TELEPHONE TRUNK I NG PROBLEMS 469 



when X < c, and 



/(,v) = /(,v - 1) f + /(. + 1) f + /(.v) [ 1 - f - f ] 



when X < c. 



For these two equations we may substitute the simpler equations 



or 



^/(x)-^-a/(x-l)^. 



The solution of these equations gives 



fix) =/(0)[^]- ^<<^ 

 and 



/(.^-)=/(0)^^ -T>c:, 



where /(O) is the arbitrary constant entering in the integration of the 

 finite difference equations. But we must have, evidently, 



Z fix) = 1. 



x=0 



Substituting in this equation the values for /(.%-) given above, we obtain 



l//(0) = e« 



1 - Pic a) + 



^)] 



Since call "X" will be delayed whenever the number x of calls he 

 encounters is equal or greater than c, we have 



a^e 



POO) = L/Cr) = 



The next question is to determine the probability, P (> /), of a 

 delay which is greater than an interval of length t. 



We will get one answer to this question if we make use of assumption 

 4A, and a different answer on the basis of assumptions 43 and 5B. 

 Therefore, from here on, it will be necessary to treat separately the 

 varying and constant holding time cases. 



