THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 483 



and the variance of the carried load can be shown to be* 



Vc = ALc ^ - ACEx,s+c{A) + Lc- L\ (32) 



On Fig. 38, Ri values are shown in solid line curves for several com- 

 binations of A and C over a small range of S trunks. The corresponding 

 losses Ri for all traffic offered the final group, where R^ = oc'/A', are 

 shown as broken curves on the same figure. The R2 values are always 

 above Ri , agreeing with the common sense conclusion that a random 

 component of traffic will receive better service than more peaked non- 

 random components. 



However, there are evidently considerable areas where the loss differ- 

 ence between the two Z^'s will not be large. In the loss range of principal 

 interest, 0.01 to 0.10, there is less proportionate difference between the 

 R's, as the A = C paired values increase on Fig. 38. For example, at 

 /?2 = 0.05, and A = C = 10, R./Ri = 0.050/0.034 = 1.47; while for 

 A = C = 30, i?2/Ri = 0.050/0.044 = 1.13. Similarly for A = 2C, the 

 R2/R1 ratios are given in Table XIV. Again the rapid decrease in the 

 R2/R1 ratio is notable as A and C increase. 



F. I. Tange of the Swedish Telephone Administration has performed 

 elaborate simulation studies on a variety of semi-symmetrical alternate 

 route arrangements, to test the disparity between the Ri and R2 types 

 of losses on the final route. f For example if g high-usage groups of 8 

 paths each, jointly overflow 2.0 erlangs to a final route which also serves 

 2.0 erlangs of first routed traffic, Tange found the differences in losses 

 between the two 2-erlang parcels, i?high usage (h.u.) —Ri, shown in 

 column 9 of Table XV. The corresponding ER calculations are performed 

 in columns 2 to 8, the last of which is comparable with the throwdown 

 \alues of column 9. The agreement is not unreasonable considering the 

 sensitiveness of determining the difference between two small prob- 

 abilities of loss. A quite similar agreement was found for a variety of 

 other loads and trunk arrangements. 



* In terms of the first two factorial moments of n : Vc is given by 



Vc = M(2) + M(i) - M(i)*, where Mw = Lc 



(leneral expressions Mu) for the factorial moments of n are derived in an unpub- 

 lished memorandum by J. Riordan. 



t Optimal Use of Both-Way Circuits in Cases of Unlimited Availability, a 

 paper by F. I. T&nge, presented at the First International Congress on the Appli- 

 cation of the Theory of Probability in Telephone Engineering and Administration, 

 June 1955, Copenhagen. 



