Waiting For The Bus: How Long Is Too Long?

Waiting for public transport can be a tedious task, especially when you're in a hurry. In theory, buses are supposed to arrive at regular intervals, but in reality, they are often delayed or early, which can be frustrating for commuters. This phenomenon is known as the waiting-time paradox. The waiting time paradox suggests that when buses are supposed to arrive at regular intervals, your average waiting time will be the same as the interval between buses. For example, if buses are scheduled to arrive every 10 minutes, your average wait will also be 10 minutes. However, this doesn't account for the fact that buses rarely arrive exactly on schedule, and you might end up waiting longer than expected.

The waiting time paradox is not limited to buses; it can also be observed in other situations like social interactions or the spread of diseases. In general, it's a good idea to arrive at the bus stop a few minutes early to avoid missing your bus, as public transportation systems are designed to be reliable and predictable.

The waiting time paradox

If buses are arriving every 10 minutes, you might expect your average waiting time to be 5 minutes. However, in reality, buses do not arrive exactly on schedule, and so you might wait longer.

For example, consider a bus schedule where buses arrive at either 4-minute or 12-minute intervals. In each hour, six buses arrive after a 4-minute interval and three buses arrive after a 12-minute interval. If you arrive at the bus stop at a random time, you are more likely to encounter a 12-minute interval than a 4-minute one, as the former occupies more numbers on a game wheel.

This paradox can be further understood through simulation and probabilistic arguments. By simulating a stream of buses that arrive at an average of 10 minutes, we can calculate the average waiting time for passengers. This simulation assumes that bus arrival times follow a Poisson process, which implies exponentially distributed arrival intervals.

However, real-world bus arrival times may not always follow a Poisson process. By analysing bus arrival data from Seattle, it was found that the average waiting time was longer than half of the average interval between buses but not equal to the scheduled interval as the waiting time paradox implied.

In conclusion, while the waiting time paradox provides interesting insights into probability and simulation, it is important to be cautious about the assumptions made during data analysis.

Bus route maps

Firstly, it is important to note that bus route maps may not always show individual bus stops. In some cases, you may need to look out for bus stop poles or shelters and ensure the bus number you are waiting for is correct. However, there are also limited bus services that stop less frequently and may be marked on the map. These services are different from express buses, which are typically used for suburban commuting. Additionally, select bus services may allow you to enter through any door and require you to purchase a ticket from a machine before boarding.

To find specific bus stop locations, you can use a bus time app or visit the bus service's website. These resources can provide detailed information about bus routes and stops, ensuring you know exactly where to wait for your bus. When using a bus time app, you can also track the real-time location of the bus and estimate your waiting time.

In some cities, such as New York, bus routes and schedules are available on the official transportation website. These schedules include individual route maps, providing a comprehensive overview of the bus network. Additionally, certain cities may offer printed maps at transit stations or hotels, which can be useful if you prefer a physical copy.

It is also worth noting that bus routes and stops can change over time, so it is always a good idea to check for the latest updates before planning your journey. By utilising bus route maps and complementary resources, you can confidently navigate your way around a new city and make the most of the bus transportation system.

Boarding the bus

Once you've found your route and have arrived at the bus stop, there are a few things to keep in mind when boarding the bus. Firstly, make sure that you arrive at the bus stop a few minutes before the bus is scheduled to arrive, as public transportation systems are often timely, and arriving just a minute or two late could mean missing your bus. As the bus approaches, check the digital banner on the front or side of the bus to ensure that it is the correct route.

Once the bus has come to a complete stop, stand back from the door to allow passengers to exit before you get on. If necessary, you can ask the driver to lower the bus to make boarding easier. After you've stepped onto the bus, you'll need to pay the required fare. If you have a bus pass, show it to the driver and/or scan it. If you're paying in cash, put the exact change into the farebox, as most bus drivers are not authorised to give change. If you're unsure about the fare, check the farebox for a sign displaying the required amount. If you think you'll need to transfer to another bus later, ask the driver for a transfer slip as soon as you board.

Riding the bus

Finding Your Route:

First, you need to figure out which bus to take. Look at a bus route map, which you can usually find online or at local schools and businesses. These maps typically have coloured lines representing different bus routes and stops. Find the route that goes to your destination and note the arrival and departure times. If you need to transfer buses, look for intersecting routes and plan your trip accordingly. Many cities also offer online trip planning tools on their public transportation websites.

Boarding the Bus and Paying the Fare:

Before you board the bus, get a bus pass or cash to pay the fare. You can usually buy a pass online or at your city's public transportation office. If you're paying in cash, bring exact change, as bus drivers may not be able to make change for you. Arrive at the bus stop a few minutes early so you don't miss your bus. As the bus approaches, check the digital banner to ensure it's the correct route. Wait for passengers to get off before boarding, and then pay the required fare. If you need to transfer buses, ask the driver for a transfer slip.

Riding and Getting Off the Bus:

Once you're on the bus, find a seat or stand out of the way, holding onto a pole or handle if you're standing. Give priority seating at the front of the bus to seniors and people with disabilities. Try to minimise the amount of space you take up, especially during crowded times. Pull the signal cord or press the stop button about a block before your stop to let the driver know you need to get off. Typically, exit through the back door unless you need to use the front door due to a disability or if you need to remove your bike from the rack. After exiting, wait for the bus to depart before crossing the street.

By following these steps, you'll be a confident bus rider in no time!

Exiting the bus

Once the bus has stopped, wait until it has pulled away and it is safe to cross the road. If you are at a busy stop, press the crosswalk button and wait for the signal to cross. If it is quieter, look both ways and cross when clear to do so.

If you are elderly or disabled, the bus driver should be able to lower the bus to make exiting easier. Simply ask the driver to do so before you get off.

Finally, if you are transferring to another bus, don't forget to ask for a transfer slip when you pay your fare. This will ensure you are not charged again when you board the next bus.

Frequently asked questions