REDUCING TIME IN RELIABILITY STUDIES 195 



h'ule R/ 



"Continue experimentation with replacement until the inequality 



k 



X 6*-(^i-'-i) ^ (1 - P*)/P* (29) 



is satisfied. Then stop and select the population with n failures as the 

 one with the largest scale parameter." 



For rule Rz" the experimenter need only specify P* and the smallest 

 value 5* of the single parameter 



8 = ^' e''''"''-''"''' = ae'^ (30) 



62 



that he desires to detect with probability at least P*. 



We shall approximate the OC and ASX function of R/' for k = 2 

 by computing them under the assumption that (27) holds at termina- 

 tion. The results will be considered as an approximation for the OC and 

 ASN functions respectively of R/ for /,■ = 2. The similarity of (29) 

 and (6) immediately suggests that we might replace a* by 5* and a by 

 5 in the formulae for (6). To use the resulting expressions for R^ we 

 would compute 5* as a function of a* and /3* by (28) and 5 as a function 

 of a and /3 by (30). 



The similarity of (29) and (6) shows that Z„ (defined in Reference 5, 

 page 170) under (27) with gr > is the same function of 5* and 5 as it 

 is of a* and a when g = 0. To complete the justification of the above 

 result it is sufficient to show that the individual increment ^ of Z„ is the 

 same function of 5* and 8 under (27) with ^ > as it is of a* and a 

 when ^ = 0. To keep the increments independent it is necessary to as- 

 sociate each failure with the Poisson life that follows rather than with 

 the Poisson life that precedes the failure. Neglecting the probability 

 that any two failures occur ^^•ithin g units of time we have two values for 

 z, namely 



^ -(.nt-g)/ei -ntl$2 



z = log^^^ = -log 5 (31) 



and, interchanging 61 and ^2 , gives z — log 5. Moreover 



