Addendum to 



Delay Curves for Calls Served at Random 



By JOHN RIORDAN 



B. S. T. J. January 1953 Pages 100 to 119 



I owe the following remarks, which help to complete the record, to Emile 

 Vaulot: 



1. The Erlang formula for delay with order of arrival service, for the proof 

 of which reference has been made to a paper by E. C. Molina, was proved earlier 

 by E. Vaulot (Application du Calcul des Probabilites a I'exploitation telepho- 

 nique. Revue G^n^rale de 1' Electricite, ^^, pp. 411-418, 1924). Indeed his seems 

 to be the first proof. 



Also, the associated Erlang C function C (c,a), for which I said there was no 

 extensive tabulation, is tabulated for n = 1 (1) 139 and an extensive but irregular 

 set of a's by Arne Jensen (Moe's Principle, Table III, Copenhagen Telephone 

 Co. Copenhagen, 1950). Also the recurrence relation for this function given in a 

 footnote has previously been given by Conny Palm (Vantetider Vid Slumpvis 

 Avverkad K6, Tekniska Meddelanden Fran. Kungl, Telegrafstyrelsen, Special- 

 nummer for Teletrafikteknik, pp. 109. Stockholm, 1946, see p. 43). 



2. The extensive treatment of delay by Conny Palm, just mentioned, includes 

 a section on random service (section 4) ; it may be noticed that this is dated May 

 16, 1946, which is only a few months after Vaulot's article on the same subject 

 (Jan. 28, 1946), and of course is an independent development. 



I owe the following to my colleague S. O. Rice. Pollaczek, in the Comptes 

 Rendus paper mentioned, has given an integral effectively for what I have called 

 F(u). Rice ha« put this in a slightly different form adapted to numerical compu- 

 tation and hjis obtained the following results for F(u) 



V = u(l-a) 



« 1 2 4 6 8 10 12 14 



0.8 0.0079 0.0039 0.0020 0.0011 



0.9 0.2866 0.1388 0.0471 0.0198 0.0094 0.0049 0.0026 0.0015 



Comparifioii with the tables of the papers shows a satisfying agreement and 

 iubstantiatofl the conjectuK; that approximation by a relatively small number 

 of exponential K \h snffioiont. 



John Riordan 



1266 



