- Unlike average queue length,
the average delay
cannot be
directly
- Fortunately, there is a simple indirect way to compute the average delay
- John Little proved
the following famous formula
(See:
click here)
- The long-term
average number of customers
in a stable system
is equal to:
- the long-term average arrival rate multiplied by the long-term average time a customer spends in the system
- In other words:
Avg. # customers in system = λ × Avg. delay of customers
(I rather remembers this more intuitive form:) Avg. # customers in system λ = ---------------------------- (Little's formula) Avg. delay of customers
Proof:
- Consider a stable system
that has an average arrival rate
of λ:
- Claim: The long term
departure rate
is also equals to
λ
- Reason:
- If the long term
departure rate
is less (<) the
long term
arrival rate
λ, then
the number of customers in
the system will
keep growing to infinity
and the system will
not be stable
- On the other hand,
if the long term
departure rate
is greater (>) the
long term
arrival rate
λ, then
the number of customers in
the system will
becomes equal to 0.
Such a system is not be stable
- If the long term
departure rate
is less (<) the
long term
arrival rate
λ, then
the number of customers in
the system will
keep growing to infinity
and the system will
not be stable
- Conclusion: in
a stable system,
long term departure rate
= long term arrival rate
- Reason:
- We can calculate the
long term departure rate
as follows:
Hence, the long term departure rate is equal to:
Avg. # customers in system long term departure rate = --------------------------------------- Avg. time a customer spends in systemwhich is the Little's formula !!!
- The long-term
average number of customers
in a stable system
is equal to:
- According to a previous analysis
(See: click here),
the
average queue length
in the
M/M/1 queueing system
is:
ρ N(M/M/1) = ------- ..... (click here) 1 - ρ - According to
the Little's formula:
T = N / λ (Little's formula in another form)Therefore, the average service time of the M/M/1 queue is
N T = --- λ 1 ρ = --- * ------- λ 1 - ρ 1 λ/μ = --- * ------- λ 1 - ρ 1 1 = --- * ------- ..... (3) μ 1 - ρ