396 BELL SYSTEM TECHNICAL JOURSAL 



The relati\c prominence of these two errors will not change very rapidly 

 with different sizes of observation periods (that is, the number of calls, n), 

 since the end effects' standard deviation (equation 31) will vary inversely 

 as n, while the standard deviation due to the error in measuring individual 

 calls (equation 32) will vary inversely as the \^n. Doubling the length 

 of the observation period would then decrease the first <j to one-half, and 

 the second a to .707 of its former value. 



Equation (36) shows that 



(ji' = fim, w, H, i, i). 



We will not know m and ic before making the switch counts but we can 

 probably substitute the average value of their sum which is 2ni/T, without 

 seriously disturbing the average value of a-f. This gives 



y n / lie T , 



{11 + i)e i - 21 -\- i 



1 - 



\ - e i , (37) 



where now ar is a function of only four variables, n, I, i and T. 



It is of interest to compare the errors predicted by equation (37) with 

 those found by the theory formulated for the particular m and w observa- 

 tions found in each hour's observations at Newark. An average of n = 

 332 calls per hour was observed with an average holding time in the order 

 of / — 145 seconds. The switch count interval, i, was approximately 60 

 seconds, and the observation period was 7' = 3600 seconds. If we take 

 P = .90, the per cent error corresponding will be ± 100 (1 .645) (ri'/i. Using 

 the above estimate of a/ , we find an error of about 3.05 per cent, or ±.0305 

 (145) = ±4.42 seconds. This point has been plotted on Fig. 17b and bears 

 about the same relationship to the observed errors as does the theoretical 

 curve in Fig. 17a in which the comparison takes into account the actual 

 calls carried beyond the start and end of each hour. The discrepancy in 

 Fig. 17b is largely accounted for by the same discussion given heretofore 

 for Fig. 17a. 



The Overall Error in Estimating the Average Holding Time 



The engineer who has the problem of devising a switch count schedule 

 will want to be able to estimate at least roughly the order of accuracy he 

 will actually obtain in the average holding time found from the data in a 

 number of observation periods. Up to this point in this section we have 

 concerned ourselves with discovering only the errors inherent in measuring 

 the average length of a particular n calls of the exponential type in an ob- 

 servation period of length T. As we saw in section III, even when such an 



