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 any process the average inventory is equal to the average flow rate times the average flow time. Now unlike most other types of things, other 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 is quite a powerful tool for you as you are going to analyze processes with respect to inventory, flow rate, and flow time. To see the intuition rate 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 the horizontal difference as a 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 with the flow rate, namely the rate at which customers come in and go out of the restaurant. This is not a formal proof of Little's law but the iteration behind it. We see that on average a process is the inventory, which is expressed in our case here as customers, is the flow rate, which is expressed in customers per hour or customers per minutes, times, so for time, which is simply hours. So hours here cancel out and you have on both sides the customer as a unit. Again, this is not a formal proof, but 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 we might be able to mess around with, and then the third one is written in law by nature. So for example if you hold the flow rate fixed for the moment. Say it's your revenue that's the number of customers you served let's hold this fixed for the moment then this will tell us 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 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 the third one. Now typically in the 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 email? This is really not a number that most of us routinely track. However, you can compute this number quite easily. If you have 240 emails in your inbox, that is an inventory, then this number is equal to the flow rates. So say you're writing 60 emails per day times the flow time that you really don't know. It tells us that your average flow time is 4 days. This is the average. Some emails you might respond faster, some you might take longer. But, on average, it will take you about 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. There are 10 babies born per day in the hospital. Now, 80% of these deliveries are easy, and they require mother and baby to stay in the hospital only for two days. 20% 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% probability it's going to be two days. [NOISE] And with a 20% probability, it's going to be five days. ] This is 1.6 plus one equals 2.6 days. To find the average occupancy, we compute littles 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. 10 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 while other days will be higher. In this session, we've seen our first equations in operations management, Little's Law. On average in a 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 or flow time. In particular, we will see an interesting application of Little's Law in the next session.