Working Curves for Delayed Exponential 

 Calls Served in Random Order 



By ROGER I. WILKINSON 



(Manuscript received December 19, 1952) 



Working curves of delays for waiting calls served at random are given 

 for a considerable range of loads and group sizes. Exponential holding time 

 calls are assumed originating at random, and served by a simple group 

 of paths. Results of a number of throwdown tests are given to illustrate 

 the effect on call delays of several modes of service, and particularly of 

 service on a random basis. For random service, these results verify the theory 

 recently developed by J. Riordan; perhaps more interestingly they show 

 the effects on delays of certain blends of queued and random service which 

 approximate methods of handling delayed calls in practical use {such as 

 gating and limited storage circuits). The use of random and queued delay 

 theory is illustrated by a number of examples. To remind the reader that 

 these results are not limited to telephony, department store and vehicular 

 traffic problems are included. 



A theory for predicting the delays which telephone calls (or other 

 corresponding types of traffic such as vehicular, aircraft, people waiting 

 in line, etc.) having exponentially distributed holding times would en- 

 counter when the delayed calls are served in a random order was pub- 

 fished in a recent issue of this Journal* by John Riordan. Mr Riordan's 

 mathematical analysis involved a determination of the first several 

 moments of the delay distributions. He then devised a method of com- 

 bining elementary exponential curves in such a way as to satisfy the 

 moments previously calculated. 



Since a limited number of moments were used in the above determina- 

 tions the curves derived are approximate only, but at the same time they 

 are believed to be good approximations. The critical cases are those of 

 paths carrying very heavy loads, in the occupancy ranges of a = 0.80 

 or higher. 



* Bell System Technical Journal, January, 1953, pages 100-119. 



360 



