Start-Up Demonstration Tests with Rejection of Units upon Observing d Failures

Waiting time distributions of competing patterns in higher-order Markovian sequences

“…Let a = 11111, b = 1011, and c = 00 and let the competing patterns under consideration be (1) = a and (2) = b ∪ c . For the realisation R 1 = 011011111101100, of length n = 15, the use of system-wide non-overlapping (SWNO) counting yields two occurrences of b and one of c (and thus three occurrences of (2) ), but no occurrences of (1) . Though there are six consecutive 1s in R 1 , only four of them occur after the first completion of 1011 (which re-starts counting), and thus a does not occur.…”

“…If overlapping counting is used, only the entire triple need be checked. Here the ending block associated with (2) will be 0, to indicate that one more 0 gives another overlapping pattern occurrence.…”

“…The ending blocks associated with the complete triple are 111 for (1) and 1 for (2) , for both overlapping and WPNO counting (with 111 being the only ending block for SWNO counting). In checking for finishing blocks at the middle locations or end of the triple, it is observed that a pattern could end after its first, second, or third 1.…”

“…In early research on startup demonstration testing [7], [17], it was assumed that the testing of the power-generation equipment continued until k consecutive successes had occurred, irrespective of the number of failures that occurred prior to it. As this setup is clearly impractical, [2] studied start-up demonstration tests with independent start-ups, where the equipment is accepted or rejected based on whether or not k consecutive successes occur before d total failures. The analysis 978 J.…”

“…Four different tests are derived by varying the simple patterns (1) and (2) , and the number of occurrences r 1 and r 2 , in the following manner. For a test with rejection of the unit if d total failures occur before k consecutive successes and acceptance if the k consecutive successes occur first, take…”