Results on the total time on test plot are usually obtained on the assumption that the number of events to be observed is fixed in advance. Here it is shown that the same large sample results hold when the number of events is random if a simple condition is satisfied.
"The Total Time on Test Plot and the Cumulative Total Time on Test Statistic for a Counting Process." Ann. Statist. 14 (3) 1234 - 1239, September, 1986. https://doi.org/10.1214/aos/1176350062