454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



(17) 



Or, 



[a + (n — a)w(?i) + ??.]/(n) 



= [a + (n -a- l)co(n - l)]/(/^ - 1) + (n + l)/(7i + 1) 



The choice of aj(n) will determine the solution of (17). Most simply, 

 co(n) = k, making the variation from the average call arrival rate directly- 

 proportional to the deviation in numbers of calls present from their 

 average number. In this case, the solution for an unlimited trunk group 

 becomes, with a' = a{l — k), 



a (a + k) -■■ [a + {n - 1)A;] 



fin) = 



n! 



^^ , , a' (a' + k) , a' (a' + k)(a' + 2k) , 

 1 + « H ^t; H ^ TT, + 



(18) 



2! ' 3! 



which may also be written after setting a" = a'/k = a(l — k)/k, as 



a'ia' + 1) • • • [a" + (n - 1)]A;" 

 fin) = 



n! 



(19); 



(1 - k)- 

 The generating function (g.f.) of (19) is 



Z/(n)r = 



(1 - kT)-"" 



n=0 (1 - k)--" 



which is recognized as that for the negative binomial, as distinguished 

 from the g.f., 



P 



(i + ? tX 



(1/g)^ 



for the positive binomial. 



The first four descriptive parameters of /(w) are: 



I 



