Final answer:
To find the average delay per vehicle from 8:00 a.m. until the initial queue clears, calculate the total delay and divide it by the number of vehicles. The time-dependent deterministic rate of arrivals is given by λ(t) = 2 + 0.1t. Use d/d/1 queuing and subtract the service rate of 6 veh/min to find the total delay. Divide the total delay by the number of vehicles to find the average delay per vehicle.
Step-by-step explanation:
To find the average delay per vehicle from 8:00 a.m. until the initial queue clears, we need to calculate the total delay and divide it by the number of vehicles.
The time-dependent deterministic rate of arrivals is given by λ(t) = 2 + 0.1t, where λ(t) is in veh/min and t is in minutes. From 8:00 a.m. to 8:07 a.m., which is a total of 7 minutes, the number of vehicles arriving is:
Number of vehicles = λ(t) * t = (2 + 0.1t) * t = (2 + 0.1(7)) * 7 = 59.5 vehicles
Assuming d/d/1 queuing, the average delay per vehicle is given by:
Average delay per vehicle = total delay / number of vehicles
Total delay = sum of (arrival rate - service rate) for each vehicle
In this case, the service rate is a constant 6 veh/min.
So, we can calculate the total delay and then divide it by the number of vehicles to find the average delay per vehicle.
Since we have calculated the number of vehicles to be 59.5, we can use this value to calculate the average delay per vehicle.