106 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



condition may be stated as 



lim Fn{u) = 1, all 2^ 



n=oo 



The probability of delay at least u of an arbitrary call is the sum on 

 n of the product of the probability that n calls are waiting when the 

 call arrives and the probability, Fn(u), that for this condition the call 

 is delayed at least u. The first probability (for statistical equilibrium) 

 is known to be 



(1 - a) dc, a)a" 



where C{c, a), as stated above, is the probability that all trunks are 

 busy; (1 — a)C(Cj a) is the probability that all trunks are busy and no 

 calls are waiting. Hence the probability in question, say /(it), is given by 



or by 



/(«) = (1 -«)C(c,a)2:«"^n(«) 



/(«) = C(e, a)F(u) 



F(.u) = (1 - a) Z a"F„(M) (2) 







F{u), like Fn{u), is then a conditional probability, the probability at 

 least w of a delayed call. Notice that, consistent with this, F(0) = 1. 



It is interesting to notice that Mellor's basic equation, which in pres- 

 ent notation may be written as 



du n + 1 



follows from (1) if first it is supposed that Fn-\{u) = Fn{u) = Fn+i(u) 

 and then, for clarity, Gn replaces Fn . Hence, as indicated by the third 

 boundary condition, it may be expected to be useful for large values of 

 n. Its solution is 



Gn{u) = e-"^^"-^^^ (4) 



A somewhat better approximation may be determined by the Mac- 

 Laurin series obtained by repeated differentiation of (la) and evaluation 



