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# Little's Law

The law states that average queue time will equal the average queue size divided by the average processing rate.
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Zettelkasten, December 06, 2021

• The law states that average queue time will equal the average queue size divided by the average processing rate.

• It also states that the long-term average number $L$ of work in a system equals the long-term average effective arrival rate $\lambda$ multiplied by the average time $W$ that the work stays in the system.

$L = \lambda W$
• MIT professor John D. C. Little proved it in 1961.

• This formula is robust; it applies to virtually all queues disciplines, arrival rates, and departure processes ($W_Q$ is the queue time for an average job, $L_Q$ number of jobs in a queue, $\lambda$ average processing rate, $W_S$ is the system time for an average job, $L_S$ number of jobs in the system).

$W_Q = \frac{L_Q}{\lambda} \quad \text{or} \quad W_S = \frac{L_S}{\lambda},$
• You can use the formula for the whole system instead of just for a queue. This use is helpful when you have trouble distinguishing which items are in the queue and which ones are in service.

### Examples

Assume customers arrive at a rate of 10 per hour in a store and stay 0.5 hours on average. The average number of customers in the store is $L = 10 * 0.5 = 5$.

## References

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