An approach to a signalized intersection has a saturation flow rate of 1800 veh/h. At the beginning of an effective red, there are 6 vehicles in queue and vehicles arrive at 900 veh/h. The signal has a 60-second cycle with 25 seconds of effective red. What isthe total vehicle delay after one cycle?

To calculate the total vehicle delay after one signal cycle, we can follow these steps:

Step 1: Determine the number of vehicles that will accumulate during the effective red time.

We know the arrival rate of vehicles is 900 vehicles per hour (veh/h) and the effective red time is 25 seconds. To find out the number of vehicles arriving during the red phase, we convert the rate to vehicles per second and multiply by the duration of the red phase.

\[ \text{Arrival rate per second} = \frac{900 \text{ veh/h}}{3600 \text{ seconds/h}} \] \[ \text{Arrival rate per second} ≈ 0.25 \text{ veh/s} \]

Therefore, during the 25-second effective red phase:

\[ \text{Vehicles during red phase} = 0.25 \text{ veh/s} \times 25 \text{ s} \] \[ \text{Vehicles during red phase} = 6.25 \text{ veh} \]

Since vehicles cannot be fractional, we can round up because even a fraction of a vehicle would mean one full vehicle is present, so there are 7 vehicles arriving during the red.

Step 2: Add the initial queue to the number of vehicles that have accumulated during the red phase.

We start with 6 vehicles in the queue, so adding the vehicles that have accumulated:

\[ \text{Total vehicles in queue} = \text{Initial queue} + \text{Vehicles during red phase} \] \[ \text{Total vehicles in queue} = 6 + 7 \] \[ \text{Total vehicles in queue} = 13 \text{ veh} \]

Step 3: Calculate the discharge rate during the green phase.

The saturation flow rate is how many vehicles can go through the intersection when it is completely unimpeded during one hour. With a saturation flow rate of 1800 veh/h, we convert this to vehicles per second:

\[ \text{Saturation rate per second} = \frac{1800 \text{ veh/h}}{3600 \text{ seconds/h}} \] \[ \text{Saturation rate per second} ≈ 0.5 \text{ veh/s} \]

The green time in the signal cycle is the total cycle time minus the effective red time (60 seconds - 25 seconds = 35 seconds).

Step 4: Determine how many vehicles will be served during the green phase.

\[ \text{Vehicles served during green phase} = \text{Saturation rate per second} \times \text{Green time} \] \[ \text{Vehicles served during green phase} = 0.5 \text{ veh/s} \times 35 \text{ s} \] \[ \text{Vehicles served during green phase} = 17.5 \text{ veh} \]

Since we can't serve half a vehicle, and because we've been rounding up on vehicle additions (assuming any fraction of a vehicle constitutes a whole vehicle for the practical purpose of delays), we wouldn't round down when calculating capacity, meaning that only 17 vehicles can actually go through during the green phase.

Step 5: Calculate the vehicle delay.

Total vehicle delay is typically calculated by adding up how long each vehicle waits in the queue. However, in this simplified scenario, where we don't have specific arrival times of each vehicle, we can estimate the delay by using the number of vehicles in queue after the entire cycle (green phase and subsequent red phase), multiplied by the cycle length, and then subtracting the free-flow travel time for the vehicles that are served.

Initially, we had 13 vehicles, but since the saturation flow can serve only 17 vehicles, there would be no queue left after the green phase. All 13 vehicles would be served, and there would be no extra vehicles in queue carried over to the next cycle.

Hence, the vehicle delay in this simplified calculation would be just the waiting time of these 13 vehicles during the effective red phase, which is 25 seconds each, meaning:

\[ \text{Total vehicle delay} = \text{Number of queued vehicles} \times \text{Red phase duration} \] \[ \text{Total vehicle delay} = 13 \text{ veh} \times 25 \text{ s} \] \[ \text{Total vehicle delay} = 325 \text{ seconds} \]

Keep in mind that this is a simplification, assuming that there's enough green time to serve all queued vehicles and that vehicles arrive at a steady rate. It doesn't take into account the varying arrival patterns that could occur in real-world traffic conditions.

Extra: Saturation flow rate is an important concept in traffic engineering. It represents the maximum number of vehicles that can pass through a given point or intersection in an hour under ideal conditions, generally expressed in vehicles per hour (veh/h). It's used for designing signal timings to ensure that traffic flows as efficiently as possible through an intersection.

On the other hand, traffic signal timing involves allocating time for each movement of traffic at an intersection. A typical traffic signal cycle consists of a green phase, a yellow (amber) phase, and a red phase. The "effective red time" mentioned in the problem is the time when vehicles on a particular approach cannot proceed through the intersection because the signal is red.

Vehicle delay at intersections is an important measure of the level of service and efficiency of an intersection. Delays can be caused by multiple factors, including traffic volume, signal timing, and intersection layout. Minimizing delays is crucial for reducing congestion and improving traffic flow.