THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 511 



^""^ Ma) = Z Ma,{m) = a'ao{x)/a,(x) (1.22) 



m=0 



Ordinary moments are found from the factorial moments by linear 

 relations; thus if Wt is the A;th ordinary moment (about the origin) 



mo = M^o) nil = M^) m-i = il/(2) + il/(i) 



mz = il/(3) + 3Af (2) + il/(i) 



Thus 



mo(m) = (ro(m)/ai(x) 



mi(m) = aai(m)(To(x) / (Ti(x)(T2(x) 



vi-iim) = aa2{m)(TQ{x) / (r'2{x)(7z{x) + a(Ji{;m)<Ta{x) / <ti{x)(T2{x) 



and, in particular, using notation of the text 



mo{x) = (ro{x)/ax{x) = Ei,xia) 



mi(x) (Tiix) a 



(Xx = — ^r = a 



mo{x) <T.(x) .T - a + 1 + aEi,,{a) (1.23) 



ni2{x) 2 aaiix) , 2 



Vx = — 7-r — (Xx = ir-^ + OCx — ax , ^ 



mo{x) csix) (1,24) 



= ax[l — ax + 2a(x + 2 -\- ax - a)~^] 



X 



Finally the sum moments: nik = ^ mk{m) are 







Wo = 1 



mi = a = a(To{x)/(yi{x) = aEi_x{a) 

 rrh = aaQ{x)/a2{x) -\- mi = mi[a{x -\- I -\- nii — a)~ +1] 



(1.25) 



(1.26) 



y = m2 — mi = mi[l — vh + a(.^' + 1 + nii — a) ] 

 In these, Ei,x(a) = (ro(:c)/(ri(.T) is the familiar Erlang loss function. 



Appendix II — character of overflow load when non-random 



TRAFFIC IS offered TO A GROUP OF TRUNKS 



It has long been recognized that it would be useful to have a method 

 by which the character of the overflow traffic could be determined when 

 non-random traffic is offered to a group of trunks. Excellent agreement 

 has been found in both throwdown and field observation over ranges of 

 considerable interest with the "equivalent random" method of describ- 



