## More on Response Time

Now, let's look at what happens when we have a four-reservoir system. Do our conclusions based on the response times of smaller systems hold when we have much larger, more complex systems? Figure 2.11 shows the results of some simple experiments with a four-reservoir system.

Figure 2.11. Two versions of larger systems show that the response of connected systems is not as easily predicted as in the case of smaller, two-reservoir systems. In larger systems, some reservoirs may overshoot their eventual steady state levels, thus delaying their eventaul arrival at the steady state. Nevertheless, it is still true that adding increasing the connectedness of a large system decreases the overall response time.

The first system shown in Figure 2.11 consists of four reservoirs linked together by draining flows; this is essentially just a larger version of the two-reservoir system shown in Figure 2.10. We can imagine this system in its disconneted state and consider the response times of the indicidual reservoirs with their draining flows; these response times, given in Figure 2.11, range from 1.67 to 5 years. If we were to simply extend our analysis of the two-reservoir system, we might hypothesize that for this four-reservoir model, the response time of the whole system would be given by:

.

When we test this hypothesis by running the model, we find that the real behavior is more complex and the overall response time of the system is difficult to define, but if you focus on reservoir M2, which appears to be the slowest to respond, you can estimate that the response tme of the whole system is in the range of 4 to 5 time units. Obviously, our simple mathematical prediction of the response time was wrong; when faced with a more complex system, it is better to define the response time of the whole system by direct observation, as we have done here.

By studying the graph that shows the evolution of this system, we can see that the reservoirs tend to overshoot their eventual steady-state values and this delays their arrival at the steady state. Note that in this case, the response time of the whole system is significantly greater than the response times of some of the reservoirs considered in the disconnected state. Recall from our examination of the simple two-reservoir system that this was not the case there - the two reservoir system had a shorter response time than either of the reservoirs considered in the disconnected state. So, a very important conclusion here is that in larger systems, the overall response time of the system may be significantly greater than the response times of the separate parts of the system.

Figure 2.11 also shows the effect of adding more flows to this four-reservoir model - this amounts to increasing the connectedness of the system. Our simple analysis would lead us to believe that making this change will decrease the response time of the system, and in fact when we run this model, we find that such is the case. This indicates that our conclusion that increased connectedness, which amounts to increased complexity, decreases the response time appears to be valid even for larger systems.