REDUCING TIME IN RELIABILITY STUDIES 183 



We shall assume that the number n of units put on test is determined 

 by non -statistical considerations such as the availability of units, the 

 availability of sockets, etc. Then the only unspecified number in the 

 above procedure is the integer r. This can be determined from a table 

 of probabilities of a correct selection to satisfy any given specification 

 (a*, P*). If, for example, a* = 2 then we can enter Table I. If n is 

 given to be 4 and we wish to meet the specification a* = 2, P* = 0.800 

 then we would enter Table I with n — 4 and select r = 4, it being the 

 smallest value for which P ^ P*. 



The table above shows that for the given specification we would also 

 have selected r = 4 for any value of n. In fact, we note that the proba- 

 bility of a correct selection depends only slightly on n. The given value 

 of n and the selected value of r then determine a particular procedure 

 of type Ri , say, Ri(n, r). 



The average experiment time for each of several procedures R\{n, r) 

 is given in Table II for the three critical values of the true ratio a, 

 namely, a = \, a = a* and a = oo . Each of the entries has to be multi- 

 plied by 6-1 , the smaller of the two d values, and added to the common 

 guarantee period g. For n = oo the entry should be zero (-\-g) but it 

 was found convenient to put in place of zero the leading term in the 

 asymptotic expansion of the expectation in powers of I/71. Hence the 

 entry for n = 00 can be used for any large n, say, n ^ 25 when r ^ 4. 



We note in Table II the undesirable feature that for each procedure 

 the average experiment time increases with a for fixed 62 . For the se- 

 quential procedure we shall see later that the average experiment time 

 is greater at a = a* than at either a = 1 or a = 00 . This is intuitively 

 more desirable since it means that the procedure spends more time when 

 the choice is more difficult to make and less time when we are indifferent 

 or when the choice is easy to make. 



PROCEDURES OF TYPE R2 — NONSEQUENTIAL, REPLACEMENT 



"Such procedures are carried out exactly as for procedures oiRi except 

 that failures are immediately replaced by new units from the same 

 population." 



To determine the appropriate value of r for the specification a* = 2, 

 P* = 0.800 when g = we use the last row of Table I, i.e., the row 

 marked n = ^ , and select r = 4. The probability of a correct selection 

 for procedures of type Ro is exactly the same for all values of n and de- 

 pends only on r. Furthermore, it agrees wdth the probability for pro- 

 cedures of type Ri with n = co so that it is not necessary to prepare a 

 separate table. 



