Response Time


Notice that in both cases shown in Figure 2.6, the system returns to the steady state in the same amount of time. Now, try to predict what will happen if we get a bit more extreme and put 100 liters in the bathtub to begin with. Do you think the system will recover and return to its steady state in the same amount of time? My initial impulse is to say that it will take longer, but I know that dynamic systems can be difficult to predict, so I am not confident in this intuitive prediction. Think about this carefully, and then do an experiment where you alter the initial conditions so that the tub starts out with 100 liters and run the model. If you compare the results with those shown in Figure 2.6 above, you will see that in fact, the system does return to the steady state in the same amount of time - quite a surprise. Let's investigate this a bit further and at the same time learn about a powerful feature of the program - sensitivity analysis.

Select Sensi Specs from the Run menu; this brings up a window that allows to set a sensitivity analysis, where you systematically change one parameter in a uniform way and monitor the system's response. Click on water tub in the Allowable column and send it over to the Selected column by pressing the >> button. Then click on water tub in the selected column to highlight it; enter 5 in the box titled # of Runs, make sure that the Incremental button is highlighted, then set the beginning and ending values of water tub at 0 and 400. When you've done this, a list of the initial values for the water tub reservoir will appear in the column on the right. Click the box next to Sensitivity On, you are ready to go. Consult Figure 2.7 to make sure you've entered the right values in the right places in the window before proceeding. If you have troubles here, consult the STELLA manual.

Figure 2.7. The sensitivity window in STELLA, used to systematically change on part of the system. You may either click on the Define Graph button in this window, or you can alter a pre-existing graph so that it plots all five sensitivity runs by making sure that the Comparative button is highlighted in the Define Graph window.

Make sure that your graph is set up to plot all of the sensitivity runs; the comparative button must be checked in the Define Graph window. Then, as you select S-Run from the Run menu, the program will cycle through five simulations, each with a different starting value for the reservoir. Your ending graph should look like Figure 2.8 below.

Figure 2.8. With a variety of initial values for the reservoir, the system returns to the steady state in which the reservoir has a value of 10 liters in the same amount of time. The response time is independent of the initial value of the reservoir; instead, it is defined as 1/k where k in our case is 0.1 seconds-1. Thus the response time here is 10 seconds, shown by the vertical dashed line.

Sure enough, we see that the time it takes to return to steady state is completely independent of the amount of water initially in the water tub. Notice that the approach to the steady state value follows what is called an exponential curve -its slope is constantly changing and as it approaches its ending value, the curve changes very slowly. This makes it a little difficult to say just when the system has returned to its steady state. For this reason, it is convenient to define the response time a bit differently. If we say that the outflow process, the drain, is defined as a rate constant, k, times the amount in the reservoir, W, then the response time is defined as:

In our case, k is 0.1 and the units are /second, so the response time is 10 seconds. Another way of saying this is that the response time is the time at which the initial imbalance in the system, the difference between the initial value and the steady state value for the reservoir, has been reduced by 63%. With this relationship defined, we can accurately predict the effect of changing the parameter k from 0.1 to 0.5; it will reduce the response time to 2 seconds instead of 10. This concept of a response time is an important one - it will help you develop a better intuitive sense for how a system will behave.