464 BELL SYSTEM TECHNICAL JOURNAL 



at random, the probability that its holding time is greater than an 

 interval of time of length t is e~^'^, where e is the base of natural 

 logarithms and h is the average holding time of all calls. Fortunately 

 there are cases in practice where the variations in length of holding 

 times are closely represented by this exponential law. This is clearly 

 shown by the following Fig. 1 entitled "Distribution of Interoffice- 

 Trunk Holding Times." The points shown on the figure are the 

 plots of actual holding times obtained from pen register records 

 made on a group of trunks running from Waverly to Mulberry in 

 Newark, New Jersey. 



Case No. 2 



Another assumption covered in this memorandum, because it checks 

 closely with cases arising in practice, is that all calls have exactly the 

 same holding time. The precise solution of the delay service problem 

 becomes extremely difficult on this assumption of a constant holding 

 time. An approximate solution is presented in this paper. Cases in 

 practice where holding times are essentially constant are those of 

 sender holding times of key indicator trunk groups and cordless B 

 boards. 



With reference to either Case No. 1 or Case No. 2 consider a group 

 of a certain number of operators handling a certain number of calls 

 having a certain average holding time. If we double the average 

 holding time and halve the number of calls (so that the operators are 

 busy for the same per cent of time on the average), the per cent of 

 calls delayed will not change, but the average delay on calls delayed 

 as measured in seconds will exactly double. Suppose, for example, we 

 wish to obtain the same average speed of answer to line signals with 

 two different groups or teams of operators, the first handling traffic 

 which requires only a short operator holding time or work interval, 

 and the second handling traffic which requires a longer work interval. 

 If the teams are equal in number and ability, we must allow the 

 second team a larger proportion of idle time than the first. If, on the 

 other hand, the proportion of idle time is to be the same for both 

 teams, the second team must be larger or more capable than the first. 

 This general effect is well known, but it is hoped that the results 

 herewith presented will supply more exact knowledge of the subject. 



Before proceeding further it will be helpful to give here the notation 

 used on the delay curves following this paper. 



h = average holding time per call. 



c = number of trunks in a straight multiple. 



a = average number of calls originating per interval of time h. 



