regression to model exponential distribution
There was an MIT group that got a lot of attention a few years back for predicting twitter trends and doing so with really good ROC-AUC Scores.  The problem you are trying to solve is kind of reminiscent of the twitter trending problem.  The crux of the solution is that they avoided the hassle of parametric modeling through a grouping of similar time series followed by a classification.

There is a pretty good writeup of the method for the general public in Stan Nikolov's Blog post.  Notice that there is a video of a presentation by his thesis advisor also.

The basic idea of the method is to find the most common shapes in the time evolution hold those as the dominant classes.  These would be various exponential shapes in your case (from Nikolov's blog post):


Then you can take your training data and choose the best shape for each of your cases.  You can then use this data to train a classification algorithm.  Finally you can use this to predict the outcome of your new data (from Nikolov's blog post):


The full dissertation is also available online for download.

There's actually an entire company that is using this methodology and trying to sell it to companies.  They have some pretty cool promotional videos on youtube also, which won't teach you the methodology, but show how powerful it can be.

Hope this helps!