198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



is satisfied. Then stop and select the population with ri failures as the 

 population with d = di". 



It can be easily shown that the greatest lower bound of the bracketed 

 quantity in (42) is 0i*/^2*. Hence for di*/d2* = a* and P* > i 2 the time 

 required by Rz*{6i*, 62*, P*) ivill always be less than the time required 

 by R,(a*,P*). 



Another type of problem is one in which we are given that 6 = di* 

 for one population and d = 62* for the A; — 1 others where 6]* > 62* are 

 specified. The problem is to select the population with 6 = di*. Then 

 (42) can again be used. In this case the parameter space is discrete with 

 k points only one of which is correct. If Rule R3* is used then the 

 probability of selecting the correct point is at least P*. 



Equilibrium Approach When Failures Are Replaced 



9 



Consider first the case in which all items on test are from the same 



exponential population with parameters (6, g). Let Tnj denote the length 

 of the time interval between the j^^ and the j + 1^* failures, (j = 0, 

 1, • • • ), where n is the number of items on test and the 0*'' failure de- 

 notes the starting time. As time increases to infinity the expected number 

 of failures per unit time clearly approaches n/(0 + g) which is called the 

 equilibrium failure rate. The inverse of this is the expected time between 

 failures at equilibrium, say E{Tn^). The question as to how the quanti- 

 ties E{Tnj) approach E(Tn^) is of considerable interest in its own right. 

 The following results hold for any fixed integer 71 ^ 1 unless explicitly 

 stated otherwise. It is easy to see that 



^^(^i) ^ E{TnJ ^ E(T„o) (43) 



since the exact values are respectively 



e /, e-^-^'^/^X ^ g+d ^ , d 



< 



^ 9+ - (44) 



n — 1 \ n / n n 



In fact, since all units are new at starting time and since at the time of 

 the first failure all units (except the replacement) have passed their 

 guarantee period with probability one then 



^(^i) ^ E(Tnj) S E{Tn,) (j ^ 0) (45) 



If we compare the case g > with the special case g = we obtain 



E{2\j) ^ - (y= 1,2, •••) (46) 



n 



