192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



terminate the experiment we obtain for the ASN function 



and, in particular, for a = 1, oo 



E(F; 1) = s- and E{F; oo) = s (14) 



It is interesting to note that for s = 1 we obtain 



E{F; a) = 1 for all a ^ 1 (15) 



and that this result is exact since for s = 1 the right-hand member S \ 



of (3) is at most one and hence the procedure terminates with certainty ' 



immediately after the first failure. ' 



As a result of the exponential assumption, the assumption of replace- ; 



ment and the assumption that ^ = it follows that the intervals between \ 



failures are independently and identically distributed. For a single popu- ' 



lation the time interval between failures is an exponential chance vari- ; 



able. Hence, for two populations, the time interval is the minimum of j 



two exponentials which is again exponential. Letting r denote the i 



(chance) duration of a typical interval and letting T denote the (chance) j 

 total time needed to terminate the procedure, Ave have 



E{T; a, 62) = E{F; a)E(r; a, d^) = E{F; a) (^^^ (f^) (16) 



I 



Hence Ave obtain from (13) and (14) 



E{T; a, 02) = - -^ ^^^ for a > 1 (17) 



n a — 1 a* + 1 



E{T; 1, d,) = ^ and E{T; <^, 0,) = ^ (18> 



For the numerical illustration treated above Avith k = 2 we have 



na) = ^-^ (19) : 



P(l+) = ^; P(2.088) = 0.95; P(oo) = 1 (20) 



EiF-a) = 4^^4^ = 4 ^--+ Vy + '^ (21) 



a— la*-f-l a*-t-l 



E{F; 1) = 16.0; /iXF; 2.088) = 10.2; E{F; 00) = 4 (22), 



