Question 4

(a) Consider a freeway with a capacity of 3,900 vehicles/hour and a constant vehicle flow of 3,000 vehicles/hour for arrival rate. An incident occurs on this freeway at time zero, which stops all traffic flow for 12 minutes. Then the freeway is partially opened with a flow of 2,100 vehicles/hour for 19 minutes. It is then restored to full capacity of 3,900 vehicles/hour when the incident is cleared (i.e., 31 minutes after time zero). Assuming D/D/1 queuing, sketch the queuing diagram (cumulative arrival and departure curves over time) from the time the incident occurred until the queue clears. (10 points)

(b) Vehicles arrive at a toll bridge at a rate of 430 vehicles/hour (the time between arrivals is exponentially distributed). Two toll booths are open, and each can process arrivals at a mean of 10 seconds (the processing rate is also exponentially distributed). What is the total time in the system spent by all vehicles in one hour? (10 points)

The queuing diagram shows the cumulative arrival and departure curves over time for the freeway incident scenario. The total time in the system spent by all vehicles in one hour at the toll bridge is approximately 0.8372 hours.

Explanation:

Queuing Diagram:

A queuing diagram is a graphical representation of the cumulative arrival and departure curves over time. It helps visualize the behavior of a queuing system. In this case, we will sketch the queuing diagram for the freeway incident scenario.

At time zero, the incident occurs, stopping all traffic flow. The arrival rate is 3,000 vehicles/hour, and the capacity of the freeway is 3,900 vehicles/hour. The queuing diagram will show the cumulative arrival and departure curves over time until the queue clears.

After 12 minutes, the freeway is partially opened with a flow of 2,100 vehicles/hour for 19 minutes. Then, it is restored to full capacity of 3,900 vehicles/hour when the incident is cleared (31 minutes after time zero).

The queuing diagram will show the cumulative arrival and departure curves during these time intervals, reflecting the changes in traffic flow.

Total Time in the System:

To calculate the total time in the system spent by all vehicles in one hour at the toll bridge, we need to consider the arrival rate and the processing rate of the toll booths.

The arrival rate is given as 430 vehicles/hour, and there are two toll booths. Each toll booth can process arrivals at a mean rate of 10 seconds, which is equivalent to 360 arrivals/hour.

Using Little's Law, which states that the average number of customers in a system is equal to the arrival rate multiplied by the average time spent in the system, we can calculate the total time in the system.

Let's denote the average time spent in the system as T. The average number of vehicles in the system is equal to the arrival rate multiplied by T. In this case, the arrival rate is 430 vehicles/hour, and the average number of vehicles in the system is also equal to the processing rate multiplied by T, which is 360 vehicles/hour.

Equating these two expressions, we can solve for T:

430T = 360T

T = 0.8372 hours

Therefore, the total time in the system spent by all vehicles in one hour is approximately 0.8372 hours.

Learn more about queuing theory and system time calculation here:

https://brainly.com/question/29368697

#SPJ14