STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 947 



220 



215 



N(t) 



210 



205 - 



200 



TIME, t *- 



Fig. 1 — A graph oi Nit). 



to be finite. It can also be argued that unlimited office capacity is ap- 

 proached by offices with adequate facilities and low calling rates, and 

 therefore, in some practical cases at least, the model is not flagrantly 

 inaccurate. 



Second, the choice of a constant calling rate for the model ignores the 

 fact that in most offices the calling rate is periodic. Thus, the applica- 

 bility of our results to offices whose calling rates undergo drastic changes 

 in time is restricted to intervals during which the normally variable 

 calling rate is nearly constant. Finally, although the assumption of a 

 negative exponential distribution of holding-time afl'ords the model great 

 mathematical convenience, it is doubtful whether in a realistic model 

 the most likeh' holding-time would have length zero, as it does in the 

 present one. 



IV SUMMARY OF NOTATIONS 



a = Poisson calling rate 

 h = mean holding-time 



7 = h^^ = hang-up rate per talking subscriber 

 h = ah = a\'erage number of busy trunks 

 N{t) = number of trunks in use at f 

 {0, T) = interval of observation 



n = N{0) — number of trunks in use at the start of observation 

 .1 = number of calls arriving in (0, T) 

 H = number of hang-ups in (0, T) 

 K = A + H 



