Question

Suppose that the two approaches to a freeway merge, labelled 1 and 2, have individual capacities of 2000 veh/h and 6000 veh/h, respectively. Suppose that the two approaches can send a combined maximum flow 6000 veh/h through the merge without queues forming. Suppose as well that if both approaches are queued, vehicles enter the merge in a 1:3 ratio; i.e., three vehicles from approach 2 for each vehicle from approach 1. At t = 0, there are no queues in neither of the approaches. From t = 0 to t =1h, demand on approach 1 is 1500 veh/h and demand on approach 2 is 5500 veh/h. After t = 1h, demand on Link 1 becomes 500 veh/h, and demand on Link 2 becomes 4500 veh/h. Sketch the input output diagram for Links 1 and 2 and calculate the duration where queue persists on these links. [5 marks]

Answer 1

Let us put the values of arrival rates , departure rates(capacities) for each of the approaches in 15 minutes intervals. So we have

So clearly in the first four hours, both the approaches individually DO NOT HAVE ANY QUEUING (see last two columns) because cumulative departures of approach 1 is less than cumulative departures for approach 1. Likewise for approach 2.

so let us check the COMBINED (APPROACH 1 +2) CUMULATIVE departures and arrivals

so basically there is a queue forming after the t= 0 mark itself. This queue keeps increasing upto 9250 vehicles and reaches this value at 1:45pm. Then at 2pm the queue length become zero

The cumulative arrivals and departure plots for each approach and for combined approaches is shown below

Assume that if aqueue forms on the combined approach, then it will also form ON BOTH APPROACH 1 AND APPROACH 2.

So now we have to CONTROL THE entering vehicles in a 1:3 ratio from approach 1 and approach 2. So make the arrival flow rate for approach 2 from arrival rate for flow 1 * 3

We see that once we do ramp metering , all queues disappear. cumulative arrivals and departures are exactly equal till 1 hour in to the system.