THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 459 



With the mean and variance of the combined overflows now deter- 

 mined, the negative binomial can again be employed to give an approxi- 

 mate description of the distribution of the simultaneous calls (p{z) offered 

 to the common, or alternate, group. 



The acceptability of this procedure can be tested in various ways. One 

 way is to examine whether the convolution of several negative binomials 

 (representing overflows from individual groups) is sufficiently well fitted 

 by another negative binomial with appropriate mean and variance, as 

 found above. 



It can easily be shown that the convolution of several negative bi- 

 nomials all with the same over-dispersion (variance-to-mean ratio) but 

 not necessarily the same mean, is again a negative binomial. Shown in 

 Table I are the distribution components and their parameters of two 

 examples in which the over-dispersion parameters are not identical. The 

 third and fourth semi-invariants of the fitted and fitting distributions, are 

 seen to diverge considerably, as do the Pearsonian skewness and kurtosis 

 factors. The test of acceptability for traffic fluctuation description comes 

 in comparing the fitted and fitting distributions which are shown on 

 Fig. 19. Here it is seen that, despite what might appear alarming dis- 



0(n) 



0.01 



O.OOI 



TRUE DISTRIBUTION 



NEGATIVE BINOMIAL 



FITTING DISTRIBUTION 



• RANDOM TRAFFIC, 8=1.9 



a = 9.6 



= 3.84 



I 2 3 4 5 6 7 8 9 10 II 12 



n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 17 — Probability density distributions of overflow traffic from 10 trunks, 

 fitted by negative binomial. 



