TELEPHONE TRUNKING PROBLEMS 



473 



serve call "X" after said trunk has served the call occupying it and 

 also the mth call in each of the n — \ waiting groups. Therefore 

 our call will suffer a delay of length 



{n — \)h + y, 



where y is the time which elapsed between the beginning of the interval 

 h and the instant at which the wth trunk was seized by the call 

 occupying it. The probability of this delay is a compound one made 

 up of two factors. 



1st — The probability that x calls are encountered. This probability 

 is, as derived above, 



fix) = m 



f{0)c'{a/c)"' /a\"'-' 



since .x = wc + m — 1. 



2d — The probability that the distance from the beginning of the 

 interval h to the instant at which the mth trunk was seized is y 

 or, in more precise terms, lies between y and y + dy. This 

 probability is, on the basis of assumption 5B, 



The product of these two probabilities gives, writing y = ////, 

 (a/c) = R, 



/(0)c-+Hi?)'" 



r - 1 

 m — 1 



iiR 



1 



(1 - uy-^ 



(in. 



But the subscribers' interest in a delay of magnitude (n — l)h + y 

 is totally independent of what value m might have. Therefore the 

 last probability expression must be summed for all permissible values 

 of m, that is from m = 1 to m = c. We then obtain for the total 

 probability of a delay of extent between {n — 1)// + y and (« — \)h 

 -\- y -\- dy: 



J{i))c'+^R"^{\ - uy-' 



Z 

 «j=i 



c - 1 

 m - 1 



iiR 



dii = 



