384 BELL SYSTEM TECHNICAL JOURNAL 



addition each time such a call length does occur we must introduce the 



i -\- .V 

 contingent probability — ^-^ that a negative and not a positive error will 



occur. The total probability of making an error of .v, where .v < 0, on 

 any call is then, 



/>.^o(-v) dx = '-t^" [/■(- -v) +/(/ - .v) + m - x) + . . .]dx. (14) 



Similarly we find the total probability of making a positive error of magni- 

 tude X, on any call, as 



p,^,{x) dx = '-^ [J{i - x) + f{2i - .v) + /(3/ - .v) ^- . . . 1 dx. (15) 



It will now be of interest to apply equations (14) and (15) to some particular 

 types of holding time distributions. 



(a). Constant Holding Times, t = h' 



If / is constant and equal to //, it will necessarily fall within some one of 

 the special cases enumerated above. Suppose /? lies between qi and (q + 1)/. 

 There will be but one value of the error .Vi possible in the negative range 

 and it will equal qi — //, with a corresponding single value .vo in the positive 

 range equal to (q + l)i — h. It will be seen that equations (14) and (15) 

 reduce simply to 



/'x,<o(.v.) = Piqi - /;) = '-X^'jiqi - .v) = '-±-' m = '^±^, (16) 



t It 



and 



p.,>^{^-2) = p[{q + 1)/ - /?] 



t ' I t 



(17) 



The mean and standard deviation of this two-valued variable are found 

 to be 



X = 0, (18) 



(Tx = V'— Xi(i + -Ti) = \/(i — Xi)X2 = V— XiX2. (19) 



It may be noted that a^ attains a maximum of i/2 when .vo = i/2, and 

 approaches for .v = 0. This is of importance when one has to choose an 

 observation interval / for switch counting constant or relatively constant 

 holding times. 



^ The particular error distributions for cases i/ and c were nt>tained bv G. W. Kenrick. 

 ,n 1923. 



