 So, unfortunately none of the models will explicitly come and say, hey I am a delay model or something. It will be general description and you just model it, it just so happens that it will be some form of delay or the other which you can recognize. If it explicitly says delay good for you, if not still there will be delay existing especially this example. The chemical company makes chemical called no bug used in pesticides, they dump the byproducts in the say river once a week in batches such that the concentration is about 420 paths per million. So, we can assume that the river has 1 million gallons of water that means, 420 gallons of no bug are dumped per week into the river. During the course of the week, the chemicals are absorbed by the river's natural clean up process. The average absorption rate is 2 days, model the above system. Go ahead, how will you model this? The model is will be quite simple, but description will be quite elaborate. Since, I just covered first order material delay, logic suggests that it should be similar to that. Hopefully, you have got the model this one or at least you are thinking of a model like this not a drawn it. So, it is simple again when I teach I may use words like inflows and outflows and stuff, but you can use some proper rates like you know dump rate which it represents what is actually happening, absorption rate is what they have told, absorb a clean natural river's natural clean up process. So, it will clean up whatever is in the stock, it will clean up with an average absorption time. So, I will just write the equations in red. So, here the average absorption time is given as 2 days right. So, ensure your time unit in the model, in your model time unit is selected as days ensure that. So, absorption rate is nothing, but no bargain river divided by absorption time because it is a rate stock over time it has to be. So, no bargain river divided by absorption time is what you write here. Initial value you can put it as 0 in stock, its initial value is 0. So, now to kind of simulate let us take a very simple case. Let us start with a dump rate as just a pulse quantity we already seen pulse before. So, pulse has two inputs one is the start time other is the duration. So, at time 0 there is one pulse pulse one unit occurs at time 1, but what we already know it is it dumps 420 gallons right. So, I can so I just multiply this by 420. So, dump rate is 420 multiplied by pulse 0 1. So, you can make this in Vensim. So, you have all the information you need. The units of dump rate will be gallons per day, no bug units is gallons that is amount of no bug in the river gallons, absorption rate again gallons per day, day comes from here gallons comes from here, it weakly dumping is happening, but we will just take it as only one beginning of this week it happens. So, let us just observe that then we will model the next part ok. So, this is how the model looks, N80 natural absorption time I just came up with some acronym looks like it. If you look at the absorption rate is no bug by N80, N80 is given as 2 this is easy one dump rate is 420 into pulse of 0 comma 1 just a pulse input. So, if we run the model we can see that no bug increases and then as exponential decay. So, the halving time would be about 70 percent of your average delay is a 0.7 into D, D is 2 so 0.7 times 2 is about 1.4 1.5 about a week a day and a half time it will come into half the quantity ok right. Any questions on this so far? Now, what do you think will happen if we dump every week, it will go up it will come and again it will go up come down go up come down is that what we expect let us see what happens. So, dump rate what did I do there is another function called as pulse strain what pulse strain does is as a name suggests it gives a pulse input periodically. So, in this case it gives a pulse input starting at time 0 of a duration of 1 and it that repeats every 7 days that is once a week every 7 days it repeats and the end time is 50 after 50 time units pass no more dumping happens that is after ok. So, that is a very simple model that we have. So, it is called pulse strain you can look it up in the help file. So, every week 420 gallons is dumped it ok. So, let us observe what happens I am going to run the model. So, you go to no bug let me do causes strip it nicely occurs as we see here the top graph is the stock, stock increases up to 400 and the day of dumping 420 and then gradually comes down absorption rate also happens and so nicely fits our model right here. And dumping happens only once a week which is verified by the bottom graph it shows there is a dumping happening every week. Now is the actual fun part with SD where we have to do sensitivity analysis just making this is ok, but then to get insight we can find out say for example, what happens if the natural absorption time instead of 2 days become 4 days because of various reasons it has like capacity to observe has reduced and now the natural absorption time is 4 days. So, let me just make it 2 to 4 in that case what do we expect to happen can you have the same behavior? Behavior has to be same right once it dumps its truck observing correct, but then let us see what happens let us simulate no bug I just made the absorption time 4 from 2 days the dumping is the same absorption also happens, but now you see steadily the peak from 420 has moved closer to 500 right and 480 is a peak and it does not go now below say 90 there is always some no bug in the air in the river constantly. So, as absorption time increases say logically speaking now if the absorption time becomes average becomes say 7 days itself then by 7 days we can expect roughly little more than half of the no bug would have been absorbed. So, it will start accumulating. So, the accumulation effects is captured in this stock right here. So, if we did not have the stock we may assume that as soon as a dump eventually it will get absorbed, but it is not going to get absorbed until I stop the inflows again and then allow it for some time to slowly absorb and decay completely that is interesting what? Average absorption is 4 days or 12 days of course, it is going to keep increasing instead of we have tried with 4 days you can try with 12 days the next is interesting instead of dumping 420 every 7 days what if we do 60 every day we expect some behavior a different behavior. So, quantity is the same they got permission with government to dump 420 gallons a week what different does it make if I do it every day or do it once a week let us see what happens. So, I am going to go back and take the natural absorption time back to 2 in 2 case it came up and then went down very consistently nice graph I got. Now, I go to the equation of dump rate and we are doing 60 gallons a week every day time step is day. So, I just write 60 it means every day it will be 60. So, let me simulate see the absorption time is for whatever quantity whatever quantity I put takes an average of 2 days to deplete correct. So, now if I dump every day it does not give enough time for the river to absorb everything. So, eventually the total amount of say no bug in the river actually increases to what is it about 120 looks like right that is a interesting thing, but if you think about it logically what we did is when we dumped it once a week we gave it gave time for the river to recover. In this case we are dumping 60 every day. So, by the time we start processing 60 the next 60 is coming in every day it takes time the river cannot process so much, but still so there is a some amount of about 120 gallons that is end up in the river even though nothing else changed right there is a simple act of the same quantity, but different times actually has a nice kind of so it is better when there is a time given for river to recover rather than when it is very less though the quantity is the same. So, these kind of insights you can get only if you play with it simulated try with different values and this is the kind of policy which is just a simple model can use to explain different policies people want to you know as the same example do you allow them dump once a week or dump every day though the quantity is the same we can find that actually analytically there is difference in behavior and it does not kind of not a very complicated model to explain that. Second order delay you can imagine as cascading two first order delays that is it that is called a second order delay. So, the tank example looks very similar. So, the stock flow also we may able to guess based on this. So, what we are going to do is when you do a second order delay we will assume the total delay time is divided evenly among both the levels. So, if you want to model the second order delay we just take up inflow stock 1 let us call it exit 1 stock 2 this is your final outflow this is affected by d by 2 and this is affected by d by 2. If you want to model it when same directly as a function we will do it as inflow stock outflow t here we will write the equations for outflow as delay n as your in d initial value of any then we will write the order as 2 this is your order here the equations for exit is nothing but S 1 by d by 2. So, the equation for exit is S 1 divided by d by 2. So, exit is S 1 by d by 2 outflow is S 2 by d by 2. So, this is second order delay do a third order delay that is nothing but your inflow stock 1 let us call it exit 1 and your stock 2 and your stock 3 outflow. So, when I say first order delay there is one stock, second order delay two stocks, third order delay I am looking at three stocks and some material is conserved. So, all the stocks and flows are getting interconnected and now the delay time is evenly divided among three among three of it. So, the equation for exit 1 is S 1 by d by 3 exit 2 is S 2 by d by 3 outflow is S 3 by d by 3. So, evenly divided. So, the average delay time continues to be d here as soon as outflow happens d by 3 amount starts flowing out of exit 1 as soon as inflow happens and after d by 3 1 third of the time this stock will start to peak and then of two thirds of the time this stock will start to peak. So, this so that is the peak of the outflow when this one peaks is when this peak outflow will occur. To model it in Wensum again we can use the similar delay n function with a third order delay. So, this third order delay I just showed you delays divided evenly among three levels. Suppose delay d is divided distributed evenly among n stages then instantaneous outflow of stock I will affect the subsequent stock the delay is evenly divided by n stocks. So, for each level I have d by n so, we have fifth order or sixth order delay.