Sept 2011 - 50th Anniversary of Little's Law
Created by ppfhost on 07/09/2011 10:11:57

‘Little’s Law’ is a restatement of Erlang’s work, now well known in contact centre worlds.  It states that that the Lead Time (LT) equals Work in Progress (WIP) divided by production rate – in non-service industries.  While there are some important limitations, Little’s Law is entirely independent of any of the detailed probability distributions involved.  In the proof published by John Little in 1961, the result applies to any system, and particularly, it applies to systems within systems.

John Dutton Conant Little was born in 1928, is considered a founder of marketing sicience and is an Institute Professor at MIT, teaching in the Sloan School.  In 1961, then at Case Western Reserve University, he published his first proof about the fundamental long-term relationship between work in progress, throughput and flow time of production systems in a steady state.  His proof which became known as Little’s Law is acknowledged as a fundamental principle in business and mathematics, and the law has applications to many real-world problems, particularly those problems concerning product development.

The law states that the Lead Time (LT) equals Work in Progress (WIP) divided by production rate – in non-service industries; lead time is the time it takes to fully complete a product. For example if you can produce 20 products per hour, the lead time would be 3 min per product on average. Thus if the lead time, for some reason or another, spiralled out of control, you would have the choice of either decreasing your WIP (slow the assembly line for example) or increasing your production rate (add more workers or improve
existing workers productivity).

The beautiful simplicity of Little’s Law is how Little managed to prove that under very general conditions, the average length of a queue, in steady state, will be equal to the arrival rate into the queue multiplied the average wait in the queue. Remarkably, this relationship is not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else. Nor does it depend on the structure of the queuing system: “Little’s Law” holds not just at the individual queue level but also at the
system level. 

Little’s Law is a restatement of the Erlang formula, based on the work of Danish mathematician Agner Krarup Erlang (1878 – 1929). Offered traffic E (in erlangs) is related to the call arrival rate, and the average call-holding time, h, by: E = λh. This was further extended by Palm’s Theorem circa 1934 who showed that if the inter-arrival times are exponentially distributed (i.e. followed a Poisson process) with mean m and the mean service time is then the number in the system at any given time follows a Poisson
distribution with mean mt (provided both are in the same units of time). Note that the result is independent of service time distribution.

Although it looks intuitively reasonable, the result it implies is that this behaviour is entirely independent of any of the detailed probability distributions involved, and hence requires no assumptions about the schedule according to which customers/patients/products arrive or are serviced. It does rely on arrivals staying in the system, servers being continuously available and the route through the system being independent of the time or number in the system.

In the proof published by John Little in 1961, the result applies to any system, and particularly, it applies to systems within systems. So in a bank for example, the customer line might be one subsystem, and each of the tellers another subsystem, and Little’s result could be applied to each one, as well as the whole system. The only requirements are that the system is stable and non-pre-emptive; this rules out transition states such as initial start-up or shutdown.

News Story from Inside OR - published by the OR Society Aug 2011

A Proof for the Queueing Formula: L=λW Operations Research: A Journal of the Institute for Operations Research and the Management Sciences, 9:383-387 (1961).


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