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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

Notes

  • 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 LL of work in a system equals the long-term average effective arrival rate λ\lambda multiplied by the average time WW that the work stays in the system.

L=λWL = \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 (WQW_Q is the queue time for an average job, LQL_Q number of jobs in a queue, λ\lambda average processing rate, WSW_S is the system time for an average job, LSL_S number of jobs in the system).

WQ=LQλorWS=LSλ,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=100.5=5L = 10 * 0.5 = 5.

References

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