Convergence tests for experimental data

I ran a simulation for the critical density at which traffic flow transitions from laminar flow to start-stop-waves. The results for different road lengths are:

From the graph we can see that the critical density decreases as the road length increases. Is there any test I can do that guides me into a point where my data may converge?

All I can be sure is that it converges at least to 0 (which physically makes sense to me). How can I test that it does not converge to any other value greater than 0?

PS: It takes me about a week of the code running to get very high values (1 single point at $10^4$ took 5 days of the program running).

PPS: I add a link to the data Data in case someone wants to try something.

• The answer is likely no. Given the spread of your data, it seems like it should be easy to fit it with many different functions which converge to different asymptotes. Commented Apr 24, 2018 at 20:28
• The plot you show involves a very high density of data points over quite a small range of the independent variable if you are worried about asymptotic behavior. You don't gain much by testing 125, 175 and 225 as well as 100, 150, 200, and 250. Spend some time to get small sets of data at higher values of the road length ($10^3$ and $3\times 10^3$, perhaps), which will put stronger constraints on the variation that @probably_someone talks about. I would be wary on including very small groups (like, say, a single data point at $10^4$): you need enough to reduce the odds of skewing the results. Commented Apr 24, 2018 at 20:37
• But even that won't really solve your convergence problem; it'll just reduce the range of asymptotes that could work with your data, given a specific model. What you're dealing with here is the mother of all extrapolation problems, because you're trying to extrapolate infinitely far from your data; unless you have a specific model that you're trying to fit with, you can construct an arbitrary function (even assuming various smoothness criteria) that will fit all of your data and do literally anything at infinity. Commented Apr 24, 2018 at 20:47
• You don't actually need infinity. Ideally, you'd like lengths long enough that they represent the longest sustained duration or region of fairly uniform traffic conditions, but it is sufficient that you can say with confidence that "over a wide range of cell lengths we can place a floor of XXX on the critical density at Y.Y sigma". Commented Apr 24, 2018 at 20:58