Final answer:
The student needs to find the minimum departure rate for a park entrance to ensure the average vehicle delay does not exceed 9 minutes, taking into account varying vehicle arrival rates and the use of a D/D/1 queuing system.
Step-by-step explanation:
The student is asking about the minimum departure rate needed at a park entrance to ensure that the average delay per vehicle is no more than 9 minutes, using a D/D/1 queuing system. To solve this problem, we need to calculate the arrival rates and determine the departure rate that would satisfy the queue clearing within the specified average delay.
Vehicles begin arriving at a rate of six per minute from 7:45 A.M to 8:00 A.M. Then from 8:00 A.M. onward, the rate changes to four vehicles per minute. Assuming the park opens at 8:00 A.M., we have 15 minutes of arrivals at six vehicles per minute, followed by the lower constant arrival rate. To maintain an average delay no greater than 9 minutes, we need to find a departure rate that clears the accumulated queue as well as the ongoing arrivals.
To calculate the required departure rate, consider the number of vehicles by 8:00 A.M.: 15 minutes × 6 vehicles/minute = 90 vehicles. Each minute after 8:00 A.M., 4 more vehicles are added to the queue. The departure rate must be higher than the arrival rate to clear the queue, hence it should be greater than 4 vehicles per minute. Furthermore, given the maximum average delay of 9 minutes, the minimum departure rate can be found by setting up an equation that accounts for both the initial 90 vehicles and the additional vehicles arriving at 4 per minute, factoring in the maximum average delay allowed.