Imagine that you are visiting Palm Springs, California in the U.S. for a trip. Here, you can watch hundreds of wind turbines all rotating their blades together as the sun sets. You enjoy the fascinating view and later tell your friends about your unusual, fantastic experience. But if you work at a large wind farm, you are more likely to be watching the sensor signals on the monitoring screen “24/7,” worrying about how to operate and maintain so many turbines. You also know that these days, utilities only want to build very large wind facilities due to the economies of scale. You wonder if it’s possible to operate these large systems in a nearly optimal way. Can anyone analyze large systems with any degree of accuracy? The answer is yes, but it requires imposing some important restrictions and assumptions.
There is a well-known method that works better, i.e., more accurately, as the size of the system is getting larger. It looks strange but there is. You may have heard of the “Law of Large Numbers (LLN)” and the “Central Limit Theorem (CLT)” in your statistics course. Roughly speaking, they mean that the underlying uncertainty of the sample average will disappear as the size of the sample becomes infinite, and that the average of a large sample can be approximated by the normal distribution. The use of so-called “asymptotic methods,” a collection of some different versions of LLN and CLT, will make an analysis more accurate when the size of the system is really large; if the size is infinite, the analysis would be exact!
Returning to the example of a large wind system, we know that individually analyzing all of its turbines and creating an integrated solution is extremely difficult. The use of asymptotic methods, however, will make the problem tractable. Equipped with this tool, you will be able to predict the behavior of large-size wind power systems quite accurately.
But are asymptotic methods easy to learn? If you are familiar with mathematics, no problem! If not, you need to think of a new method that requires a less mathematical background. Let’s think about it. The fundamental difficulty in dealing with a large-scale system arises from integrating many elements into a single model. What the asymptotic analysis does is to simplify the integration procedure using continuous models (they are also known as fluid and diffusion limits). Unlike asymptotic analysis, the basic idea of the new approach is to give the elements the authority to control themselves with predefined policies based on local information. In other words, the size of a system will be less important, since the elements will control themselves autonomously.
Here’s an example. Consider a very large data center consisting of thousands of heterogeneous servers running hundreds of different applications. Your manager wants you to figure out how to conserve energy. Thinking about the problem, you start by constructing and solving a huge centralized mathematical model that will produce each server’s clock speed and routing information according to fluctuating demand over time. Quickly, you realize that solving this big problem within a time limit (possibly almost real time) is not feasible, and even if you could solve the problem, it is impossible to change the clock speed of thousands of servers every few (milli) seconds. Recalling what you learned in class and conducting in-depth research on it, you try to develop an algorithm that guarantees the convergence to the globally optimal solution while allowing each server to decide its clock speed autonomously based on only local information such as current workload, clock speed, etc. Hopefully, if you successfully develop such an algorithm, your manager will give you a raise! (In fact, the development of such an algorithm is under way in my laboratory and will be completed sooner or later.)
There are numerous ways to understand and optimize large-scale systems, and I briefly introduced just a few of them here. Because analyzing large-scale systems is a big challenge, solving them gives you and other people great satisfaction. Are you interested? Please contact me via the IME department Web site.