3e2a1b56-dafb-454d-87ad-86adea3e7b86

Supplement to Chapter 11 Analytical queuing models 337
again, using Little’s law
u
WIP q = r a × W q = t e r a
(1 − u)
and since
r a
u = = r a t e
r e
u
r a =
t e
then,
u u
WIP q = × t e ×
(1 − u)
=
u 2
(1 − u)
t e
For M/M/m systems
When there are m servers at a station the formula for waiting time in the queue (and therefore
all other formulae) needs to be modified. Again, we will not derive these formulae but
just state them.
W q =
u 2(m+1)−1
m(1 − u)
From which the other formulae can be derived as before.
t e
For G/G/1 systems
The assumption of exponential arrival and processing times is convenient as far as the
mathematical derivation of various formulae are concerned. However, in practice, process
times in particular are rarely truly exponential. This is why it is important to have some idea
of how a G/G/1 and G/G/m queue behaves. However, exact mathematical relationships are
not possible with such distributions. Therefore some kind of approximation is needed. The
one here is in common use, and although it is not always accurate, it is for practical purposes.
For G/G/1 systems the formula for waiting time in the queue is as follows.
W q =
c 2 2
A a + c e D
C 2 F
A u D
C(1 − u) F
t e
VUT formula
There are two points to make about this equation. The first is that it is exactly the same as the
equivalent equation for an M/M/1 system but with a factor to take account of the variability
of the arrival and process times. The second is that this formula is sometimes known as the
VUT formula because it describes the waiting time in a queue as a function of:
V – the variability in the queuing system
U – the utilization of the queuing system (that is demand versus capacity), and
T – the processing times at the station.
In other words, we can reach the intuitive conclusion that queuing time will increase as
variability, utilization or processing time increases.