452 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



as shown by ?; = 1.30, and Vx = 1.95. In all cases the variances v and Vx 

 will exceed the variance of corresponding Poisson traffic (which would 

 have variances of a and ax respectively). 



7.2.1. A Prohahility Distribution for Overflow Traffic 



It would be of interest to be able, given the first several descriptive 

 parameters of any traffic load (such as the mean and variance and skew- 

 ness factors of the overflow from a group of trunks), to construct an 

 approximate probability distribution d{n) which would closely describe 

 the true momentary distribution of simultaneous calls. Any proposed 

 fitting distribution for the overflow from random traffic offered to x 

 trunks, can, of course, be compared with . ^ 



X 



determined from (7) or (8). 



Suitable fitting curves should give probabilities for all possitive in- 

 tegral values of the variable (including zero) , and have sufficient unspeci- 

 fied constants to accommodate the parameters selected for describing 

 the distribution. Moreover, the higher moments of a fitting distribution 

 should not diverge too radically from those of the true distribution ; that 

 is, the "natural shapes" of fitting and true distributions should be simi- 

 lar. Particularly desirable would be a fitting distribution form derived 

 with some attention to the physical circumstances causing the ebb and 

 flow of calls in an overflow situation. The following argument and der- 

 ivation undertake to achieve these desiderata.* 



A Poisson distribution of offered traffic is produced by a random arrival 

 of calls. The assumption is made or implied that the probability of a new 

 arrival in the next instant of time is quite independent of the number 

 currently present in the system. When this randomness (and correspond- 

 ing independence) are disturbed the resulting distribution will no longer 

 be Poisson. The first important deviation from the Poisson would be 

 expected to appear in a change from variance = mean, to variance ^ 



* A two-parameter function which has the ability to fit quite well a wide variety 

 of true overflow distributions, has the form 



t(n) = Kin + l)''e-^(''+i) 



in which K is the normalizing constant. The distribution is displaced one unit 

 from the usual discrete generalized exponential form, so that ^(0) 9^ 0. The ex- 

 pression, however, has little rationale for being selected a priori as a suitable 

 fitting function. 



I 



