REDUCING TIME IN RELIABILITY STUDIES 197 



and hence, in particular, letting 8 = 8* in (38) we have 



P{8*;R,(a*)} ^ P{8*;R,"(8*)] ^ P* (40) 



since the right member of (34) reduces to P* when W is an integer and 



5 = 5*. The error in the approximations above can be disregarded when 

 g is small compared to 02 . Thus we have shown that for small values of 

 g/d2 the probability specification based on (a*, ^*, P*) is satisfied in the 

 sense of (40) if we use the procedure Rsia*, P*), i.e., if we proceed as if 



It would be desirable to show that w^e can proceed as if g = for all 

 values of g and P*. It can be shown that for swfficiently large n the rule 

 Ri{a*, P*) meets it specification for all g. One effect of increasing n 

 is to decrease the average time E{t) between failures and to approach 

 the corresponding problem without replaceme^it since g/E{T) becomes 

 large. Hence we need only show that Ri{a*, P*) meets its specification 

 for the corresponding problem without replacement. If we disregard the 

 information furnished by Poisson life and rely solely on the counting of 

 failures then the problem reduces to testing in a single binomial whether 



6 = di for population IIi and 6 = do for population 112 or vice versa. Let- 

 ting p denote the probability that the next failure arises from 111 then 

 we have formally 



tia'-V = -. — ; — versus Hi-.p = 



1 + a ^ 1 + a 



For preassigned constants a* > I and P* (V2 < P* < 1) the appropri- 

 ate sequential likelihood test to meet the specification: 



"Probability of a Correct Selection ^ P* whenever a ^ a*" (41) 

 then turns out to be precisely the procedure Rsia*, P*). Hence we may 

 proceed as if gr = when n is sufficiently large. 



The specifications of the problem may be given in a different form. 

 Suppose 01* > 02* are specified and it is desired to haxe a probability of a 

 correct selection of at least P* whenever ^1 ^ 0i* > 02* ^ 02 . Then we 

 can form the following sequential likelihood procedure R3* which is 

 more efficient than Rsia*, P*). 



Rule /?3*.- 



"Continue experimentation without replacement until a time t is 

 reached at which the inequality 



