5.12 A toll booth on a turnpike is open from 8:00 A. M. to 12 midnight. Vehicles start arriving at 7:45 A. M. at a uniform deterministic rate of six per minute until 8:15 A. M. and from then on at two per minute. If vehicles are processed at a uniform deterministic rate of six per minute, determine when the queue will dissipate, the total delay, the maximum queue length (in vehicles), the longest vehicle delay under FIFO, and the longest vehicle delay under LIFO.

The time which the queue dissipate is 52.5 minutes, the total delay is 3037.5 vehicles-minute. The longest vehicle delay is in FIFO is 15 minutes and the longest vehicle delay in LIFO is 52.5 minutes

The data given;

Arrival rate

Departure rate

Arrival curve between  7:45am to 8am

Q = 6t

Arrival curve starting from 8:15am to the rest to the rest of the day.

Q = 120 + 2t

Departure Curve

Q = -90 +6t

set Q = Q in order to get the value of t; which is the time when the queue dissipates.

Q  =  Q

120 + 2t = -90 + 6t

120 + 90 = 6t - 2t

t = 52.5 minutes.

This implies that at 8:37am, the queue completely dissipates.

To calculate the total delay, we may need to integrate

This becomes

Imputing the upper and lower limits;

The total delay is 3037.5 vehicle-minute.

If we want to find the longest queue, following inspecting, the longest queue should be at t =30 minutes.

The delay using FIFO (first in, first out) is 15 minutes to arrive

The delay in using LIFO (last in, first out) is 52.5 minutes which is also the time the queue dissipate.

Learn more on rates, FIFO and LIFO here;

link

link

Total delay is 3037.5 vehicle-min

Maximum queue length is 90

Longest delay under FIFO is 15 min

Longest vehicle delay under LIFO is 52.5 min

Explanation:

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