Kingsman’s Equation and Your Plant’s Utilization

Thursday, March 2nd, 2017 : 0 Comments

Our last blog post provided a general summary of Kingsman’s Equation and how it relates to your manufacturing operation. Today we’re going to delve a little deeper into the equation to prove that when it comes to lowering the Average Queue Time (or Average Wait Time) of your resources, Utilization is King.

For every system whether it’s a single machine or an entire factory, the time a resource spends as raw material or work-in-progress can be divided into two parts.

  1. -Process Time. This is value-added time it takes per machine or machines to process the resource and churn out a finished product.
  2. -Queue/Wait Time. This is non-value added time the resources are set aside to wait at the queue of machine or bottleneck, on a machine’s setup, etc

Lead Time = Queue Time + Process Time. In most manufacturing systems, the Queue time can comprise 80-85% of the lead time. This is all non-value added time that should be reduced in order to maintain a Lean operation.

Now let’s take another look at Kingsman’s equation:


AQT = Average Queue Time.
p = Utilization, expressed as decimal
Ca2 + Cs2 = arrival and process coefficient of variations.
τ = average process time

so what does this mean exactly? First let’s look at some examples where utilization(p) is at .25, .5, .7, .9, and .99. For these examples, we’ll just assume: (Ca2 + Cs2)/2 = 1 and process time(τ) = 60 minutes.

Case 1: If p = .25, Average Queue Time (AVQ) = (.25/(1-.25) * 60 = 20 mins
Case 2: If p = .5, Average Queue Time (AVQ) = (.5/(1-.5) * 60 = 60 mins
Case 3: If p = .7, Average Queue Time (AVQ) = (.7/(1-.7) * 60 = 140 mins
Case 4: If p = .9, Average Queue Time (AVQ) = (.9/(1-.9) * 60 = 520 mins
Case 5: If p = .99, Average Queue Time (AVQ) = (.99/(1-.99) * 60 = 5940 mins or 99 hours

In this example since (Ca2 + Cs2)/2 = 1, the STDEV of the respective variables would have to equal the mean to get 1. While such high variability does exist in the real world, it’s not very common. But regardless, this example is enough to illustrate two very important take-aways.

1.) Despite such high variability in both arrival and process times. If the utilization of the machine(s) is low, as it is in case 1 or case 2, the Average Queue Time is still manageable. This is because in terms of reducing your AQT, machine utilization is by far the most important factor.

2.) As you can see from the exponential increases in AQT over the cases, it is very, very wasteful to run your machines at such high utilization because your Average Queue Time goes through the roof. Can you imagine an 99 hour Average Queue Time for each of your products? And the screeching of your customers or sales reps? No thanks.

One final thing I should note is that the AQT in Kingsman is not the exact AQT, but more of a likely upper bound of your real AQT. Though it’s accurate enough to get the point across.

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