8/13/2019 Principle Little's Law

1/3

Principle (Little's Law): the fundamental long-term relationship between Work-In-Process,

throughput and flow time of a production system in steady state is:

Inventory =Throughput !low Time

Motivation

Little's law is both fundamental and simple. Because it relates three critical performance

measures of any production system, it is a basic manufacturing principle. But, although it hasdeep mathematical roots, it is extremely intuitive. If we observe a milling machine that cuts

100 parts per hour with a ueue of !00 parts in front if it, we say that it has "! hours of #I$".

In spea%ing of #I$ in terms of time, we are ma%ing use of Little's law, which can be thought

of as a conversion of units. Inventory is measured in pieces, flow time in hours, andthroughput in pieces per hour. &ence, if we divide inventory by throughput we get flow

time. o, to convert !00 pieces to the time it will ta%e to process them, we divide by the

throughput (100 pieces per hour) to get ! hours. *his conversion is useful when diagnosing aplant. If we see what physically loo%s li%e a large amount of inventory in the system, we

cannot tell whether this is a signal of trouble until we %now how much time is represented by

the inventory. +or instance, if we see 000 pieces of #I$ in a system that produces 10 perday, this is probably a disastrous system, while 000 pieces in a system that produces 1000

per hour is probably extremely lean. (-ote however that the efficiency of a system depends

on more than how much #I$ is present. *o see how, go to diagnostictools.)

But Little's law is actually much deeper and more general than a simple conversion of units.It applies to single stations, production lines, factories, and entire supply chains. It applies to

systems with and without variability. It applies to single and multiple product systems. It

even applies to nonproduction systems where inventory represents people, financial orders,or other entities. /enerally spea%ing, there are only two reuirements for Little's law to hold

1. inventory, throughput and flow time must represent longterm averages of a stable

system, and

. inventory, throughput and flow time must be measured in consistent units.

*o appreciate the first reuirement, suppose we have a production system that has ust startedup, so that it has inventory but no throughput yet, then inventory will be positive, throughput

will be 2ero and flow time will be undefined. 3learly Little's law will not hold. But over

time, as the system stabili2es and we get averages for the three uantities, it will hold.

*he reuirement of consistent units is simple for the single product case. +or instance, ifthroughput is measured in pieces per hour, then flow time must be in hours (not wee%s or

months). But it is more subtle if we want to apply Little's law to a multiproduct system. +or

instance, we might have a wor%station that processes several part types, each with differentinventory, throughput and flow time. #hile it is perfectly consistent to apply Little's law to

the parts one at a time, we can also spea% of them in aggregate terms if we use the proper

units. +or instance, we could measure all inventories in units of dollars and spea% of

http://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Diagnostics.htmhttp://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Diagnostics.htm

8/13/2019 Principle Little's Law

2/3

throughput in terms of dollars per hour. &ere, however, we must be careful to maintain

consistency by measuring throughput in terms of cost of goodssold (instead of in terms of

revenue), so that the dollars used match those used to measure inventory. 4easuringinventories and flows in dollars ma%es Little's law applicable to a vast number of situations.

Examples

1. Estimating Waiting Times:If are in a grocery ueue behind 10 persons and estimate

that the cler% is ta%ing around 5 minutes6per customer, we can calculate that it willta%e us 50 minutes (10 persons x 5 minutes6person) to start service. *his is essentially

Little's law. #e ta%e the number of persons in the ueue (10) as the "inventory". *he

inverse of the average time per customer (165 customers6minute) provides us the rateof service or the throughput throughput. +inally, we obtain the waiting time as eual

to number of persons in the ueue divided by the processing rate (106(165) 7 50

minutes).

. Planned Inventory Time:uppose a product is scheduled so that we expect it to wait

for days in finished goods inventory before shipping to the customer. *his two days

is calledplanned inventory timeand is sometimes used as protection against systemvariability to ensure high delivery service. 8sing Little's law the total amount of

inventory in finished goods can be computed as

+/I 7 throughput 9 planned inventory time

:. Tracking Flow Time: #hat is the flow time of an automobile; *he answer, ofcourse, is "it depends".

8/13/2019 Principle Little's Law

3/3

reduce cycle time. &owever, we must be careful. @educing #I$ in a line without

ma%ing any other changes will also reduce throughput (see the discussion of 3losed

=synchrounous Linesfor an explanation of the throughput vs. #I$ relationship). +orthis reason, simply reducing inventory is not enough to achieve a lean manufacturing

system. =n integral part of any lean manufacturing implementation is a variability

reductioneffort, to enable a line to achieve the same (or greater) throughput with less#I$.

http://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Principle.htmhttp://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Principle.htmhttp://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Principle.htmhttp://www.factoryphysics.com/principle/Station/Asynch/Closed/InfinClosed/IBCL_Principle.htm