**Little’s law, queuing theory, links the waiting time, the in-progress and the flow of the system.**

## Introduction

The Little Law (1961), “*Queuing theory*“, links the waiting time, the in-progress and the flow of the system. The origin of this law is in the management of the telephone networks in Copenhagen, by the work of the Danish engineer A. K. Erlang, in the years 1900-1920. It studies the arrival systems in a queue and the priorities of each new entrant. It observes the “*Poissonnien”* character of the call arrivals at a telephone exchange and the exponential character of the call times: in other words, *at the most there are incoming calls, at most the length of the call is long*. It then deduces the likelihood of having a rejected call, and proposes a theory of balance between the inputs and exits.

From the years 1930, thanks to mathematicians known as Markov or Kolmogorov, the theory developed via operational applications: traffic flows of vehicles, people, scheduling of production, management of patients in emergencies, Sizing Bank counters…

But it will be an American mathematician, John Little, who identified and popularized the formula in 1954^{1}. Having no “* proof *” of his law, mathematician P. M. Morse asked anyone to prove otherwise.^{} In 1961, John Little republished^{3} his formula by proving it by the fact that one cannot contradict it…

**The Erlang model**

The original Erlang model is still very much used in telecommunications to size systems. His formula is as follows (published in 1917):

With:

**B:**the size of the system**m:**the number of circuits**α:**The average number of calls in the absence of blockage, which is equal to λ/μ**λ:**Call density at the entrance**μ:**Average call duration

## The principle

Little’s law applies to any kind of process, regardless of its variability. It says:

**WIP = T * LT**

With:

**WIP :**Work In Progress**T:**The throughput per unit of output time of our system**LT:**Lead time (average Cycle time spent in the system) that corresponds to the wait time plus processing time.

It should be noted that to make the corollary with the **accounting** of the Theory of constraints :

**WIP = Inventory****T = Throughput**

It should also be noted that eliminating **transports Muda**, Lean implicitly uses this law.

This expression is very interesting. It shows us that if we want to reduce the Lead Time, we have two choices:

- Reduce the WIP.
- Increase the flow.

As a result of the fact that a system will inevitably have a bottleneck, we’ll reduce the Lead Time, then choose to calculate the maximum outstanding. Set up a **Kanban** allows you to manage this in-progress.

## The 3 learning the law of Little

1. Little’s law puts forward that the time of passing of tasks is proportional to the stock of in-progress. They therefore oppose the habits of “*putting pressure*” to *push* the work by saying that the more there is, the more it comes out.

2. It also highlights the fact that to speed up the processes the only process optimization track is not sufficient. We also need to think about better managing what goes into the process.

3. Finally, it has the challenge of demonstrating that the mastery of the courses is a very important element if we want to optimize its processes. In-progress:

- Take up space.
- Generate non-quality.
- Make planning more complex.
- Represent money that sleeps.
- Provide variability in cycle times.

**In short, Little’s law teaches us that we will not go any faster by pushing an entry into a process.**

## Example

We’re a manager. The staff complained about the response time we have to answer the emails. For the moment, we have about 200 unread mails in our box, and we know how to process an average of 25 mails a day.

Our average Lead time is therefore 200/25 or 8 days of average time spent by an email in the box.

We would like to reduce this time by at least 3, or go from 8 days to just over 2 and a half days. Knowing that we can not process more than 25 e-mails per day, Little’s law allows us to calculate the maximum outstanding, either in our case:

**WIP max = 2.66 * 25 = 65 to 70 mails.**

In other words, we must make sure that we have at most only 65 to 70 unread mails in our box and therefore set up standards to manage email submissions.

**A little humor on the theory of queues**

## Source

1 – A. Cobham (1954) – Priority assignment in waiting line problem

2 – M. P. Morse (1958) – Tails, inventories and maintenance: the analysis of operational system with variable demand and supply

3 – J. D. C. Little (1961) – A proof of queuing formula: L = ΛW

A. K. Erlang (1909) – The theory of probabilities and telephone conversations

F. On (2011) – Queues

L. Melese (2014) – Kanban for IT

M. Samuellides (2014) – Probabilities for engineering sciences

J. F. Hêche, T. M. Liebling, D. De Warra (2003) – Operational research for the engineer

T. Bonald, M. Sheet (2011) – Performance of networks and computer systems