Add new vignette or orther support material to describe distribution of WLs

[chrismainey]

Reference is made in the simulation vignette to proportions of the distributions. It would be helpful to have a website section, so a vignette that discusses the arrivals being Poisson processes and the exponential distribution of the queue. This can then be cast as NHS 18 weeks and proportions of queue meeting target requiring fixed mean waits. @ThomUK has a nice plot for this already, which I'll wrap something around.

[ThomUK]

Here is the plot, and the code for it (I updated it to add the mean wait lines for each distribution)

# exponential density data
distributions_df <- tibble::tibble(
  x = seq(0, 52, length.out = 1000),
  `6` = dexp(x, 1/6),
  `9` = dexp(x, 1/9),
  `12` = dexp(x, 1/12),
  `15` = dexp(x, 1/15),
  `18` = dexp(x, 1/18)
) |>
  pivot_longer(-x) |>
  mutate(
    `Mean waiting time (weeks)` = factor(
      name,
      levels = c("6", "9", "12", "15", "18"),
      labels = c(
        "6 weeks  (95% meets 18wk standard)",
        "9 weeks  (86% meets 18wk standard)",
        "12 weeks  (77% meets 18wk standard)",
        "15 weeks  (69% meets 18wk standard)",
        "18 weeks  (63% meets 18wk standard)"
      )
    )
  )

ggplot(distributions_df, aes(x = x), group = `Mean waiting time (weeks)`) +
  scale_color_viridis_d(direction = -1) +
  scale_fill_viridis_d(direction = -1) +
  scale_y_continuous(labels = scales::label_percent()) +
  geom_vline(aes(xintercept = as.numeric(name), colour = `Mean waiting time (weeks)`), linetype = "longdash") +
  geom_line(aes(y = value, colour = `Mean waiting time (weeks)`), linewidth = 1) +
  geom_area(aes(y = value, fill = `Mean waiting time (weeks)`), alpha = 0.1, position = 'identity') +
  geom_vline(xintercept=18, linewidth = 1, colour = "coral")+
  annotate("text", x=18, y=0.075, hjust=-0.2,label="18 weeks",  colour = "coral")+
  theme_minimal() +
  theme(legend.position = c(0.80, 0.5),
        legend.background = element_rect(fill = "white", color = "grey")) +
  labs(
    title = "Distribution of waiting times by mean wait",
    subtitle = "Each curve represents the same number of patients",
    x = "Weeks",
    y = "Proportion of waiting list",
  )