404 BELL SYSTEM TECHNICAL JOURNAL 



period is made. If no such corrections are to be made, equation (41) is 

 used instead of (35). The arithmetic mean, equation (43), of the values 

 obtained in the various periods will then be the best available estimate 

 of the unknown true average holding time. The reliability of this figure 

 should be substantially that which the schedule was designed to produce. 



Switch Co II III Errors for Non-Exponential Holding Times 



If switch counts are made on calls with other than exponential holding 

 times the resultant errors may be greater or less than those shown by Figs. 

 19, 20 and 21. The comparison of typical o-x's calculated from equations 

 (23) and (27) would suggest that for varying holding times the error in the 

 measurement of individual calls is not greatly dependent on the form of the 

 holding time distribution as long as the average call length covers several in- 

 tervals i. In such a case the charts developed for exponential holding times 

 can probably be used with little allowance for the discrepancy present. 



On the other hand for calls with an unusual or extreme fluctuation about an 

 average / less than i, the errors due to assuming the situation to be equiva- 

 lent to the exponential case may be no longer negligible. The only pro- 

 cedure then would appear to be either to work out the errors actually pres- 

 ent, reverting to the basic error equations (14) and (15), and approximating 

 the new end effect corrections, or to revise the switch counting program to 

 materially shorten the interval i. 



For relatively constant holding times the value of cr^- can be reduced to a 

 small figure by choosing the switch count interval i so that it is contained 

 in the average holding time I closely a whole number of times. Then by 

 equation (19) the error in individual call measurements must, of necessity, 

 be small in nearly every instance. It will be noted that the above specifi- 

 cation permits choosing i = i; moreover, it may readil}^ be seen that in 

 this case the end effect corrections will tend to disappear, giving a highly 

 accurate measurement with relatively few observations. Just how many, 

 of course, will depend on how constant is the quantity measured, and how 

 closely the switch count interval i approaches the true average /. 



V — Gener.\l Summary 



The general problem of determining the average holding times of sul)- 

 scribers' or other calls by sampling methods has been discussed. The need 

 for a proper apportionment of the sample is emphasized and examples are 

 given from telephone experience to illustrate typical analysis procedures. 

 Methods for estimating the reliability of these sampling results for both 

 directly measured holding times and for switch count studies are given along 

 with various curves and charts calculated to assist the traffic engineer in 

 devising a working schedule for the sampling of holdingtimes, particular!)- 

 those of an exponential character. 



