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
where N is the random variable for the number of jobs or customers in a system, λ is the arrival rate at which jobs arrive, and T 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:
α ( t ) ≡ number of arrivals in the interval (0,t ) δ ( t ) ≡ number of departures in the interval (0,t )
N ( t ) ≡ number of jobs in the system at time t = α (t ) − δ( t )
γ ( t ) ≡ accumulated customer - seconds in (0,t )
These functions are graphically shown in the following figure:
The shaded area between the arrival and departure curves is γ (t ) .
λ t = arrival rate over the interval (0,t )
α (t ) t
N t = average # of jobs during the interval (0,t ) =
γ (t) t
Tt = average time a job spends in the system in (0,t )
γ (t) α (t)
⇒ γ ( t ) = Ttα ( t ) T α (t ) ⇒ Nt = t = λt Tt t
Assume that the following limits exist:
€ lim λt = λ
lim Tt = T
lim N t = N
also exists and is given by N = λT .
Keywords: Little's Law Little's Theorem Steady state
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