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In this session we're going to look at Little's Law.

First we will derive Little's Law based on a very intuitive

example in the next couple of slides.

And we'll see how this basic intuition that we have is

basically what shows up in Little's Law.

Little's Law is about the interrelationships between three

basic elements of process flow analysis.

Flow time, flow rate, and inventory.

So, we'll see the concept of it first, and then we will see some applications of it.

So we'll apply Little's Law in some practical applications.

So you get some sense of how it can be used and how this

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idea can be used to come up with information that's missing.

Or to look at processes and say, how should they be performing,

based on these metrics that we do have?

All right, so let's take this simple example of two activities.

So it's a very simple process, two activities.

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First activity A has a cycle time of 10 minutes,

activity B has a cycle time of eight minutes.

So, flow rate for each of these activities, cycle time of 10 minutes so

flow rate is going to be the reciprocal so it's going to be one tenth of A.

Unit per minute for A and for activity B, cycle time of 8 minutes.

So the flow rate is gonna be one eighth of a unit per minute of B.

So we have one tenth of a unit per minute of A and

one eighth of a unit per minute of B.

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Flow rate for the process, as you probably remember, the flow rate for

the process is determined by the activity that has the largest cycle time.

So the larger of the two is ten minutes,

that will determine the flow rate of the whole process, so

the flow rate process is going to come from the flow rate of A.

Or it's going to be 0.1 unit per minute, one-tenth of a unit per minute.

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So this is the part that's really simple.

Now what you can also think about is how much inventory

there is going to be in this system at any point in time.

So notice that we're not looking at Little's law yet.

We're not looking at any kind of a formula yet.

We're not looking at any kind of a theory yet.

All I'm asking you to think about is how much inventory there would

be at any point in time in this system.

So it's an imbalanced system, Activity A takes more time so

Activity A is obviously going to be busy all the time.

Right? There's going to be one unit of inventory

in activity A all of the time, 100% of the time.

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But for every ten minutes of time,

activity B is going to be idle for two minutes.

It's going to starve for material to come from A.

For two minutes in every ten minutes so you start the process at 9:10.

Activity A starts the process at 9 AM, activity A passes something

to activity B at 9:10, activity B finishes that at 9:18 and

has to wait until 9:20 to get the second units.

For every ten minutes, activity B is going to be waiting for two minutes.

So with the continuous process working in every 10 minute period,

how busy will A and B be?

So, what you've seen is,

in this system there are two units in the system 80% of the time.

80% of the time both A and B are busy.

And 20% of the time, A's busy while B is starving for product to come to it.

So B is sitting idle.

So if you were to calculate the average inventory, you can calculate

it based on this idea of there being two units in the system 80% of the time.

20% of the time there being one unit in the system, and that gives you 1.8 units.

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The same thing, you could have got if you were to multiply the throughput

time of this process, the flow time of the process with the throughput rate.

So what's the flow time of this process?

If you recall it's 18 minutes.

What's the flow rate of the process?

You're producing 0.1 of a unit per minute of time.

So take the product of those two and you get the same answer that you got by

looking at average inventory from that perspective,

from the long perspective that we took to get there.

And it works out to be the same number, 1.8 units.

So this little formula, and

no pun intended, is what is known as Little's Law.

It's known as Little's Law because John Little Came up with a proof for this law.

Although this was a concept that was known even earlier,

he was the first one to provide a mathematical proof for it.

And that's why it's known as Little's Law.

So Little's Law is basically,

inventory in the system is a product of the flow time times the flow rate.

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So more formally stated, the average number of items in a system is

the product of the time that the item spends in the system and

the average flow rate of the system.

I = T * R.

Now, once you know I = T * R,

you can also express it as I = T divided by the cycle time.

You may recall that flow rate and cycle time are the reciprocal of each other.

So, the inventory can also be calculated as flow time divided by cycle time.

One note of caution here that you should pay attention to.

It's a very simple formula, a very simple concept.

But where a lot of us get stumbled is, we take the units, we

may take something in units per hour, and then, we may take the flow time in days.

Or take the throughput rate in units per minute, and take the flow time in hours.

And if you're doing that, you're going to mess up the calculations, so

you have to be careful.

When you're using Little's Law to use consistent units

in all of the things you're using to make the calculations.

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So the first assumption is that only averages count, that there's no variance,

there's no standard deviation that we are incorporating into the calculations here.

That we're basing everything on averages.

There's no uncertainty, everything acts as predicted.

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The other assumptions are that the arrival rate equals departure rate,

and what you should be thinking about there is that the number of units that

enter the system are the units that exit the system.

Every unit that enters, exits the system, there's no yield loss.

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So if you're thinking about a line for

security at an airport, there's an arrival rate and

there's a departure rate, and they're going to be equal.

People entering the line, joining the line,

is going to equal the rate at which they leave the line to go to their aircraft.

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Over a period of time that you're observing the system,

the average inventory in the system stays constant, and

the average age of the inventory in the system is constant.

What that translates into in lay person's terms is that

you're going to have a first in first out system.

So, people who join the line first, leave the line first and that's how it works.

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And the last thing that you have here is that Little's Law

applies when the system is in steady state which means that if

you're starting the system without any units within it,

the law is that I equals T times R will not exactly work out.

You have to wait for

the system to get filled for the system to be in steady state.

When you look at the security line first thing in morning at a airport and

you see that there's nobody there and then a lot of people start accumulating,

you want to wait for that process to be in steady state in how it's

usually performing before you take these measurements, and

that's when the I = T * R will apply.

So, we know I = T * R.

What does that mean?

How does that apply?

How is it practically useful?

And the way it is practically useful is that let's say that you can only calculate

two of these aspects.

The third one becomes easily determined.

So if you know the flow rate and the flow time, if you know at what rate

you are selling stuff to customers, that becomes your throughput rate,

that becomes your flow rate, you know how much time it spends in the system.

That becomes your flow time.

You know on average how much inventory there is in the system,

and that's kind of a useful tool when you are not able to measure three things.

So just to give you a few concrete examples of this,

let's say that you care about the average response time for an order.

But you have information on average number of orders in the system.

You have information on average rate at which orders are being delivered.

You can calculate the average response time.

You can calculate T based on knowing the average number of orders, I,

the inventory, and the average rate at which the orders are being delivered or

R, throughput rate.

Similarly, if you're talking about

average number of people that are waiting in a security line.

And you know the rate at which passengers come and leave to get to their aircraft,

and you know the average time that customers or passengers spend in the line.

You can get the average number of people waiting in line.

So, if you're planning for, say, how much space you will need for passengers to

wait at the security area, you can get to that based on these other two metrics.

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You can be thinking about the bottles of wine on your wine rack.

And you can calculate the average age of wine in terms of how long you've had it,

based on your consumption rate and

the average number of bottles that are on that wine rack.

So knowing two of these makes the third one determined,

you can calculate the third one.

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Now, let's do an exercise here to end this session.

And what you're going to do in this exercise is apply Little's Law,

and it's gonna seem very mechanical.

You're going to apply the formula to many different aspects,

many different types of processes.

And hopefully, what you'll get from this exercise is,

you'll be able to see the various applications,

the various areas in which this kind of a calculation can be useful.

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So we have this table of information in front of us.

And you can see that the three columns,

the first one represents the rate, what we've been calling R so far.

The second one represents I, and the third one represents T.

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And your R is going to be determined by I / T.

So we can do these calculations based on the information about the other two

that's given to us.

So let's just do a couple before we get to the solution.

For the first one, for the semiconductor factory,

the whip is 45,000 wafers, the rate is 1,000.

45,000 divided by 1,000 will give us 45 days time and inventory.

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And for the second one, similarly, it's gonna be 150 divided by 50 so

that gives us 3 days time in inventory.

For the third one,

it's a maternity ward where you're trying to determine how many rooms you need.

And that's where you need your I, so it's gonna be T * R.

You have the throughput rate given to you, and you have the throughput time.

90% of moms stay 2 days, and 10% stay 7 days, so

you can get the average throughput time based on that.

And the inventory, or the number of rooms that you will need,

is going to be based on an average of 12.5 mothers.

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Right, and

you can see here that you can work through these solutions in your own time.

But what you can see is there are many different applications right from

a semiconductor factory that is making wafers to a maternity ward that's planning

for how much room they need to have for the number of patients that they expect.

To a toll booth looking at how much time is spent waiting

in queue at the tollbooth, to real estate,

which is looking at the rate at which houses are being sold in that market.

And finally, a donut shop

that is looking at the rate at which customers come and leave the the shop.

So hopefully you've gained an appreciation for how this idea of

Little's Law can be used to analyze processes based on three basic metrics.