468 BELL SYSTEM TECHNICAL JOURNAL 



43. The holding times of all calls are equal to a constant h. 



SB. If, at any instant, 5 of the c trunks are busy, the distribution in 

 time of the instants at which said 5 busy trunks were seized is 

 identical with the distribution of 5 points picked individually 

 at random on a straight line of length h. 



Assumptions 1 and 2 together imply that the number of sources of 

 calls is so large that any blocking of calls due to limitation of sources 

 need not be considered. Assumptions 1, 2, 3 and 4:A were made in 

 deriving the formulas for Case No. 1. Assumptions 1, 2, 3, 45 and 

 5B were made in deriving the formulas for Case No. 2. Assumption 

 5B is not strictly compatible with the physics of the constant holding 

 time case. The distribution in time of the 5 calls mentioned in SB 

 will, to a certain extent, depend on the history of previous calls. It 

 is because this dependence is ignored that the solution for Case No. 2, 

 presented in the Appendix, is only approximate. 



APPENDIX 



Mathematical Theory of Delay Formulas 



The following mathematical analysis is based on the assumptions 

 given above on pages 467 and 468. 



Consider the state of affairs at the instant a particular call "A^" 

 originates. Suppose call "X" encounters x other calls; if x is less 

 than c, call "X" will be served immediately but, if not, "X" will 

 have to wait. Our problem is to determine the probability that the 

 delay which "X" may suffer shall have a specified value. 



We begin by determining the relative frequency with which the 

 number of calls encountered by "X" has the value x. Let f(x) be 

 the relative frequency with which x calls are encountered by "X." 

 At an instant of time /, x calls will be encountered if at the preceding 

 instant (/ — dt) either x, (x + 1) or (x — 1) calls would have been 

 encountered. We ignore as of too rare occurrence the cases where 

 more than (x + 1) or less than (x — 1) calls would have been en- 

 countered at time (/ — df) with x at time /. Now in passing from time 

 (/ — dt) to time / the probability of an increase of one call is propor- 

 tional to the difference in time, dt, and to the average number of calls, 

 a, falling per holding time interval. Likewise the probability of a ■ 

 decrease of one call is proportional to the time difference, dt, and to I 

 the number of calls occupying trunks (a decrease must be due to a ■ 

 busy trunk becoming idle). Therefore M 



/(x) = j{x - 1) ^ +/(x + 1) ^ +/(x) ^ 1 - _ - _ J 



