188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



due to the sequential feature alone is greatest for large a and is inde- 

 pendent of n. Hence if the initial sample size per process n is large we 

 can disregard the replacement techniciue. On the other hand the true 

 value of a is not known and hence the advantage of sequential experi- 

 mentation should not be disregarded. 



The formulas used to compute the accompanying tables are given in 

 Addendum 2. 



ACKNOWLEDGEMENT 



The author wishes to thank Miss Marilyn J. Huyett for considerable 

 help in computing the tables in this paper. Thanks are also due to 

 J. W. Tukey and other staff members for constructive criticism and 

 numerical errors they have pointed out. 



Addendum 1 



In this addendum we shall consider the more general problem of select- 

 ing the best of k exponential populations treated on a higher mathemati- 

 cal level. For k = 2 this reduces to the problem discussed above. 



DEFINITIONS AND ASSUMPTIONS 



There are given k populations H, (^ = 1, 2, • • • , k) such that the life- 

 times of units taken from any of these populations are independent 

 chance variables with the exponential density (1) with a common (known 

 or unknown) location parameter g ^ 0. The distributions for the k popu- 

 lations are identical except for the unknown scale parameter 6 > which 

 may be different for the k different populations. We shall consider three 

 different cases with regard to g. 



Case 1 : The parameter g has the value zero (g = 0). 



Case 2: The parameter g has a positive, known value (g > 0). 



Case 3: The parameter g is unknown (g ^ 0). 

 Let the ordered values of the k scale parameters be denoted by 



di^ e.-^ ■■■ ^ dk (4) 



where equal values may be regarded as ordered in any arbitrary manner. 

 At any time / each population has a certain number of failures associated 

 with it. Let the ordered values of these integers be denoted by ri = ri{t) 

 so that 



I 



ri g r2 ^ • • • ^ r-fc (5) ^ 



i 



