104 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



service is increased (over order of arrival) but this is achieved at the 

 cost of making other customers wait much longer. 



Service in order of arrival has the advantage to the customer that his 

 delay is independent of all who come after him, and this is particularly 

 appreciated in times of heavy crowding when long delays are possible 

 for random service. In Table I, these crowded conditions correspond to 

 small values of F{u) or large values of a, or both. In this picture it 

 seems intuitively clear that much longer delays are possible for random 

 service, for those unlucky customers who keep missing their turn. 

 (Of course, a more realistic model would also include the effects of cus- 

 tomers leaving before service, a factor of considerable telephone interest 

 also.) 



As noted at the start of this section, F(u) is a conditional probability, 

 the probability of delay at least it of a call that is surely delayed. To 

 obtain unconditional probabilities of delay, F(u) is multiplied by the 

 probability that all trunks are busy, which is the probability that a call 

 is delayed. This probability is given by a well-known formula due to 

 Erlang and customarily written as 



c r 2 c-i 



C(o.a) =,-_,^^^_^ !_! + - + 2,+ ••• + 



(c - 1) ! (c - a) L 1! 2! ■ ' (c - 1)! 



+ 



(c - 1) ! (c 



a)J 



Tables of this function are available*. 



Finally it may be noticed here that for random service and light traf- 

 fic (roughly, a less than 0.7), with sufficient approximation 



with 7/1 = 1- VW^y 2/2=1 + VoA 



* But there seems to be no extensive tabulation. However, the table for the 

 Erlang B function made by Conny Palm (Stockholm, 1947) may be used with the 

 relations 



1 1 1 



C(c, a) B(c, a) B(c - 1, a) 



^ a , 1 - (g/c) 

 " c B(c, a) 



Notice that C(c, a) also has the recurrence relation 



1 -1 . (c - a){c - 1) 



C{c, a) c - I - a a(c - 1 - a)C(c - 1, o) 



