462 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



P5n 



P^n 



CONVOLUTION OF 3 NEGATIVE BINOMIAL 

 VARIABLES WITH PARAMETERS: 



AVG WR 



1 I 



2 3 

 2 6 



, -FITTING NEGATIVE BINOMIAL 



6 8 10 12 



n= NUMBER OF CALLS PRESENT 



I 



-I I l_^ 



14 



16 



Fig. 19 — Fitting sums of negative binomial variables with a negative binomial. 



traffic a and trunks .Ti which would have produced ai and Vi . Then pro- 

 ceeding in the forward direction, using a and Xi + X2 , one consults the 

 a and v charts to find txi and Vz . Thus, within the limitations of straight 

 group traffic flow, the character (mean and variance) of any overflow 

 load from x trunks can be predicted if the character (mean and variance) 

 of the load submitted to them is known. 



Curves could be constructed in the manner just described by which the 

 overflow's a' and v' are estimated from a load, a and v, offered to x trunks. 

 An illustrative fragment of such curves is shown in Appendix II, with an 

 example of their application in the calculation of a straight trunk group 

 loss by considering the successive overflows from each trunk as the 

 offered loads to the next. 



Enough, perhaps, has been shown in Section 7.2 of the generally ex- 

 cellent descriptions of a variety of non-random traffic loads obtainable 

 by the use of only the two parameters a and v, to make one strongly 

 suspect that most of the fluctuation information needed for traffic engi- 

 neering purposes is contained in those two values. If this is, in fact, the 

 case, we should then be able to predict the overflow a', v' from x trunks 





