DELAY CURVES FOR CALLS SERVED AT RANDOM 105 



3. BASIC FORMULATION 



As noted above, the following notation is used: c is the number of 

 trunks, h is the average holding time (the distribution of holding times 

 is exponential) and a is the average number of calls arriving in time in- 

 terval h. Then, if Fn(t) is the probability of delay at least t of a call 

 arriving Avhen n other calls are waiting, the differential recurrence 

 relation given by Vaulot is 



^ = ^ ^ F„_,(0 - ^ FM + ^ F„+.(0 (1) 



dt n+l/i h h 



This may be derived as follows. Consider the interval dt after the epoch 

 of arrival of the call in question. In this interval three events may occur: 

 (i) a call may arrive, (ii) a trunk may be released, or (iii) neither of 

 these. The probability of a call arrival is {a/h)dt and if a call arrives 

 the delay function is Fn+i{t — dt). The probability of a trunk release, 

 because of the assumption of exponential holding time, is {c/h)dt, and 

 if a trunk is released the number of waiting calls is reduced by one; the 

 probability that the call seizing the waiting trunk will not be the call 

 in question is n/{n + 1). Finally the probability of the third event is 

 1 — (c + a)dt/h. All this is summarized in the differential relation 



FM) = ^dt Fn+i{t - dt) + -^ I dt Fn-iit - dt) 

 h n -\- 1 h 



+ (i - ^ dty^ - 



dt) 



Passing to the limit gives equation (1). 



Using new variables : tt = ct/h, a = a/c, equation (1) may be written 

 more simply as 



dFniu) 



du n + 1 



Fn-l{u) - (1 + a)Fn(u) + aFn+lW (l^) 



This equation is a mixed differential-difference equation of the first 

 order as a differential equation and of the second order as a difference 

 equation; hence three boundary relations are required. For the differen- 

 tial part, it is clear that Fn{0), which is the probability of some delay 

 of the test call, is unity for all n in question, that is, for all integral non- 

 negative n. Also Fniu) = for all negative n, is an obvious necessity, 

 and, since Fn is a distribution function Fn{^) = 1- Finally the third 



