DELAY CURVES FOR CALLS SERVED AT RANDOM 101 



Considerable further theoretical work has been necessary to obtain 

 the results given here. Vaulot's differential recurrence relation, which 

 formulates the probability of delay at least t of a call which arrives when 

 n other calls are waiting, has no simple solution. By approximate meth- 

 ods, it was possible to use a differential analyser to determine these 

 probabilities for small values of n. But it was not feasible in this way to 

 cover the whole range of interest, and these results were supplemented 

 by approximations for large n, which are described below. Finally the 

 delay for an arbitrary call was obtained by summing on n* 



These results are not reported here, because the attempt to verify 

 the accuracy attained led to formulation of the moments of the delay 

 curves and this in turn to the representation of the curves as sums of 

 exponential curves, mth great simplification of the calculations re- 

 quired. As will appear, two exponentials furnish a sufficient approxima- 

 tion except for heavy traffic. 



2. DELAY CURVES 



The delay distribution on calls delayed for occupancy levels (defined 

 below) from 0.1 to 0.9 in steps of 0.1 is shown in Fig. 1. The abscissae 

 are derived time units which seem to be natural to the problem: u = 

 ct/h, with c the number of trunks and h the average holding time. The 

 ordinates, on a logarithmic scale, are conditional probabilities that a 

 call delayed will be delayed at least u, that is, values of a function 

 F{u); the logarithmic scale is chosen to emphasize the dominantly ex- 

 ponential character of the curves. The occupancy level a is the ratio 

 a/c where a is the average call input in average holding time h. 



Fig. 1 is a master curve for all eventualities and may be changed to 

 working curves for various sizes of trunk groups. For the construction 

 of these curves Table I, from which Fig. 1 was made, and which also 

 compares present results with those for calls served in order of arrival, 

 is convenient. A more elaborate table will be given later. For the con- 

 venience of the reader, it may be noticed that for order of arrival serv- 

 ice F(u) = e-"(i-«>. 



The striking feature of Table I is the increase in delay time for ran- 

 dom service, which becomes more pronounced with decreasing F(u) 

 and increasing occupancy (or traffic) level, a. The increase throughout 

 the table is an effect of the limitation to small values of F{u). For given 



* Thanks are due to George W. Abrams for directing this work, to Dr. Richard 

 W. Hamming for transforming the equations into forms suitable for the differen- 

 tial analyser and for supervising its operation, and to Miss Catherine Lennon for 

 a great deal of calculation. 



