time with a time and a distance so with that we can calculate we can this slope of this line gives you the speed so with the time space diagram we can calculate flow we can calculate density with the flow we can here q it is calculated between t1 and t2 so three vehicles are there between a time interval of t1 and t2 we can calculate what is the flow and in density between x1 and x2 length we can calculate is the four vehicles are spaced in the density 1 2 3 4 and divided by the x2 minus x1 gives the density k these are the references I have used for presenting this presentation thank you