Not many equations in this course, and so let's greet every one of them that we encounter with great joy. Time for the first equation in this course. This equation is known as Little's Law. Little's Law states that, in any process, the average inventory is equal to the average flow rate times the average flow time. Now, unlike most other things are the equations in mathematics that I know that were proven by some ancient Greeks, Little's Law is actually a fairly recent mathematical discovery. We'll see how to apply Little's Law, and that it's quite a powerful tool for you as you're going to analyze processes with respect to inventory, flow rate, and flow time. To see the intuition behind Little's Law, let's take another look at some of the analysis that we have done in an earlier session on the Subway restaurant. Remember, how a couple of sessions ago, we were sitting in front of the Subway restaurant keeping track of the inflow of customers, as well as of the outflow. We refer to the vertical difference between these graphs as the inventory, the number of customers in the system, as well as the horizontal difference as the flow time. How long a specific customer stayed in the restaurant. If I smoothen these graphs here a little bit, the slope of this line corresponds to the flow rate. Namely, the rate of which customers come in and go out of the restaurant. This is not a formal proof of Little's Law, but simply the intuition behind it. We see that on average in a process, the inventory, which is expressed in our case here as customers, is the flow rate, which is expressed in customers per hours or customers per minute times the flow time, which is simply hours. And so, the hours here cancel out, and you have on both sides the customer as a unit. Again, this is not a formal proof, but the basic intuition behind Little's Law. What are the implications of Little's Law? First of all, Little's Law tells us that from the three fundamental performance measures that we care about in this course, inventory, flow rate, and flow time, two of them you might be able to mess around with. But then, the third one is written in law by nature. So, for example, if you hold the flow rate fixed for the moment, that's your revenue, that's the numbers of customers you serve. Let's hold this fixed for the moment. Little's Law tells us that whatever you're doing to inventory, you're doing to flow time. The second thing that we do with Little's Law is often times we find that we might know two of the performance measures in a process, but it's hard to observe the third one. Little's Law can help us with two given performance measures to compute a third one. Now, typically in a process, flow rate and inventory are relatively easy to observe. Flow time in contrast, is not. Let me give you an example. Think about, how long will it take you on average to respond to your eemail? This is really not a number that most of us routinely track. However, you can compute this number quite easily. If you have 240 e-mails in your inbox, that is in inventory. Then, this number is equals to the flow rate. So say, you're writing 60 e-mails per day, times the flow time that you really don't know. Well, it tells us that your average flow time is four days. This is the average, some e-mails you might respond faster, some you might take longer, but on average, it takes you four days to respond to an e-mail. Here's another interesting example of Little's Law. Imagine you're working for a large hospital, And there are ten babies born per day in the hospital. Now, 80 percent of these deliveries are easy, and they require mother and baby to stay in the hospital only for two days. Twenty percent of the cases are more complicated and require a five day stay. Now, what is the average occupancy in this department here? Well, it turns out that you can solve this matter by Little's Law. First of all, observe that the flow rate here is simply ten babies per day. The flow time is the average time a baby spends in the department. With an 80 percent probability, it's going to be two days. And with a twenty percent probability, it's going to be five days. As this 1.6 plus one equals to 2.6 days. To find the average occupancy, we compute Little's, by Little's Law the inventory in the process, that's the number of babies in the hospital, is the flow rate times the flow time. Ten babies a day times 2.6 days is equals to 26 babies. Again, notice that the occupancy might vary And you might be ill advised to build the department with only 26 beds. This is the average occupancy, each day might vary and some days will be lower, but other days will be higher. In this session, we have seen our first equations in operations management, Little's Law. On average in the process, inventory equals flow rate times flow time. Little's Law is not an empirical law. To prove Little's Law, we have to turn to stochastic optimization, and do some heavy lifting math. The strength and the weakness of Little's Law is that it deals with averages. Averages are very powerful at the aggregate level perspective on an operation. But, at the micro-level, they can be misleading. I, as a patient, care about my wait time, not about the average wait time, i.e., the wait time of everybody else. We will use Little's Law going forward, typically, to compute inventory of flow time. In particular, we'll see an interesting application of Little's Law in the next session.
