482 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



with that received by non-random parcels overflowing to it from high 

 usage groups. 



7.5 Expected Loss on First Routed Traffic Offered to Final Route 



The congestion experienced by the first-routed traffic offered to the 

 final group in a complex alternate route arrangement [such as the right 

 hand parcels in Figs. 10(c) and (d)] \vill be the same as encountered in a 

 series of random tests of the final route by an independent observer, 

 that is, it will be the proportion of time that all of the final trunks are 

 busy. As noted before, the distribution of simultaneous calls n (and hence 

 the congestion) on the C final trunks produced by some specific arrange- 

 ment of offered load and high usage trunks can be closely simulated by 

 that due to a single Equivalent Random load offered to a straight group 

 of aS -f C trunks. Then the proportion of time that the C trunks are 

 busy in such an equivalent system provides an estimate of the corres- 

 ponding time in the real system ; and this proportion should be approxi- 

 mately the desired grade of service given the first routed traffic. 



Brockmeyer has given an expression (his equation 36) for the pro- 

 portion of time, Rx , in a simple S -\- C system with random offer A, 

 and "lost calls cleared," that all C trunks are busy, independent of the 

 condition of the *S-trunks: 



R, = f{S,C,A) 



= Ii,x,s+cKA) — — 



where 



m=o \ m / (S — m 



However, (rdS) is usually calculated more readily step-by-step using 

 the formula 



<Tc{S) = aciS - 1) -f CTc-liS) , 



starting with 



crc(O) = 1 and ao(S) = A^Sl 

 The average load carried on the C paths is clearly 



Lc = A[Ei,sU) - Ei,s+c{A)], (31) 



