TELEPHONE TRUNK I NG PROBLEMS 471 



By the multinomial theorem 



Imi -\- m^ -\- • • ■ ni 



E I , r " ' ; - — (a"" b"" ■ ■ ■ K'"-) = {cty 



\ \mi \m2 ■ • • Wc / 



ior a = b = ■ ■ ■ K = t and (mi + Wo + • • • wj = x — c. 

 The expression above reduces to 



(e-0 



kr — c 



■rdt. 



Now all this is on the assumption that "X" encountered x other calls. 

 Therefore we must multiply by f{x) and sum for all values of x from 

 c to CO . We obtain 



Y.f{x)e-^'f^cdt = 



x=c I X C 



•^-^ J \ J ^x~c \ r I -v r 



/(o) ("^Vf-^'^/i: 



X — c 



/(O) ( — ) cdte-^'e'^' = /(O) (" ^ ) ce-^'^-^^'dt. 



This is the probability that "X" will be delayed for an interval of 

 length t. To obtain the probability that the delay will be greater 

 than t we must integrate with reference to t from / to oo . But 



r 



^— (f— a) t 

 e-(c-a)t^f = 



Thus, finally, 



= P (> 0)6-^'-"^'. 



This formula for P (> /) has been deduced by taking h as the unit 

 of time. Evidently we would have obtained 



P (> /) = P(> 0)e-('^-°)'/'' 



if h had not been taken as unit of time. 

 For the average delay on all calls we have 



I = £/J^. = .(>o)(^-A_), 



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