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The P-K Formula
The Pollaczek-Khintchine (called P-K,
for obvious reasons) formula gives the
expected average WIP at a single-tool
workstation where arrivals to the
workstation are highly variable, and
process times are somewhat less variable.
More specifically, the formula applies
when interarrival times to the
workstation are exponentially
distributed, and process times follow a
general distribution (M/G/1 queues). For
tools that fit this description, the
expected WIP can be computed from the
mean interarrival time, the mean process
time, and the variance of the process
time distribution.
It turns out that in a wafer fab,
interarrival times to a given workstation
usually are highly variable, and some
research does suggest modeling them as
exponential. We usually think of process
times as being fairly constant. However,
if you look at process time as the time
from when a lot gets to the front of the
queue to when it finishes processing,
then things like setups, equipment
downtimes, operator delays, and different
operations processed on the same tool all
add variability to the process times as
seen by successive lots. And this
variability, as shown by the P-K formula
below, can drive up WIP and cycle
time.
WIP = average number in queue and in
process (units)
l = arrival
rate (units per hour)
m = service
rate (units per hour)
r = l /m (traffic intensity
or utilization)
s2
= variance of service time distribution
(0 for constant service times)
WIP = {r} +
{[r2]/[2*(1 -
r)]} + {[l2*s2]/[2*(1
- r)]}.
If you look at the above formula for
WIP, you see that it is first of all a
function of traffic intensity. We know
this. As a tool is loaded more heavily,
the number of wafers in queue increases.
As r
approaches one, the denominator of the
last two terms approaches zero, and the
WIP approaches infinity. This is why
capacity planners always plan for a
capacity buffer on each tool group - to
keep the WIP from becoming very large.
You’ll also notice in the P-K
formula as stated above, that the last
two terms in curly brackets have the same
denominator, and could be combined. We
separated them to highlight the influence
of process time variability. If you have
constant process times, the whole last
term drops off. If you have highly
variable process times, that term can
become significant. A graph illustrating
this is shown below:
The graph shows that it is mostly
equipment loading that drives cycle time
at individual tools. If we just care
about reducing cycle time, we can
decrease start rates or increase
capacity, and cycle times will go down.
However, either of these approaches costs
money. The nice thing about variability
reduction is that it also reduces cycle
time, without requiring costly equipment
purchases or decreased start rates.
What the P-K formula tells us is that,
if we look at individual tools in the
fab, anything that we can do to reduce
variability in the process times seen by
successive lots will directly act to
reduce WIP at these tools, without
requiring any reduction in tool loading.
And, as predicted by Little’s Law,
cycle time will go down at the same time.
The P-K formula is the mathematical
justification for variability reduction
efforts in a wafer fab.
For a derivation of the P-K formula,
see Fundamentals of Queueing Theory:
Second Edition, by Donald Gross and
Carl Harris (Wiley), page 254-256.
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