THE INTERCONNECTION OF TELEPHONE SYSTEMS 561 



portant arrangements of gx + y trunks for a terminal assignment of 

 X -\- y = 20, g varying from two to seven subgroups, at a probability 

 of loss of P = .01. This same figure portrays vividly the relative 

 inefficiency of a straight subgrouped multiple compared with a like 

 graded multiple, and again, the eminent superiority of the complete or 

 full-group multiple over both of these. Table VI shows the same 

 information as Fig. 23, recorded in the more familiar tabular form 

 ready for engineering use. 



Summary 



We have sketched briefly some of the general principles underlying 

 the furnishing of an adequate and economical telephone exchange 

 service. One of the several practical means of interconnection is 

 through the employment of special trunking arrangements of the type 

 known as graded multiples. The common-sense theory of this plan 

 has been discussed in some detail after which an approximate mathe- 

 matical formula is presented. 



A group of graded multiple tests run in Chicago in 1927 serves to 

 indicate what modifications should be made in this theoretical trunking 

 schedule before it is used for engineering purposes. A typical table 

 of the loads which, on this basis, may properly be submitted to attain 

 a specified grade of service over a wide variety of arrangements and 

 numbers of trunks, is shown as an example of what appear as the 

 most satisfactory graded multiple capacity figures for Bell System 

 practice at the present time. 



As a more detailed and accurate knowledge is acquired concerning 

 the behavior of telephone traffic over increasingly complex and non- 

 symmetrical graded trunking arrangements, some slight modification 

 in the conclusions we have reached here may be expected. There is 

 ample opportunity, then, for additional theoretical analysis of the 

 graded multiple problem (as well as of many other multiple problems), 

 an analysis that perhaps will overcome the limitations which have 

 thus far been levied in order that a working result might be obtained. 



Appendix I ^■^ 

 Mathematical Theory of the Simple Graded Multiple 

 The mathematical analysis given in this appendix is based on the 

 following assumptions: 



1. Constant holding time per call. 



2. " Lost calls held." 



3. The load submitted by each subgroup of selectors varies about its 



average value " a " in accordance with the Poisson Law 

 " Prepared by E. C. Molina. 



