536 BELL SYSTEM TECHNICAL JOURNAL 



efficiency (maximum average load per trunk) with a given hunt or 

 access. We have seen that the ultimate efficiency is obtained when 

 the total trunks are arranged in a complete or straight grouping. 

 We have also noted how greatly the efficiency is reduced by splitting 

 the total trunks into distinct individual subgroups. It may well be 

 that by certain rearrangements we shall be able to increase this low 

 efficiency without increasing the access. 



Graded Multiple Theory 



The "graded multiple" means for improving the efficiency of trunk- 

 ing multiples requiring more channels than a single switch can profit- 

 ably hunt over was proposed in 1905 by E. A. Gray.^ We may gain 

 an insight into the manner of working of graded multiple by considering 

 a very simple example. 



Suppose we have two 10-trunk subgroups as in Fig. 6{A), each carry- 

 ing an average load corresponding to some predetermined probability of 

 loss, say P = .001. Then the approximate average load carried by 

 each individual trunk, provided all calls hunt over the trunks in the same 

 order, is shown in Table I.^ The important point to observe here is 

 that the first trunk is busy a goodly proportion of the time [a(l) = .748], 

 the second trunk a somewhat shorter time, and so on down to the 

 last trunks which are comparatively lightly loaded, the tenth trunk 

 being busy only about one-half of one per cent of the time. 



The same distribution is approximately maintained in both sub- 

 groups of 10 trunks, each of which on the average presents all of its 

 10 trunks as busy to one out of each thousand calls submitted to it; 

 but it is quite unlikely that this busy condition would occur simul- 

 taneously on the two groups. Hence, if on those occasions when a call 

 is being lost in one group it could be allowed to hunt over, say, the 

 last half of the other group, in many cases it would find an idle trunk 



3 E. A. Gray, assignor to the American Telephone and Telegraph Company, filed 

 application July 30, 1907. The patent. No. 1002388, was granted September 5, 

 1911, for "A Method of and Means for Connecting Telephone Apparatus." 



* By Erlang's statistical equilibrium method, upon the assumption that "lost" 

 calls are immediately cleared and do not reenter the system, we find the average 

 carried on the rth trunk is 



a{r) = alB{r - \, a) - B{r, o)], 



where B(.v, a) is the proportion of traffic passing beyond the .vth trunk and may be 

 expressed, 



Jx 



B{x, a) = -2 ^r- -^ • 



In all cases "a" refers to the average number of simultaneous calls being submitted 

 to an individual set of trunks. 



