Delay Curves for Calls Served at Random

One of t h e classic results in t h e study of telephone traffic is the formula for delay given by the Danish engineer A. K. Erlang 1 in 1917. This is for random call input to a fully accessible simple t r u n k group with t h e 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 determine delays on another basis. T h e contrasting assumption is of calls picked at random, which is again an idealization but in large offices appears to be called for, as a bound for the service actually given. T h e first a t t e m p t to formulate t h e last seems to be t h a t of J. W. Mellor. 3 While his basic formulation is incomplete, it offers a useful approximation to the complete results, particularly in the most interesting region of heavy traffic, and will be referred to here as the "Mellor approximation." A complete formulation due to E. Vaulot 4 appeared in 1946 and included both the fundamental differential recurrence relation and formulas for delay probabilities for small delays. For completeness, these are repeated below. F. Pollaczek 5 has given a development of Vaulot's work directed toward determining an asymptotic delay formula. 100