A Paradox of the Average Waiting Time for the Case of a Single Bottleneck on the Commuters’ Route

Table 1

For constant arrival/departure rates during phase-1 (, vehicles per minute) phase-2 duration and average waiting time for all vehicles crossing the bottleneck zone are given for various values of (departure rate being unchanged).

(vehicles per minute)	Queue formation start at	Queue reaches peak at	Queue disappears at	(min)	Number of arrivals in phase-1	Total waiting time of arrivals in phase-1 (min)	(min)	Number of arrivals in phase-2	Total waiting time of arrivals in phase-2 (min)	Average waiting time in phase-1 (min)	Average waiting time in phase-2 (min)
48	07 : 30 : 00 A.M.	08 : 30 : 00 A.M.	10 : 10 : 00 A.M.	60.00	4,799	47,990.00	100.00	4,799	47,990.00	10	10
36	07 : 30 : 00 A.M.	08 : 30 : 00 A.M.	09 : 20 : 00 A.M.	60.00	4,799	47,990.00	50.00	1,799	17,990.00	10	10
24	07 : 30 : 00 A.M.	08 : 30 : 00 A.M.	09 : 03 : 20 A.M.	60.00	4,799	47,990.00	33.33	799	7,990.00	10	10
15	07 : 30 : 00 A.M.	08 : 30 : 00 A.M.	08 : 56 : 40 A.M.	60.00	4,799	47,990.00	26.67	399	3,990.00	10	10
10	07 : 30 : 00 A.M.	08 : 30 : 00 A.M.	08 : 54 : 00 A.M.	60.00	4,799	47,990.00	24.00	239	2,390.00	10	10