Previously I have theorized that the equation below holds true. 

I have spent some time exploring the idea that depth is proportional to power, and depth is inversely proportional to velocity.  However I have not tried to prove that there is in fact a relationship between power and passes, or power and velocity.

In theory if we hold velocity constant than a cut at a power of 4 and pass of 1 should result in the same depth as a cut with a power of 1 and a pass count of 4.  To prove this I wrote a program that runs over the same line 4 times, with one segment of the line being cut at full power in one pass, another at 1/2 power in two passes, 1/3 power in 3 passes and 1/4 power in 4 passes.  

This actually can scale to any number of passes, the cut on the right uses 10 passes and seems to look roughly the same as the 4 pass cut on the left.  In the above image the full power single pass is on the left while the lowest power, highest number of passes is on the right.

You can see that things are not exactly 1:1 here, there is a bit of a loss of depth as you increase passes and reduce power.  To work this out I modified the program to scale up the power a tiny bit for each increase in pass count.  After some back and forth I found that adding the following correction factor seems to normalize things:

power = (BasePower / passes) + (BasePower * 0.02 * (passes - 1))

This works fine for my laser, but it is probably highly dependent on the laser you are using. I have no explanation for this, The laser power is in fact linear so the error is not coming from there.  There is a small amount of overlap between passes, it is possible that the spot size grows with power and that is causing the error. In other words this could be a flaw in my test, not a flaw in the formula.

Anyway I hope to run the same test again, but this time varying speed and power to try and maintain the same depth of cut. Hopefully that will work out well and show there is a solid relationship between speed and power as well.