THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 453 



mean. Corresponding changes in the higher moments would also be 

 expected. 



WTiat would be the physical description of a cause system with a vari- 

 ance smaller or larger than the Poisson? If the variance is smaller, there 

 must be forces at work which retard the call arrival rate as the number 

 of calls recently offered exceeds a normal, or average, figure, and which 

 increase the arrival rate when the number recently arrived falls below 

 the normal level. Conversely, the variance will exceed the Poisson's 

 .should the tendencies of the forces be reversed.* This last is, in fact, a 

 rough description of the incidence rates for calls overflowing a group of 

 trunks. 



Since holding times are attached to and extend from the call arrival 

 instants, calls are enabled to project their influence into the future; that 

 is, the presence of a considerable number of calls in a system at any in- 

 stant reflects their having arrived in recent earlier time, and now can be 

 used to modify the current rate of call arrival. 



Let the probability of a call originating in a short interval of time dt be 



Po.n = [a + (n — a)co(n)] dt 



where n = number of calls present in the system at time t, 



a = base or average arrival rate of calls per unit time, and 

 w(n) = an arbitrary function which regulates the modification in 

 call origination rate as the number of calls rises above 

 or falls below a. 

 Correspondingly, let the probability that one of n calls will end in the 

 short interval of time dt be 



which will be satisfied in the case of exponential call holding times, with 

 mean unity. Following the usual Erlang procedure, the general statistical 

 equilibrium equation is 



(16) 



Jin) = /(n)[(l - Po.n){l - Pe,n)\ + /(« " l)Po,n-l(l " Pe.n-l) 



-Vj{n+ 1)(1 - Po.„+i)P,,„+i 

 which gives 



(Po,„ + P.,„)/(n) = Po,«-i/(n - 1) + Pe,n+xKn + 1) 



i ignoring terms of order higher than the first in dt. 



* The same thinking lias been used by Vaiilot^ for decreasing the call arrival 

 I rate according to the number momentaril}^ present; and by Lundquist^ for both 

 increasing and decreasing the arrival rate. 



