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.
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:
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
.
