Help with figuring out how to model a theoretical decay problem

[JasonJM]

First post here so I hope this is in the right spot.

I am working on a theoretical problem at the moment and it involves a decay issue that I cant get my head around how to correctly model. I have included a rough drawing of the problem but here is the description.


Take a theoretical element that has a characteristic of 'Efficiency'. Maximum efficiency rate is 100% and minimum efficiency rate is a variable 'floor'. The initial efficiency of the element is always 100%, but as usage of the element increases the efficiency drops. The problem is that I want the efficiency to drop at a very slow rate to start with, say only a 10% reduction until it reaches a set threshold, say when the usage of the element hits 70 out of 100. Once this threshold is surpassed the efficiency drops at a more accelerated rate until it hits the pre-determined 'floor' when usage reaches 100.

The key variables in the equation are:

Initial Decay = Rate of decay from initial usage until threshold is reached
Accelerated Decay = Rate of decay after the threshold is reached
Threshold Value = Usage value at which the rate of threshold changes
Floor Value = The lowest efficiency that the element will have. This will always occur when usage equals 100.

So basically starting point is: usage = 1 and Efficiency = 100. Final point is: usage = 100 and efficiency = floor value

I did think of applying two different decay functions to this problem, as in one for before and after the threshold, but I was hoping there might be a way that I could model it all as one function.

Note: While a solution would be great I am mainly after any guidance on the best way to approach solving this problem.


Many Thanks

Jason

 

[Ishuda]

First post here so I hope this is in the right spot.

I am working on a theoretical problem at the moment and it involves a decay issue that I cant get my head around how to correctly model. I have included a rough drawing of the problem but here is the description.


Take a theoretical element that has a characteristic of 'Efficiency'. Maximum efficiency rate is 100% and minimum efficiency rate is a variable 'floor'. The initial efficiency of the element is always 100%, but as usage of the element increases the efficiency drops. The problem is that I want the efficiency to drop at a very slow rate to start with, say only a 10% reduction until it reaches a set threshold, say when the usage of the element hits 70 out of 100. Once this threshold is surpassed the efficiency drops at a more accelerated rate until it hits the pre-determined 'floor' when usage reaches 100.

The key variables in the equation are:

Initial Decay = Rate of decay from initial usage until threshold is reached
Accelerated Decay = Rate of decay after the threshold is reached
Threshold Value = Usage value at which the rate of threshold changes
Floor Value = The lowest efficiency that the element will have. This will always occur when usage equals 100.

So basically starting point is: usage = 1 and Efficiency = 100. Final point is: usage = 100 and efficiency = floor value

I did think of applying two different decay functions to this problem, as in one for before and after the threshold, but I was hoping there might be a way that I could model it all as one function.

Note: While a solution would be great I am mainly after any guidance on the best way to approach solving this problem.


Many Thanks

Jason

For problems like this, one might also think of how one could rotate (in and out of plane) and translate other curves. For example, your curve looks like half a parabola (or maybe a y=a*x²ⁿ+b) type curve turned on its side. So change the dependent and independent variables around and fit to a curve of that type.

Another example is just a small modification to a classic curve. That is, the classic slow growth at first and faster & faster growth later on is the exponential, so modify that to a a-b*e^c*x. If b and c are positive, the larger the x the more the value decreases.

 

[JasonJM]

Thanks Ishuda, I will give those a try and see how it looks.


Many thanks

Jason

For problems like this, one might also think of how one could rotate (in and out of plane) and translate other curves. For example, your curve looks like half a parabola (or maybe a y=a*x²ⁿ+b) type curve turned on its side. So change the dependent and independent variables around and fit to a curve of that type.

Another example is just a small modification to a classic curve. That is, the classic slow growth at first and faster & faster growth later on is the exponential, so modify that to a a-b*e^c*x. If b and c are positive, the larger the x the more the value decreases.