42 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



Table II 



Formula 



Generalized 



Erlang B . . 

 Poisson .... 



Erlang C . . 



Assumption Concerning the Disposal of Calls that do not Obtain 

 a Line Finder Immediately 



Waiting calls are cleared out at a rate j times the rate at 



which calls are terminated when served by the line 



finders. 

 Calls are cleared out of the system immediately, that is 



no calls wait (j = «). 

 Waiting calls are cleared out at a rate equal to that with 



which calls are terminated when served by the line 



finders (j = 1). 

 Calls wait until served (j =0). 



When X ^ c a new situation is encountered, "c" calls are engaged in 

 conversation and x — c calls are waiting for service. If the waiting calls 

 are forced to wait for an unduly long period of time so that in effect 

 they are being denied service, it can be expected that they will wait for 

 some average period, say H, and then abandon their attempts. On this 

 basis the corresponding equation is: 



nfix) = jjix + 1) + ^^-+i-^/U + 1) 



(2) 



It has been assumed in the above equations that the distribution of 

 the holding times is exponential, an assumption which is found in most 

 local systems to be reasonably justified. The distribution of the waiting 

 times is also taken to be exponential. By introducing a factor j, where 

 j = h/H, equation (2) can be written in the simpler form: 



af{x) = [c + jix + 1 - cMx + 1) 



Solving this system of simultaneous equations, we obtain: 

 when x < c, 



(3) 



m = ^/(o) 



when a: ^ c, 



fix) = 

 where 



c\(c + j){c + 2j) 



[c + (x - c)j] 



(4) 



/(O) (5) 



^^^^ ^ VS ^ ^ .Si c!(c+i)(c + 2i).--[c+ (x 



- c)j]) 



(6) 



