Predict the outcome of a new time series (mailing)
Think about this problem in two parts.  

Part 1) 
How many samples do you need in order to have the desired confidence in the Null hypothesis?

Part 2) How long does it take to get that many samples?

Part 1 can be answered with A-priori power analysis.  Use this technique to determine the sample size needed to get the p-value desired.  

Part 2 can be addressed with an algorithm applied to the data you have collected thus far.  Your predictors in this case would be: 

Column 1: the probability that your statistic of interest has leveled off (is not changing significantly with additional input).  

Column 2: date/day/time start. You probably will need to codify this.

Column 3: number of samples at finish.  These should be the number which you determined from your prior analysis.

Your response/output in this case would be:

Column 4: the date/day/time finish.

Of course would need to do some data analysis to determine the appropriate algorithm to use on this data set.  The end product should be a function that takes as input a target probability, date/day/time start and a target sample size, while outputting a date/day/time finish.

I hope this at least gives you some ideas on how to proceed.