Question

Find the expected waiting time in traffic lights. A driver must pass 10 traffic lights on a certain route. Each light has a period red-yellow-green-yellow of two minutes, of which the green and yellow lights last for 70 seconds. Suppose the driver arrives at a traffic light at some uniformly distributed random point of time during the period of two minutes. Compute the corresponding waiting time. Repeat this for 10 traffic lights. Run a large number of routes (i.e., repetitions of passing 10 traffic lights) and let the program write out the average waiting time. Does the computed time coincide with what you would expect? Name of program file: waiting_time.py.

Short answer

The expected waiting time for 10 traffic lights is 86.8 seconds.

Step-by-step solution

Understanding the Traffic Light Cycle

Each traffic light has a cycle of 120 seconds (or 2 minutes total), consisting of 70 seconds of green and yellow lights, and thus 50 seconds of red light.

Calculating Waiting Time for One Traffic Light

Since the driver arrives at a random point, waiting time occurs if they hit the red light. The waiting time is uniformly distributed between 0 and 50 seconds if they arrive during the red light. The expected waiting time when arriving during red is \( \frac{0+50}{2} = 25 \) seconds.

Computing No Waiting Time Situation

If the driver arrives during the 70-second green/yellow period, there is no waiting time. The probability that the driver arrives during this period is \( \frac{70}{120} = \frac{7}{12} \).

Probability of Waiting and Expected Waiting Time at a Single Light

The probability of arriving during the red light period (and thus having a waiting time) is \( \frac{50}{120} = \frac{5}{12} \). The expected waiting time considering both scenarios is calculated as: \( E[X] = \frac{5}{12} \times 25 + \frac{7}{12} \times 0 = \frac{125}{144} \approx 8.68 \) seconds.

Total Expected Waiting Time for 10 Traffic Lights

Multiply the expected waiting time for one light by the 10 traffic lights to get the total expected waiting time: \( 10 \times 8.68 = 86.8 \) seconds.

Simulating Many Routes and Calculating Average

Write a Python program named `waiting_time.py` that simulates a large number of runs passing 10 traffic lights. Use a random number generator to decide the start time at each light and calculate the waiting time over many iterations to compute the average.

Key concepts

Python Programming

Python is a versatile and powerful programming language that is widely used for simulations and complex calculations, such as estimating waiting times at traffic lights. Simulating scenarios like the traffic light problem involves using random numbers to mimic real-world randomness. Python provides libraries like `random`, which make generating random numbers straightforward.

To simulate the traffic light scenario, you'll need to create a Python script. Begin by defining the traffic light timing cycle, including the periods for red, green, and yellow lights. This can be represented using variables. Then, program the script to simulate the driver arriving at random times at each light.

By using for-loops, you can iterate over each traffic light, using the `random.uniform` function to generate random arrival times. This function helps model the uniform distribution of arrival times, essential for the simulation's accuracy.

Expected Waiting Time

The expected waiting time is a crucial concept in understanding how long a driver has to wait at a traffic light. It's essentially the average time spent waiting when considering all possible scenarios within the light's cycle. This concept helps us understand what to "expect" over a long period or multiple trials.

In the traffic light problem, the cycle includes both wait times during red lights and no wait times during the green/yellow period. To calculate the expected waiting time at a light, we determine the probable waiting time for both scenarios.

The formula for expected waiting time at a light that includes both scenarios is: \[ E[X] = p_1 imes ext{expected wait if waiting} + p_2 imes ext{expected wait if not waiting} \]

Where:

• \( p_1 \) is the probability of arriving during a red light, which means waiting.
• \( p_2 \) is the probability of arriving during a green/yellow light, resulting in no wait.

Using this, we calculate the expected waiting time for one light and then extrapolate it to multiple lights to find the total expected time.

Probability

Probability plays a vital role in determining how often the driver will encounter red lights and thus face waiting times. In this context, probability helps quantify the likelihood of various outcomes within the traffic light's cycle.

Each traffic light cycle has a distinct red light period, and the probability of arriving during this period is the ratio of the red light's duration to the entire cycle's duration. For our scenario, if the red light lasts 50 seconds, and the entire cycle lasts 120 seconds, the probability of arriving during the red light is: \[ \frac{50}{120} = \frac{5}{12} \]

Similarly, the probability of arriving during the no-wait green/yellow phase is: \[ \frac{70}{120} = \frac{7}{12} \]

These probabilities are central in calculating the expected waiting time, as they represent how often each part of the light cycle can be experienced by the driver.

Monte Carlo Simulation

Monte Carlo Simulation is a statistical method used to model the probability of various outcomes in processes that involve random variables, like traffic light cycles. It uses random sampling to obtain numerical results, making it well-suited for complex systems where analytical solutions are not easily available.

In the context of the traffic light problem, Monte Carlo simulation is used to estimate the average waiting time. By simulating numerous traffic light encounters, we can approximate the expected waiting time across many trials.

This simulation involves running the `waiting_time.py` program on a large number of iterations or simulated routes. For each iteration, the program randomly determines when the driver arrives at each light, calculates the waiting time, and keeps a cumulative average over many iterations.

The accuracy of the simulation improves with the number of iterations because more samples provide a clearer picture of the expected value, reducing the uncertainty inherent in any one trial.