• Poisson Process: A method that helps to count independent, random events over a specified time interval

  • Poisson Distribution: A distribution that counts the number of times the event occurs over a specified time interval

• Requirement: \(\lambda\) | The average number of events within a certain time interval

• Assumption: Events are discrete, do not overlap, and each event is independent

• Application: The Poisson Process can help us model and estimate bus arrival times given the estimated frequency. It’s minimal requirements pose a resource-efficient method of modelling station frequencies

The Poisson Distribution is determined with the following equation:

\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\]

• \(k\): The number of occurrences

• \(\lambda\): The average number of events per interval (for this, we’ll use hours)

Explanation: This formula calculates the probability that the number of occurrences in a specific window of time will equal the number of events that we want (\(k\)).

• Expected value is equivalent to the parameter \(\lambda\):

\[E[X] = \lambda\]

• Variance is also equivalent to the average occurrence:

\[Var(X) = \lambda\]

• Naturally, we can derive the standard deviation:

\[\sigma = \sqrt{\lambda}\]

• Essentially, if we expect 3 busses to arrive in an hour, we are uncertain that those three will actually show up. No specific outcome is guaranteed.

We see that when \(\lambda = 3\), the most expected number of arrivals hovers around 3 as well.

With a lower theoretical arrival rate (\(\lambda = 2\), blue), there is less variance and a much higher probability of the theoretical arrival rate being met. Higher theoretical arrival rates (\(\lambda = 8\), red) yield more uncertainty, and a lower chance of meeting the theoretical rate.

This demonstrates the concept from the previous slide. The lower the anticipated rate, the more likely it is to be met.

To find a Poisson Distribution for arrivals in R, we use the dpois() function:

k_values <- 0:15 # This must be a vector
lambda <- 4 # Replace this with the average arrival rate

prob <- dpois(k_values, lambda)

• This creates a starting point from which to visualize, simulate and experiment.

To generate single events following the poisson distribution, use rpois():

vars_to_return <- 20 # Each yields a single event, 
                     # based on the poisson distribution
lambda <- 4

prob <- rpois(vars_to_return, lambda)

We can simulate the arrival and departure of transit using randomly generated values based on the poisson distribution:

Note that as the sample size increases, the simulation (red) tends towards the ideal distribution (pink).

• We can use the Poisson Distribution to model the arrival frequencies of transit vehicles at stations.

• To model a real world system, we only need \(\lambda\), the theoretical average of arrivals over a time interval.

  • This can be obtained using GTFS data, typically available through a transit agency or municipal data portal.

Continuing Work: Curious readers could verify the accuracy of the poisson distribution as a model for station frequencies at specific localities. Perhaps differences in planning and geography affect this distribution’s utility as a model.