利特尔法则

或者，我们可以用一个代数式来表达： L=λ W

利特尔法则可用来确定在途存货的数量。此法则认为，系统中的平均存货等于存货单位离开系统的比率（亦即平均需求率）与存货单位在系统中平均时间的乘积。

虽然此公式看起来直觉性的合理，它依然是个非常杰出的推导结果，因为此一关系式“不受到货流程分配、服务分配、服务顺序，或任何其他因素影响”。

此一理论适用于所有系统，而且它甚至更适合用于系统中的系统。举例来说，在一间银行里，顾客等待的队伍就是一个子系统，而每一位柜员也可以被视为一个等待的子系统，而利特尔法则可以套用到任何一个子系统，也可以套用到整个银行的等待队伍之母系统。

唯一的条件就是，这个系统必须是长期稳定的，而且不能有插队抢先的情况发生，这样才能排除换场状况的可能性，例如开业或是关厂。

In mathematical queueing theory, Little's law (also result, theorem, lemma, or formula) is a theorem by John Little which states that the long-term average number of customers (L) in a stationary system is equal to the long-term average effective arrival rate (λ) multiplied by the average time that a customer spends in the system (W). Expressed algebraically the law is L=λ W.

The relationship is not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else. In most queuing systems, service time is the bottleneck that creates the queue.

The result applies to any system, and particularly, it applies to systems within systems. For example in a bank branch, 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 thing. The only requirement is that the system be ergodic.

In some cases it is possible not only to mathematically relate the average number in the system to the average wait but even to relate the entire probability distribution (and moments) of the number in the system to the wait.

History

In a 1954 paper, Little's law was assumed true and used without proof. The form L = λW was first published by Philip M. Morse where he challenged readers to find a situation where the relationship did not hold. Little published in 1961 his proof of the law, showing that no such situation existed. Little's proof was followed by a simpler version by Jewell and another by Eilon. Shaler Stidham published a different and more intuitive proof in 1972.

Examples

Finding response time

Imagine an application that had no easy way to measure response time. If the mean number in the system and the throughput are known, the average response time can be found using Little’s Law:

mean response time = mean number in system / mean throughput

For example: A queue depth meter shows an average of nine jobs waiting to be serviced. Add one for the job being serviced, so there is an average of ten jobs in the system. Another meter shows a mean throughput of 50 per second. The mean response time is calculated as 0.2 seconds = 10 / 50 per second.

Customers in the store

Imagine a small store with a single counter and an area for browsing, where only one person can be at the counter at a time, and no one leaves without buying something. So the system is:

entrance → browsing → counter → exit

If the rate at which people enter the store (called the arrival rate) is the rate at which they exit (called the exit rate), the system is stable. By contrast, an arrival rate exceeding an exit rate would represent an unstable system, where the number of waiting customers in the store would gradually increase towards infinity.

Estimating parameters

To use Little's law on data, formulas must be used to estimate the parameters, as the result does not necessarily directly apply over finite time intervals, due to problems like how to log customers already present at the start of the logging interval and those who have not yet departed when logging stops.

Applications

Little's law is widely used in manufacturing to predict lead time based on the production rate and the amount of work-in-process.

Software-performance testers have used Little's law to ensure that the observed performance results are not due to bottlenecks imposed by the testing apparatus.

Other applications include staffing emergency departments in hospitals.

Lastly, an equivalent version of Little's law also applies in the fields of demography and population biology, although not referred to as "Little's Law". For example, Cohen (2008) explains that in a homogeneous stationary population without migration, P=B× e, where P is the total population size, B is the number of births per year, and e is the life expectancy from birth. The formula P=B× e is thus directly equivalent to Little's law ( L=λ × W). However, biological populations tend to be dynamic and therefore more complicated to model accurately.

Distributional form

In some cases, it is possible not only to relate averages but to relate the entire probability distribution of the number in the system to the time in the system under a first come, first served discipline. A distributional relation for many FIFO/Poisson-class systems was derived by Keilson and Servi (1988) and further developed in related work, including links to the Fuhrmann–Cooper decomposition; later, Bertsimas and Nakazato (1995) provided a new proof and surveyed applications.