436 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



after meeting a busy signal will have a higher probability of again find- 

 ing all paths busy, than would a randomly originated call. 



The curves (3) show the TF-formula previously developed in this sec- 

 tion, which contemplates exponential return times on all NC attempts. 

 The average return time here is also taken as 1 .67 holding times. These 

 curves lie higher than Kosten's values for two reasons. First, the altered 

 congestion due to return calls is allowed for; and second, with exponential 

 returns nearly two-thirds of the return times are shorter than the aver- 

 age, and of these, the shortest ones will have a relatively high probability 

 of failure upon re-trying. If the customers were to return with exponen- 

 tial times after waiting an average of only 0.2 holding time (e.g., 30 

 seconds wait for 150-second calls) the TT^-curves would rise markedly to 

 the positions shown by (4). 



Curves (5) and (6) give the proportions of time that all paths are busy 

 (equation 4) under the T'F-formula assumptions corresponding to NC 

 curves (3) and (4) respectively; their upward displacement from the 

 random return curves (2) reflects the disturbance to the group congestion 

 produced by the non-random return of the delayed calls. (The limiting 

 position for these curves is, of course, given by Erlang's E2 (or C) delay 

 formula.) As would be expected, curve (6) is above (5) since the former 

 contemplates exponential returns with average of 0.2 holding time, as 

 against 1.67 for curve (5). Neither the (5)-curves nor the open dots of 

 constant 30-second return times show a marked increase over curves (2). 

 This appears to explain why the relationship of load carried versus "NC 

 existing" (as charted in Figs. 3 and 4) was found so insensitive to vari- 

 able operating procedures in handling subsequent attempts in toll ring- 

 down operation, and again, why it did not appreciably change under 

 operator dialing. 



Finally, through the two fields of curves on Fig. 7 is indicated the 

 Poisson summation P{c, L) with load carried L used as the entering 

 variable. The fact that these values approach closely the (2) and (3) sets 

 of curves over a considerable range of NC's should reassure those who 

 have been concerned that the Poisson engineering tables were not useful 

 for losses larger than a few per cent.* 



4. SERVICE REQUIREMENTS FOR DIRECT DISTANCE DIALING BY CUSTOMERS 



As shown by the TF-curves (3) on Fig. 7, the attempt failures by cus- 

 tomers resulting from their tendency to re-try shortly following an NC 



* Reference may be made also to a throwdown by C. Clos (Ref. 3) using the 

 return times of Fig. 6; his "% NC" results agreed closely with tlie Poisson pre- 

 dictions. 



