21.4.4: Algorithmic Thinking

Problem:

If the inter-arrival time is Exponentially distributed with average time of 20 minutes, then what is the distribution of the number of people who show up in an 8 hour period?

Algorithmic thinking is crucial in statistics as it enables a structured approach to solving complex problems by breaking them into smaller, logical steps. This mindset is particularly valuable when designing workflows for data analysis, from data preprocessing and visualization to modeling and interpretation. By thinking algorithmically, statisticians can systematically address challenges such as cleaning messy data, optimizing computational efficiency, or automating repetitive tasks.

Moreover, algorithmic thinking facilitates the translation of statistical concepts into code, allowing for reproducibility, scalability, and adaptability in analysis. In a field increasingly reliant on computational tools, developing this skill ensures that statisticians can efficiently tackle problems and adapt to new methods or technologies. This example illustrates algorithmic thinking to arrive at an interesting result.

Solution:

This is a tough question but let's break it down into its parts, then simulate it.

First, let's simulate the inter-arrival time (the time between arrivals).

iat = rexp(100, rate=3)     ### 'iat' is in hours

From this, we can see that the total time between the 1st and the 20th arrival would be the sum of 20 of those inter-arrival times:

iat = rexp(100, rate=3)
sum(iat[1:20])

On the other hand, the number of arrivals in an hour would be

iat = rexp(100, rate=3)
iatCS = cumsum(iat)
max(which(iatCS <= 1))

So, the number of arrivals in eight hours would be

iat = rexp(100, rate=3)
iatCS = cumsum(iat)
max(which(iatCS <= 8))

To get the distribution of those "number of arrivals in eight hours," you just need to repeat the code many times, saving the number of arrivals each time:

arrNum = numeric()

for(i in 1:1e6) {
  iat = rexp(100, rate=3)
  iatCS = cumsum(iat)
  arrNum[i] = max(which(iatCS <= 8))
}

The histogram below shows the estimated distribution of the number of customers arriving in those 8 hours.

Closely examining the histogram and thinking about the distributions you should have already learned, it is clear that the number of arrivals in 8 hours follows this distribution

Number of Arrivals ~ Poisson(λ=24)

To make it even more manifest, overlaying the histogram with a graph of that Poisson distribution illustrates this (the dots on the histogram).

hist(arrNum, freq=FALSE, breaks=seq(1,50)-0.5)
points(1:50, dpois(1:50, lambda=24), pch=16)

Note that this is not proof of the relationship between the two distributions. It merely suggests the relationship. Your probability theory course will give you the tools to actually prove the relationship.

Extensions:

Extension 1: Change all of the time measurements in the previous example from hours to minutes. Make sure that the conclusions are the same.

Extension 2: Change the inter-arrival time to 10 minutes. What is the distribution of the number of arrivals in three hours? Show it using the histogram.