THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 443 



to solve more than a few of the infinite variety of arrangements by means 



of "throwdowns." 



However, for both engineering (planning for future trunk provisions) 

 I and administration (current operating) of trunks in these multi-alternate 



routing systems, a rapid, simple, but reasonably accurate method is 

 (required. The basis for the method which has been evolved for Bell 



System use will be described in the following pages. 



7.1. The "Peaked" Character of Overflow Traffic 



The difficulty in predicting the load-service relationship in alternate 

 route systems has lain in the non-random character of the traffic over- 

 flowing a first set of paths to which calls may have been randomly 

 offered. This non-randomness is a well appreciated phenomenon among 

 traffic engineers. If adecjuate trunks are provided for accommodating 

 the momentary traffic peaks, the time-call level diagram may appear 

 as in Fig. 11(a), (average level of 9.5 erlangs). If however a more limited 

 j number of trunks, say a: = 12, is provided, the peaks of Fig. 11(a) will be 

 Ichpped, and the overflow calls will either be "lost" or they may be 

 j handled on a subsequent set of paths y. The momentary loads seen on 2/ 

 then appear as in Fig. 11(b). It will readily be seen that a given average 

 i load on the y trunks will have quite different fluctuation characteristics 

 i than if it had been found on the x trunks. There will be more occurrences 

 of large numbers of calls, and also longer intervals when few or no calls 

 are present. This gives rise to the expression that overflow traffic is 

 "peaked." 



Peaked traffic requires more paths than does random traffic to operate 

 at a specified grade of delayed or lost calls service. And the increase in 

 paths required will depend upon the degree of peakedness of the traffic 

 involved. A measure of peakedness of overflow traffic is then required 

 which can be easily determined from a knowledge of the load offered and 

 the number of trunks in the group immediately available. 



In 1923, G. W. Kendrick, then with the American Telephone and 

 I Telegraph Company, undertook to solve the graded multiple problem 

 ■through an application of Erlang's statistical eciuilibrium method. His 

 i principal contribution (in an unpublished memorandum) was to set up 

 I the equations for describing the existence of calls on a full access group 

 \oi X -{- y paths, arranged so that arriving calls always seek service first 

 iu the .T-group, and then in the ^/-group when the x are all busy. 



Let f{m, n) be the probability that at a random instant m calls exist 

 j on the x paths and n calls on the y paths, when an average Poisson load 



