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BELL SYSTEM TECHNICAL JOURNAL 



which fails to get a trunk immediately is unknown. It is obvious, 

 however, that when the number of trunks is such that the liability of 

 the call failing to get a trunk immediately is very small — for example: 

 of the order of one in one hundred — the reaction of these calls on other 

 calls must be negligible independently of whatever assumption ^ is 

 made in place of C. 



Problem I 



Referring to Fig. 1 consider a group of 269 subscribers' lines each 

 equipped with a 20-point line switch. When a subscriber removes his 

 receiver his line switch revolves and picks up the first idle trunk which , 



I 2 





269 Subscriber Lines 

 3 I 269 



m 



Line Switches 



fi 



Group of 

 20 Trunks 



Fig. 1 



it comes to. The 20 points of all switches are multipled together so 

 that a single group of 20 trunks must handle the calls originating from 

 these 269 lines. 



What is the probability that when a particular subscriber X calls he 

 fails to obtain a trunk immediately? 



Referring to Fig. 2 let point P represent the unknown instant 

 within the hour at which X calls. Consider the two minutes imme- 

 diately preceding the instant P. Evidently, by assumption C, calls 

 falling outside of this particular two-minute interval can not prevent 

 X from obtaining a trunk. 



* It is well known that the Erlang formula which is based on an assumption dia- 

 metrically opposed to assumption C, namely that calls which find all trunks busy 

 do not wait for a trunk to become idle, gives essentially the same results (for small 

 probabilities, which are the only ones of interest in practice) as the Poisson furmula 

 which assumes C. 



