THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 455 



Since only two constants, a and k, need specification in (18) or (19), 

 the mean and variance are sufficient to fix the distribution. That is, with 

 the mean /7 and variance v known, 



a = ,7 or a' = n(l - k) = if/v, or a" = n(l - k)/k (24) 



A: = 1 - a/y = 1 - n/v. (25) 



The probability density distribution f(n) is readily calculated from 

 (19); the cumulative distribution G(^n) also may be found through use 

 of the Incomplete Beta Function tables since 



G(^n) = hi7i - l,a") 



(26) 

 = h(n - l,a(l - k)/k) 



The goodness with which the negative binomial of (19) fits actual dis- 

 tributions of overflow calls requires some investigation. Perhaps a more 

 elaborate expression for co(n) than a constant k in (17) is required. Three 

 comparisons appear possible: (1), comparison with a variety of 0«(n) 

 distributions with exactly m calls on the x trunks, or d{n) with m unspeci- 

 fied, (obtained by solving the statistical equilibrium equations (7) for a 

 divided group) ; (2), comparison with simulation or "throwdown" results; 

 and (3), comparison with call distributions seen on actual trunk groups. 

 These are most easily performed in the order listed.* 



Co7nparison of Negative Binomial with True Overflow Distributions 



Figs. 14 to 17 show various comparisons of the negative binomial dis- 

 tribution with true overflow distributions. Fig. 14 gives in cumulative 

 form the cases of 5 erlangs offered to 1, 2, 5, and 10 trunks. The true 



j = n 



distributions (shown as solid lines) are obtained by solving the difference 

 equations (7) in the manner described in Section 7.1. The negative bi- 

 nomial distributions (shown dashed) are chosen to have the same mean 

 and variance as the several F{^n) cases fitted. The dots shown on 



* Comparison could also be made after equating means and variances respec- 

 tively, between the higher moments of the overflow traffic beyond x trunks and 

 the corresponding negative binomial moments: e.g., the skewness given by (15) 

 can be compared with the negative binomial skewness of (22). The difficulty here 

 is that one is unable to judge whether the disparity between the two distribution 

 functions as described by differences in their higher parameters is significant or 

 not for traffic engineering purposes. 



