368 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



b. Holding times are exponentially distributed. 



c. Congestion loss from the group is negligible. 



d. Observation periods are in statistical equilibrium. 



How do departures from these assumptions affect the reliability of 

 usage measurements? 



a. Holding Time Distribution 



Experience in application of delay and loss formulas has shown that 

 theories based on exponential holding times are often applicable to other 

 holding time distribution cases which have a wide range. However, for a 

 constant holding time distribution special theories often are called for. 

 The average and standard deviation of smtch count estimates of carried 

 load when holding time is constant, are given in part 2 of the Appendix. 

 It is shown there that for estimates of carried load, 



X = 



r > 1 T^, = 100 /j/~ (r - 1) (10) 



r < 1 minimum T'^ = (/' = 1, |, ^, j, etc.) 



maximum T^x = 100 - /</-rT^ ('' = I, I, f. etc.) 



Since constant holding times found in practice are often verj- short, the 

 case of r > 1 is the most hkely to be met. For all values of r greater than 

 one, the error given by formula (1) for exponential holding times is some- 

 what greater than the error given by formula (10) for constant holding 

 times, so use of formula (1) for the constant holding time case is con- 

 servative. For values of r less than 1, the error is an oscillating function 

 of r. The coefficient of variation varies from zero to 23 per cent above 

 that for exponential holding times. Where r may not be accurately known 

 the formula for exponential holding times again seems appropriate. 



In making estimates of the source load when the holding time is con- 

 stant, if r > 1, each scan is uncorrected with any other, since no call 

 can be counted twice, and may be considered a random sample of traffic. 

 There are a total of Nc scans which have an average scan of a and standard 

 deviation Va- The average error in estimating a is, therefore: 



s = 

 with coefficient of variation 



'■• = ^°° V^c = ^™ l/^ ('2' 



