Vehicles arrive at a single toll booth beginning at 8:00 A. M. They arrive and depart according to a uniform deterministic distribution. However, the toll booth does not open until 8:10 A. M. The average arrival rate is 8 veh/min, and the average departure rate is 10 veh/min. Assuming D/D/1 queuing, when does the initial queue clear and what are the total delay, the average delay per vehicle, longest queue length (in vehicles), and the wait time of the 100th vehicle to arrive (assuming first-in-first-out)?

It is given that :

i). Average arrival rate = 8 vehicle/min

ii). Average departure rate = 10 vehicle/min

We have to assume : D/D/1 queuing

Now let us consider departure starts after 10 minutes of the vehicles arrival,  

i.e.      

Therefore,

The time required to clear the queue is given as :

 minutes

Therefore, time required to clear the queue is 50 minutes.

Now we plot a graph that shows the timing and the number of vehicles.

The total number of vehicles delay is :

So from the graph we get,

The area of triangle AOB =

 Area of triangle ACB =

 Therefore, we get :

The total number of the vehicles arriving :

 vehicles

Now the average delay per vehicle is :  

minutes  

So the average delay per vehicle,  

 minute

The longest queue is :

 vehicles  

Now the arrival time of the   vehicle = 12.5 min

The departure time of the   vehicle = 20 min

Therefore, the wait time for the vehicle is = 20 min - 12.5 min

                                                                     = 7.5 min

So,    minute

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