FabTime Cycle Time Management for Wafer Fabs
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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:

WIP vs. traffic intensity for different process time variances, calculated using the P-K formula

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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