462 BELL SYSTEM TECHNICAL JOURNAL 



The object of this paper is to present for consideration the results 

 of some theoretical studies made with reference to calls handled on a 

 delay basis. 



It is felt that these results may be applied with but slight modifica- 

 tions to many of those traffic problems in which calls are subjected to 

 delay rather than loss when idle mechanisms for advancing them are 

 not immediately available. Such items as service to the subscribers, 

 wear on selecting mechanisms, reliability of circuit operation, etc., are 

 often dependent on the magnitudes of the delays encountered in such 

 cases. Their application to problems in manual traffic is probably 

 less immediate and precise due to the human element which enters 

 into the reckoning. Such factors as an operator's ability to speed up 

 at times of heavy traffic and the facility with which she may reach 

 distant signals appearing before other positions make the problems 

 rather more involved than those dealing with mechanisms whose 

 reactions under various circumstances are more possible to predict. 

 Nevertheless in such cases as these, as well as in problems re'ating to 

 the delays encountered in clearing trouble conditions, installing tele- 

 phones, awaiting elevator service, and many other problems of interest 

 to engineers in general, the results and methods discussed here, though 

 probably not directly applicable, may prove highly suggestive in a 

 qualitative way. 



Obviously the average number of calls to be handled per unit of 

 time, the average length of holding time per call, and the number 

 of trunks assigned to handle the traffic are factors entering into the 

 mathematical results. A knowledge of these three quantities is not 

 sufficient, however, for the solution of the problem. Quite different 

 results will be secured according to the assumption made as to how 

 the holding times of individual calls vary about their given average, 

 i.e., one set of results follows from the assumption that holding times 

 are all of the same length — other sets of results if holding times vary, 

 the precise set depending on the particular law of variation assumed. 



The choice made of laws representing holding time variations must 

 be governed by two considerations : 



1. An assumed law must agree at least approximately with the 

 points obtained if we plot the way holding times vary as found from 

 observations. 



2. The form of the law must lend itself to the mathematical solution 

 of the delay problem. 



Case No. 1 



An assumption which permits of an easy and exact mathematical 

 solution of the problem may be stated as follows: If a call is picked 



I 



