# Little’s Law – Application in Lean

 John Little – MIT Professor

I was recently teaching some value stream mapping concepts to some lean leaders and we were talking about the relationship between WIP and Lead Time. I shared with them the important concept of Little’s Law in relation to lean manufacturing. Little’s Law is most commonly used by service organizations to explore wait and response times. To paraphrase the Little’s Law Wikipedia page a bit, Little’s Law is that “Average # of customers in a stable system L is equal to average arrival rate λ multiplied by the average time a customer spends in the system W”. Relating this to a mathematical formula yields L = λW.

Well how does this relate to manufacturing and value streams? If we break down the statements in the definition and relate them to lean terms, then this relationship can be easily understood.
“Average # of customers in a stable system L”: In a service organization, Average # of customers in a stable system is essentially system WIP. In a value stream map that would be the parts in the system. A stable system in service would describe any time there isn’t a built in ramp up and ramp down like opening time and closing time perhaps. A stable system in value stream mapping would be similar – any time you are in normal production and not new production introduction. In our new formula we’ll define L = WIP
“Average arrival rate λ”: How often a customer walks into a service process is your arrival rate – well what would that be in lean? How often should parts arrive into a value stream – theoretically at the customer demand rate. The customer demand rate we will call throughput which will be a measure of units over time (e.g 2 parts per day, 1 part per week, etc). For our new formula, λ = T (throughput).
“Average time a customer spends in the system W”: For a service organization this is exactly what it sounds like – basically how long he/she waits for a response. If its McDonalds it’s the time from when the customer walks in to when they order food and leave the queue. Relating this back to manufacturing, if our customers are our WIP, then we can think about how long the WIP is in our system. This is of course leadtime! So for our new formula W = LT (leadtime)
So after breaking down Little’s Law we now have a modified formula for manufacturing and value streams: WIP = T x LT. So the inventory in your value stream is a function of throughput times the leadtime. This is a very powerful relationship to understand and can prove useful in many situations. My boss recently told me that my inventory needed to be no more than \$1.2m on a critical value stream. By knowing the average value of a part in the value stream, I could tell that this represented approximately 30 pieces of WIP. I knew based on required customer demand that throughput was 2 per day. So:
Target WIP = 30 pieces
Throughput = 2 per day
30 = 2 * LT
LT = 15 days
So based on this, I know my target leadtime is now 15 days. I know based on system actuals that my current leadtime is 18.5 days. Based on this knowledge I know how much of a leadtime improvement I need to make in order to achieve my bosse’s targets. Little’s Law is also useful in doing sanity checks while doing value stream mapping.
Any thoughts about Little’s Law and experience applying it to manufacturing situation?
Edit: I recommend the Lean Fundamentals quiz where I’ve added some Lean Math questions based on Little’s Law

## One comment

1. shindeajay says:

Nice article.
Simpler explanation with practical implications

Regards
Ajay