THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 447 



I distributions for 6{n) mid dx(;n). Since each such overflow is independent 

 I of the others, they may be combined additively (or convokited), to ob- 

 [tain the corresponchng total distribution of calls appearing before the 

 , I alternate route (or common group) . It may further be possible to obtain 

 I [an approximate fitting distribution to the sum-distribution of the over- 

 flow calls. 



The ordinary moments about the point of the subgroup overflow 

 distribution, when m of the x paths are busy, are found by 



V 



ta'im) = 2 njim, n) (10) 



When an infinite number of |/-paths is assumed, the resulting expres- 

 sions for the mean and variance are found to be:* 

 Number of x-paths busy unspecified :'\ 



Mean = a = a-Ei,^{a) (11) 



Variance = v = a[l — a -{- a{x -\- I -\- a — a)'^] (12) 



All x-paths occupiedi 



Mean = a^^ = a[x - a + 1 -\- aEiMf^ (13) 



Variance = v^ = ax[l — ax + 2a(x + 2 + a^ — a)~^] (14) 



Equations (11) and (12) have been calculated for considerable ranges 

 1 of offered load a and paths x. Figs. 12 and 13 are graphs of these results. 

 i For example when a load of 4 erlangs is submitted to 5 paths, the aver- 

 I age overflow load is seen to be a = 0.80 erlang, the same value, of 

 I course, as determined through a direct application of the Erlang Ei 

 formula. During the time that all x paths are busy, however, the over- 

 flow load wdll tend to exceed this general level as indicated by the value 

 of ax = 1.41 erlangs calculated from (13). Similarly the variance of the 

 overflow load will tend to increase when the x-paths are fully occupied, 



* The derivation of these equations is given in Appendix I. 

 t The skewness factor may also be of interest : 





ilz 



l^i: 



3/2 



^" + "-"^"' +a^ (15) 



x+1 +a- a \x + 2\{x-a)'^^-2{x-a) + x + 2 + {x^-2-a)a 



+ 3(1 -a) I + a(l - a)(l - 2a) 



