Page 60 - A First Course In Stochastic Models

THE FORMULA OF LITTLE 51

D n = the amount of time spent by the nth customer in the queue
(excluding service time),

U n = the amount of time spent by the nth customer in the system
(including service time).

Let us assume that each of the stochastic processes {L q (t)}, {L(t)}, {D n } and {U n }
is regenerative and has a cycle length with a finite expectation. Then there are
constants L q , L, W q and W such that the following limits exist and are equal to
the respective constants with probability 1:

1 t
lim L q (u) du = L q (the long-run average number in queue),
t→∞ t
0
1 t
lim L(u) du = L (the long-run average number in system),
t→∞ t 0
n
1
lim D k = W q (the long-run average delay in queue per customer),
n→∞ n
k=1
n
1
lim U k = W (the long-run average sojourn time per customer).
n→∞ n
k=1
Now define the random variable
A(t) = the number of customers arrived by time t,
It is also assumed that, for some constant λ,

A(t)
lim = λ with probability 1.
t→∞ t
The constant λ gives the long-run average arrival rate of customers. The limit
λ exists when customers arrive according to a renewal process (or batches of
customers arrive according to a renewal process with independent and identically
distributed batch sizes).
The existence of the above limits is sufficient to prove the basic relations

L q = λW q (2.3.1)

and
L = λW (2.3.2)

These basic relations are the most familiar form of the formula of Little. The reader
is referred to Stidham (1974) and Wolff (1989) for a rigorous proof of the formula
of Little. Here we will be content to demonstrate the plausibility of this result. The