DELAY CURVES FOR CALLS SERVED AT RANDOM 



103 



dom curve. This is shown in Fig. 2 for a = 0.9, but the logarithmic 

 scale for F(u) obscures the equaUty of area. 



The character of the comparison may be clearer if the picture is 

 changed. Consider a department store counter with c clerks (correspond- 

 ing to c trunks) in attendance. The time for a sale corresponds to the 

 trunk holding time, and the rate of arrival of customers is like that of 

 call input. For service in order of arrival customers are given serially 

 numbered tickets on arrival; for random service, these tickets may be 

 supposed drawn from a hat, or numbered from a series of random num- 

 bers, or since aggressiveness and the clerks' attention are subject to 

 devious rule, it may be that no attention at all to order of service is 

 equivalent to random service. 



The fact that the average delay is independent of the order of service 

 may be explained roughly by saying that the average rate at which wait- 

 ing lines are removed depends only on the average rate of arrival of 

 customers and the rate at which they are served. Notice however that 

 service at random causes more variable delays (the second and all 

 higher moments are larger than for order of arrival service). Thus with 

 random service the proportion of waiting customers receiving quick 



1.0 



0.6 

 0.4 



0.06 

 — 0.04 

 LL 



0.02 



0.01 

 0.008 



0.004 



0.002 



0.001 . . . . . , 



to 20 30 40 50 60 70 80 90 100 110 120 130 140 



U 



Fig, 2 — Comparison of delay curves for order of arrival and random service; 

 a = 0.9. 



