Little's Theorem 2.4 2002/07/19 2003/03/05 Bart Sinclair bs@rice.edu Charlet Reedstrom charlet@rice.edu Bart Sinclair bs@rice.edu (Blank Abstract) 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 λ T 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. 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 T t average time a job spends in the system in [0,t] γ t α t ⇒ γ t T t α t N t T t α t t λ t T t Assume that the following limits exist: t λ t λ t T t T Then t N t N also exists and is given by N λ T .