DELAYED EXPONENTIAL CALLS SERVED IN RANDOM ORDER 367 



with this method of handhng than when a random selection of the wait- 

 ing calls is followed, the very long delays are markedly reduced, and on 

 this account the queueing procedure is generally preferred. These effects 

 are particularly evident at the higher occupancies. As illustrated in 

 Fig. 1, the ''queued" and "random" delay curves at an occupancy of 

 a = 0.4 show Httle difference down to the P = 0.001 delay level. 



IMPERFECT QUEUEING 



Interest has often centered in questions as to what form the delay 

 curves might take in a system in which queueing of the calls is main- 

 tained to a limited extent, and beyond which the record of order of 

 arrival would be lost. Such an instance might occur with a team of toll 

 recording operators who were able to keep Avell in mind the order of 

 arrival of signals up to a certain number waiting, whereupon they would 

 lose track and not regain this ability until the number of waiting calls 

 had again dropped below some small number. Other situations with 

 actual or equivalent limited delay storage arrangements can readily be 

 imagined. 



To study a case of limited queueing, a short subsidiary throwdown was 

 next run on the c = 2 case, using the 1000 calls of Runs 1 and 2 of Figs. 

 3 and 4 (which comprised the 1000-call sequence most closely approach- 

 ing the theoretical distribution). Three rules for delayed call handling 

 were tested: 



(1) Delayed calls are served in random order. 



(2) Delayed calls are queued (served in order of arrival). 



(3) Delayed calls are queued until more than w are waiting at which 

 time their arrival order is lost and they are served at random. When the 

 number waiting again drops below w, newly arriving calls are queued 

 behind those randomized calls still waiting. Note that case 1 corresponds 

 to ly = 0, and case 2 to w = qo . 



The comparative results are sho^\'n on Fig. 7, with w given successively 

 values of 0, 8, 20, 25, 30 and <» . The w = ^ curve, of course, is taken 

 directly from Fig. 4 for Runs 1 and 2 combined. Although this curve 

 does not agree particularly well with theory (Curve A), its movement 

 with changes in w is nevertheless instructive. As seen, queueing as far 

 as If; = 8 waiting calls produced practically no improvement in the 

 delay distributions. (Perhaps with the occurrence of such large numbers 

 of waiting calls, reaching a maximum of 35, one could not expect queueing 

 of so few as 8 to have much effect.) The next selection oi w = 20, how- 

 ever, still showed only a relatively sUght improvement, particularly in 



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