A toll booth on a turnpike is open from 8:00 AM until midnight. Vehicles start arriving at 7:45 AM at a uniform rate of 6 veh/min. At 8:15 AM the rate drops to 2 veh/min. If vehicles are processed at a uniform rate of 6 veh/min, determine when the queue will end, the total delay, the maximum queue length, and the longest vehicle delay.

a) The queue will end at 8:15 + 22.5 mins = 8:37:30 AM

b) Total time delay = 52 mins 30 sec

c) maximum queue length = 90 vehicles

d) longest vehicle delay = 15 mins

Explanation:

At 7:45 AM, uniform rate = 6 veh/min

At 8: 15 AM, uniform rate = 2 veh/min

c) The maximum queue length will be between 8:00 AM to 8:15 AM I. e. 15 mins

The number of vehicles = uniform rate * duration

Number of vehicles = 15 * 6 = 90 vehicles

Maximum queue length = 90 vehicles

a) From 8:15 and above:

Incoming rate = 2 veh/min

The vehicles are processed at a uniform rate of 6 veh/min,

i. e. outgoing rate = 6 veh/min

Dissipation rate = Outgoing rate - incoming rate = 6 - 2 = 4 veh/min

Therefore, 90 vehicles will dissipate at 90/4 = 22.5 min

The queue will end at 8:15 + 22.5 mins = 8:37:30 AM

b) The total delay will be between the time the queue begins and the time it dissipates

The time queue begins = 7:45 AM

The time queue dissipates = 8:37:30

Total delay = 8:37:30 AM - 7:45 AM

Total time delay = 52 mins 30 sec

d) Longest vehicle delay

The longest delay will be experienced by the 90th vehicle between 8:00 AM and 8:15 AM

The longest vehicle delay = 15 mins