194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



3. The same inequality (24) can also be used if experimentation is 

 carried on without replacement, one advantage of the latter being that 

 there is less bookkeeping involved. In this case there is a possibility 

 that the units will all fail before the inequality is satisfied so that the 

 procedure is not yet completely defined for this case. One possibility 

 in such a situation is to continue experimentation with new units from 

 each population until the inequality is satisfied. Such a procedure will 

 terminate in a finite time with probability one, i.e., Prob{ T > To} -^0 

 as To — > 00, and the probability specification will be satisfied. 



4. A procedure R3 (ni , n-z , ■ • • , rik , ti , t2 , • • • , tk) using the same 

 inequality (24) but based on dilTerent initial sample sizes and/or on 

 different starting times for the initial samples also satisfies the above 

 probability specification. In the case of different starting times it is 

 required that the experimenter wait at least g units of time after the last 

 initial sample is put on test before reaching any decision. 



0. One disadvantage of R3 is that there is some (however remote) 

 possibility of terminating while ri = r2 . This can be avoided by adding 

 the condition r^ > n to (24) but, of course, the average experiment time 

 is increased. Another way of avoiding this is to use the procedure R3 

 which depends only on the number of failures; the effect of using R3 

 when g > will be considered below. 



6. The terms of the sum in (24) represent likelihood ratios. If at any 

 time each term is less than unity then we shall regard the decision to 

 select the population with n failures and Li units of Poisson life as opti- 

 mal. Since (1 — P*)/P* < 1 then each term must be less than unity at 

 termination. 



Properties of Procedure Rz for k = 2 p 



The OC and ASN functions for Rs will be approximated by comparing 

 R3' with another procedure R/ defined below. We shall assume that P* 

 is close to unity and that g is small enough (compared to d^) so that the 

 probability of obtaining two failures within g imits of time is small 

 enough to be negligible. Then we can write approximately at termination 



Li^nT - r,g {i = 1, 2, • • • , A:) (26) 



and 



Li - Li ^ (r, - r,)g (i = 2, 3, • • • , A:) (27) 



Substituting this in (24) and letting 



5* = a* c^*" (28) 



suggests a new rule, say R/' , which we now define. 



