458 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



with mean and variance, gives a satisfactory jfit to the distribution of 

 traffic overflowing a group of trunks. It is now possible, of course, to 

 convohite the various overflows from any number of groups of varying 

 sizes, to obtain a combined overflow distribution. This procedure, how- 

 ever, would be very clumsy and laborious since at each switching point 

 in the toll alternate route system an entirely difl"erent layout of loads and 

 high usage groups would require solution; it would be unfeasible for 

 practical working. 



We return again to the method of moments. Since the overflows of 

 the several high usage groups will, in general, be independent of one 

 another, the iih semi-invariants Xi of the individual overflows can be 

 combined to give the corresponding semi-invariants A, of their total, 



Ai — iXi + 2X1 + 



(27) 



Or, in terms of the overflow means and variances, the corresponding 

 parameters of the combined loads are 



Average = A' = ai -{- az + ■ ■ ■ (28) 



Variance = V = vi + V2 + • • - (29) 



TRUE DISTRIBUTION 



NEGATIVE BINOMIAL 



FITTING DISTRIBUTION 



0.001 



2 3 4 5 6 7 8 9 10 II 12 13 14 15 

 n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 16 — Probability distributions of overflow traffic: 3 erlangs offered to 2 

 trunks, and 9.6 erlangs offered to 10 trunks. 



