TELEPHONE TRUNKING PROBLEMS 465 



P (>/) = probability that a call is delayed for an interval of time 

 greater than t. In other words, 100 times P (>t) gives the 

 per cent of calls which will, on the average, be delayed for 

 an interval greater than /. 



Charts 



The two series of charts following this paper embody, for Case No. 

 1 and Case No. 2, respectively, curves giving for different values of c 

 and the ratio a/c the probability that a call will be delayed to an 

 extent which will exceed a given multiple of the average holding time. 

 These curves may consequently be read to determine what proportion 

 of the calls will be delayed on the average by an interval as measured 

 in holding times. For example, consider the particular varying holding 

 time chart which corresponds to c — 10; we see from the curve 

 marked a/c = 0.50 that there is a probability of 0.001 that a call will 

 be delayed for an interval of time which will exceed 0.72 of the average 

 holding time; or, put another way, that 0.1 per cent of the calls will 

 be, on the average, delayed this amount. Again there is a probability 

 of only 0.000019 for a call being delayed at least 1.5 times the average 

 holding time; or 0.0019 per cent of the calls will be delayed by this 

 amount. If, on the same chart, we consider the curve marked 

 a/c = 0.70, we find that 22 per cent of the calls will be delayed, 1 

 per cent will be delayed at least 1.04 times the average holding time, 

 0.01 per cent will be delayed 2.58 times the average holding time, or 

 more. 



The dotted line on each chart gives, at its points of intersection with 

 the curves, the average delay on calls delayed as a multiple of the 

 average holding time interval. For example, on the c = 10 varying 

 holding chart we note that for a/c = 0.70 the average delay on calls 

 delayed is 0.33/?. To obtain the average delay on all calls we multiply 

 by the proportion of calls delayed, P (> 0) = 0.22, and obtain 0.073 

 times the average holding time. 



A glance at the formulas, given in the Appendix, for the average 

 delay on calls delayed shows that this delay does not reduce to zero 

 when a/c becomes zero. It approaches a lower limit which has the 

 value h/c in Case No. 1 and the value h/{c + 1) in Case No. 2. The 

 latter limit may readily be anticipated from physial considerations 

 as follows. Assume that the group consists of a single trunk; we 

 have to show that the average delay when a call is delayed approaches 

 the limit /?/(l + 1) = h/2 as the load approaches zero. Now when 

 the load is very low, those cases where two or more calls have to wait 

 for the trunk to become idle are quite negligible; we only have to 



