366 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1953 



line spikes correspond to observations when all calls in the system were 

 being served, that is x ^ c. The dotted spikes show those proportions of 

 observations when one or more calls were waiting, that is x > c. The 

 theoretical values of f{x) are indicated by the smooth curves where 

 they pass over discrete values of x. The theory and observations are 

 seen to be in quite good agreement. 



Referring again to the theoretical delays (and the throwdown checks) 

 on Figs. 1 and 2, very much larger delays can obviously be obtained 

 when delayed calls are handled at random than when they are handled 

 in a strict first-come-first-served, or queued, order, the latter distri- 



CALLS BEING SERVED 



CALLS WAITING 



0.16 

 0.14 

 0.12 



IZ 0.10 

 i-iu 

 ><" 

 to: 0.08 



0.04 



2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 



X= NUMBER OF CALLS BEING SERVED OR WAITING 



Fig. 6 — Distribution (/(x) of simultaneous calls. Theory versus throwdown, 

 c = 10 paths, OL = 0.80, 1500 throwdown calls, 



butions being shown by the straight lines which start at nearly the same 

 ordinates at delay as the random handling curves, and cut down across 

 the lower part of the charts.* Although fewer very short delays occur 



* Delay curves for exponentially distrubuted holding time calls in systems 

 where delayed calls are handled in order of arrival, are given by E. C. Molina in 

 "Application of the Theory of Probability to Telephone Trunking Problems," 

 Bell System Technical Journal, Vol. 6, p. 461, July, 1927. They are calculated from 

 the Erlang equation 



P(>0 = P(>0)e-(^)' 



a'=e-^ 



c! c — a 



T=i x\ c\ c — a 



g-(c-o)< 



(1) 



where the delay t is expressed in multiples of the average holding time. Values of 

 P(>0) = C(c,a) can be read approximately from Figure 21. 



