A Formula for Overall System Load

Thu
26
Feb 2026

Some time ago I've shared my thoughts about coming up with a nice formula to calculate memory fragmentation. I've recently had another such math puzzle, this time themed around "system load". Below you will find the formula I developed, this time with an interactive demo!

The problem is this: Imagine a system that consists of several sub-systems, each having a gauge indicating the current load, 0...100%. It may be some real device like a computer (with CPU busy %, GPU busy % and the current memory consumption), some vehicle (a car, an airplane), or some virtual contraption, like a spaceship in a sci-fi video game (with the current engine load, force field strength, temperature that can make it overheat, etc.) Apart from the load of the individual subsystems, we want to show a main indicator of the overall system load.

What formula should be used to calculate it? All the load values, displayed as 0...100%, are internally stored as numbers in the range 0.0...1.0. The key principle is that if at least one subsystem is overloaded (at or near 100%), the entire system is considered overloaded. A good formula for the overall system load should have the following properties:

The output is 0 if and only if all inputs are 0. In other words, if any subsystem has load >0%, the overall system load is also >0%.
The output is 1 if at least one input is 1. In other words, it's enough for one subsystem to be fully loaded at 100% to consider the entire system loaded at 100%.
If the output is <1, any increase in any input value should result in some increase in the output value.

Note that these requirements don't specify what happens between 0 and 1. For example, should the output become 0.5 if just one input reaches 0.5, or only after all inputs reach 0.5? We have full freedom here.

#1. My first idea was to use AVERAGE(input[i]). It meets requirement 1 and 3, but it doesn't meet requirement 2, because it requires all inputs to be 1 for the output to become 1.

#2. My second idea was to use MAX(input[i]). It meets requirement 1 and 2, but it doesn't meet requirement 3, because for any input other than the largest one, a change in its value doesn't change the output.

#3. There is a more complex formula that meets all 3 requirements. It may be called the "inverse product" and it looks like this:

output1 = 1.0 - ( (1.0 - input[0]) * (1.0 - input[1]) * ... )

You can think of it as multiplying the "headrooms" left in each subsystem.

#4. Unfortunately, the formula shown above has a tendency to give very high values nearing 100% even for low input values, which is not very user-friendly. A result that is closer to MAX() reflects the overall system load better. Considering this, but still wanting to preserve our requirement 3, I ended up blending AVERAGE() with the inverse product 50/50:

finalOutput = (AVERAGE(input[i]) + output1) * 0.5

Here is an interactive demo of all the formulas described above. You can move the sliders to control the input values.

Further modifications are possible:

A different blend than 50/50 between the average and the inverse product could be used.
The final output could be shaped somehow, making it higher or lower in mid values to meet specific design needs, e.g. by raising it to some power, like 1.5.
While a specific input value increases gradually, there is a sharp transition in the behavior of the output when it becomes the highest of all inputs. It could be smoothed out somehow.

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