Probability Distribution of the Process Duration Time based on the Process Status Measurements

I have data of execution of several processes. Each execution is identified by the process_id. The duration of the execution is measured by external agents, testing with some polling logic if the process is still executing or if it is finished.

Sample data for one process execution looks like follows:

   process_id  time status
<dbl> <dbl> <chr>
1       1001     1 execute
2       1001     2 execute
3       1001     3 finish


There are typically 0..n observation of the execute state, followed by only one observation of the finish state. The time is measured relative in seconds from the execution start.

It is important to note, that there are several observation agents using different polling strategies. Some measure each second as in the example above, other only once per 10 seconds. Some agent use such a high interval that in most case the first measurement delivers the status finished. There are also agents with aperiodical behaviour (i.e. with increased intervals).

The question is, is it possible with such a data to define the distribution of the probability (or possible the confidence interval of the probability distribution) of the process execution time?

I.e. I want to know the probability that the process will be completed within N seconds.

• You could compute empirical CDF for lower and upper bounds of the random variables that you have. I found Bott, Devroye, Kohler, Estimation of a distribution from data with small measurement errors, but there should be more on this. Feb 15 '19 at 19:50
• @ModeratorAttention - please could this question be moved to CrossValidated. Thanks! Apr 19 '19 at 6:52