THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 487 



the last trunk of a straight high usage group of any specified size, carry- 

 ing either first or higher choice traffic or a mixture thereof.* 



The Equivalent Random theory readily supplies estimates of the loads 

 carried by any trunk in an alternate routing network. After having found 

 the Equivalent Random load A offered to *S + C trunks which corresponds 

 to the given parameters of the traffic offered to the C trunks, it is a simple 

 matter to calculate the expected load i on any one of the C trunks if 

 they are not slipped or reversed. The load on the ith trunk in a simple 

 straight multiple (or the S + jth. in a divided multiple of *S lower and C 

 upper trunks), is 



A- = Is+j = A[E^,s+j-M) - Ex,s+j{A)] (33) 



where Ei,n(A) is the Erlang loss formula. A moderate range of values of 

 ■Ci versus load A is given on Figure 40. f 



Using this method, selected comparisons of theoretical versus observed 

 loads carried on particular trunks at various points in the Murray- 

 Hill-6 throwdown are shown in Fig. 41 ; these include the loads on each 

 of the trunks of the first two OST groups of Fig. 32, and on the second 

 and third alternate routes, crossbar and suburban tandem, respectively. 

 The agreement is seen to be fairly good, although at the tail end of the 

 latter two groups the observed values drop aw^ay somewhat from the 

 theoretical ones. There seems no explanation for this beyond the possi- 

 bility that the throwdown load samples here are becoming small and 

 might by chance have deviated this far from the true values (or the 

 arbitrary breakdown of OST overflows into parcels offered to and by- 

 passing XBT may well have introduced errors of sufficient amount to 

 account for this disparity). As is well known, (33) gives good estimates 

 of the loads carried by each trunk in a high usage group to which random 

 (Poisson) traffic is offered; this relationship has long been used for the 

 purpose in Bell System trunk engineering. 



8. PRACTICAL METHODS FOR ALTERNATE ROUTE ENGINEERING 



To reduce to practical use the theory so far presented for analysis of 

 alternate route systems, working curves are needed incorporating the 



* The proper selection point will be where the circuit annual charge per erlang 

 of traffic carried on the last trunk, is just equal to the annual charge per erlang 

 of traffic carried by the longer (usually) alternate route enlarged to handle the 

 overflow traffic. 



t A comprehensive table of /< is given by A. Jensen as Table IV in his book 

 "Moe's Principle," Copenhagen, 1950; coverage is for / ^ 0.001 erlang, z = 1(1)140; 

 A = 0.1(0.1)10, 10(1)50, 50(4)100. Note that n + 1, in Jensen's notation, equals i 

 here. 



