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 = ? T where N is the random variable for the number of jobs or customers in a system, ? 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: ? ( t ) ? number of arrivals in the interval (0,t ) ? ( t ) ? number of departures in the interval (0,t ) N ( t ) ? number of jobs in the system at time t = ? (t ) ? ?( t ) ? ( 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 ? (t ) . ? t = arrival rate over the interval (0,t ) ? (t ) t Elec 428 Little’s Theorem N t = average # of jobs during the interval (0,t ) = ? (t) t Tt = average time a job spends in the system in (0,t ) € = ? (t) ? (t) € ? ? ( t ) = Tt? ( t ) T ? (t ) ? Nt = t = ? t Tt t Assume that the following limits exist: € lim ? t = ? t >? lim Tt = T t >? Then € lim N t = N t >? also exists and is given by N = ? T . € Keywords: Little’s Law Little’s Theorem Steady state Page 2 of 2