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Little's Theorem

Module by: Bart Sinclair

Summary: (Blank Abstract)

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Theorem 1: Little's Theorem

Little's Theorem (sometimes called Little's Law) is a statement of what was a "folk theorem" in operations research for many years:

N¯=λT¯ N λ T (1)
where NN is the random variable for the number of jobs or customers in a system, λλ is the arrival rate at which jobs arrive, and TT is the random variable for the time a job spends in the system (all of this assuming steady-state). What is remarkable about Little's Theorem is that it applies to any system, regardless of the arrival time process or what the "system" looks like inside.

Proof

Define the following:

  • αt α t ≡ number of arrivals in the interval 0t 0 t
  • δt δ t ≡ number of departures in the interval 0t 0 t
  • Nt N t ≡ number of jobs in the system at time t=αt−δt t α t δ t
  • γt γ t ≡ accumulated customer - seconds in 0t 0 t
These functions are graphically shown in the following figure:

Figure 1
Figure 1 (Littles_Law.png)

The shaded area between the arrival and departure curves is γt γ t . λt=arrival rate over the interval [0,t]=αtt λ t arrival rate over the interval [0,t] α t t N t ¯=average # of jobs during the interval [0,t]=γtt N t average # of jobs during the interval [0,t] γ t t T t ¯=average time a job spends in the system in [0,t]=γtαt T t average time a job spends in the system in [0,t] γ t α t ⇒ γt= T t ¯αt ⇒ γ t T t α t ⇒ N t ¯= T t ¯αtt= λ t T t ¯ N t T t α t t λ t T t Assume that the following limits exist: limt→∞ λ t =λ t λ t λ limt→∞ T t ¯=T¯ t T t T Then limt→∞ N t ¯=N¯ t N t N also exists and is given by N¯=λT¯ N λ T .

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