The relationship between intra-arrival times and count models.

durations_to_counts.R

library(tidyverse)
# https://www.kellogg.northwestern.edu/faculty/weber/decs-430/Notes%20on%20the%20Poisson%20and%20exponential%20distributions.pdf
# http://www.mscs.mu.edu/~jsta/issues/11(3)/JSTA11(3)p2.pdf

# Case 1: memoryless intra-arrival times. They're not correlated.
tibble::tibble(
  durations_between = rexp(3e4, rate = 20)
) %>%
  mutate(time = cumsum(durations_between)) %>%
  mutate(time_interval = cut(time, breaks = seq(0, 1e4, 1), labels = FALSE)) %>%
  count(time_interval) %>%
  pull(n) %>%
  head(-1) %>%
  fitdistrplus::fitdist("pois") %>%
  plot()

# Case 2: same, Gamma(shape = 1, rate) ~ Exp(rate)
tibble::tibble(
  durations_between = rgamma(3e4, shape = 1, rate = 20) # equivalent
) %>%
  mutate(time = cumsum(durations_between)) %>%
  mutate(time_interval = cut(time, breaks = seq(0, 1e4, 1), labels = FALSE)) %>%
  count(time_interval) %>%
  pull(n) %>%
  head(-1) %>%
  fitdistrplus::fitdist("pois") %>%
  plot()

# Case 3: clustered intra-arrival times.
tibble::tibble(
  durations_between = rgamma(3e4, shape = 0.5, rate = 20) # now with clustered times
) %>%
  mutate(time = cumsum(durations_between)) %>%
  mutate(time_interval = cut(time, breaks = seq(0, 1e4, 1), labels = FALSE)) %>%
  count(time_interval) %>%
  pull(n) %>%
  head(-1) %>%
  fitdistrplus::fitdist("nbinom") %>% # notice negative binomial
  plot()