372 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



indicated in Fig. 7. Only that part of a call which falls within the obser- 

 x'ation poi-iod contrilxites to the usage of that period. First the error 

 made by switch counts in measuring liiis contribution will be deri\'ed. 



Type I 



Consider a call which is already in progress at the start of the ol)ser- 

 vation period. Its duration beyond that point, according to theory, will 

 be exponentially distributed about an average of i. 



If this duration, t, is between and i, the call will be counted once (a 

 measured contribution of i erlang hours) and a positive error of x = i — t 

 Avill be made. The same error will be made if t = 2i — x so that the call 



(C-2)(C-I) C 



Fig. 7 — Graphical indication of the two types of calls with their two sub- 

 divisions. 



is counted twice and so forth. Summing all the ways of making an error 

 X, we have : 



P(x) dx = f{i - x) -\- fi2i - x) + • • • f(ci - x) (1) 



where f{i — x) is the probability of t ^ i — x and 



m^le-r- 



dx 



Calls lasting beyond ci neither start nor end in the observation period 

 so that their contribution is measured without error. For these: 



P(0) = P(t > ci) = e~ 



(2) 



Therefore : 



I — x 



Px>oix) dx = ^ c t 



dx 



+ k 



2i — X 



ci 



t dx + 



+ r" 



dx 



(3) 



