REDUCING TIME IN RELIABILITY STUDIES 189 



For each unit the life beyond its guarantee period will be referred to 

 as its Poisson life. Let Li{t) denote the total amount of Poisson life 

 observed up to time t in the population with Vi failures (z = 1, 2, • • • , fc). 

 If two or more of the r^ are equal, say Vi = rj+i = • • • = r^+y , then we 

 shall assign r, and L; to the population with the largest Poisson life, 

 ri+i and L^+i to the population with the next largest, • • • , ri+_, and Lj+,- 

 to the population with the smallest Poisson life. If there are two or more 

 equal pairs (ri , Li) then these should be ordered by a random device 

 giving equal probability to each ordering. Then the subscripts in (5) as 

 well as those in (4) are in one-to-one correspondence with the k given 

 populations. It should be noted that Li(t) ^ for all i and any time 

 t ^ 0. The complete set of quantities Li{t) {i = 1, 2, • • • , k) need not 

 be ordered. Let a = 61/62 so that, since the 6i are ordered, a ^ 1. 



We shall further assume that : 



1 . The initial number n of units put on test is the same and the start- 

 ing time is the same for each of the k populations. 



2. Each replacement is assumed to be a new unit from the same popu- 

 lation as the failure that it replaces. 



3. Failures are assumed to be clearly recognizable without any chance 

 of error. 



SPECIFICATIONS FOR CASE 1 : gf = 



Before experimentation starts the experimenter is asked to specify two 

 constants a* and P* such that a* > 1 and l'^ < P* < 1. The procedure 

 Ri = Rsin), which is defined in terms of the specified a* and P*, has 

 the property that it will correctly select the population with the largest 

 scale parameter with probability at least P* whenever a ^ a*. The initial 

 number n of units put on test may either be fixed by nonstatistical con- 

 siderations or may be determined by placing some restriction on the 

 average experiment time function. 



Rule Rs : 



"Continue experimentation with replacement until the inequality 



k 



^ ^*-(^.-a) ^ (1 _ p*)/p* (6) 



i=2 



is satisfied. Then stop and select the population with the smallest num- 

 ber of failures as the one having the largest scale parameter." 



