278 BELL SYSTEM TECHNICAL JOURNAL 



Let pi{t)dt be the probability that the speaker in control of the circuit 

 will resume speaking in the interval t to t -{- dt after pausing and let 

 p2{t)dt be the probability that the speaker not in control of the circuit 

 will start speaking^^in the interval t to t -\- dt after hearing the other 

 speaker pause. In the latter case / may be negative. Then the proba- 

 bility that, following a pause, a resumption will occur in the interval 

 X to X -\- dx and a response will occur in the interval y to y -\- dy \s 

 given by 



p\{x) pi{y) dx dy, 



and the probability of lockout following a pause is given by 



P = j j pi(x) piiy) dx dy, 



in which the integration is to be performed over the region in the xy 

 plane which contains those values of x and y for which lockout occurs. 



Assuming that either subscriber is equally likely to have control 

 of the circuit at any instant, the probability of a lockout following a 

 pause by either party is given by the average of the probabilities for 

 the two parties. 



In determining the limits of integration there are three cases to be 

 considered. Assuming a pause by E 



I. //,. < //,„ + r, 

 II. //,„ + r < //„ < h,, -i- T -\- t', 

 III. /;,„ + T + r' < h. 



In case I only lasting lockouts can occur while in cases II and III 

 both lasting and releasing lockouts can occur. Case I will be used to 

 illustrate a method of determining the limits of integration which can 

 also be applied to cases II and III for which the results will be stated 

 without proof. 



In Fig. 11 which is based on the circuit of Fig. 1, time is represented 

 horizontally and distances vertically, upward from the central line, 

 which represents the east end of the circuit, for transmission from W 

 to E and downward for transmission from E to W. Consider a 

 pause by E occurring at x = at ^. The line ABDG represents the 

 transmission of this pause to W. The point H obtained by projecting 

 G to the top line determines the point y — 0. The points C and F 

 represent the instants at which he and /?„, release. If E resumes in 

 the interval AI, the resumption will arrive at the input of hw before 

 hw has released as determined by the point F, and E retains control 

 of the circuit. If, on the other hand W responds at any time prior to 



