Delay Curves for Calls Served at Random 



By JOHN RIORDAN 



(Manuscript received March 11, 1952) 



This paper presents curves and tables for the probability of delay of calls 

 served by a simple trunk group with assignment of delayed calls to the 

 trunks at random and with pure chance call input. These are contrasted 

 with the classic results of Erlang {^'Erlang C") which are based on service 

 in order of arrival. Trunk holding times for both have an exponential dis- 

 tribution. The theoretical development for computation of the curves is di- 

 rected to the determination of the moments, which seem to be a natural means 

 of simplification. 



1. INTRODUCTION 



One of the classic results in the study of telephone traffic is the for- 

 mula for delay given by the Danish engineer A. K. Erlang^ in 1917. This 

 is for random call input to a fully accessible simple trunk group with 

 the trunk holding time exponential and calls served in the order of arrival. 

 A proof for this formula and a set of curves for its use have been given 

 by E. C. Molina.2 



In many switching systems it is not feasible to fully realize this ethical 

 ideal of first come, first served, and it has long been of interest to de- 

 termine delays on another basis. The contrasting assumption is of calls 

 picked at random, which is again an idealization but in large offices ap- 

 pears to be called for, as a bound for the service actually given. 



The first attempt to formulate the last seems to be that of J. W. 

 Mellor.' While his basic formulation is incomplete, it offers a useful 

 approximation to the complete results, particularly in the most interest- 

 ing region of heavy traffic, and will be referred to here as the "Mellor 

 approximation." A complete formulation due to E. Vaulot^ appeared 

 in 1946 and included both the fundamental differential recurrence rela- 

 tion and formulas for delay probabilities for small delays. For complete- 

 ness, these are repeated below. F. Pollaczek^ has given a development of 

 Vaulot's work directed toward determining an asymptotic delay for- 

 mula. 



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