Markovian Distributions

Markovian distributions, though few meet this classification, are a very important in many queueing models and theory. They have one important property that is memorylessness. For distributions that are memoryless, the current sample from the distribution is not dependent on the prior sample. In queueing context, an entity’s current processing time is independent on how much longer it will take to process. Markovian distributions are the exponential, geometric, and poisson. The exponential and poisson are the most commonly used and are closely related. The shorthand notation is M.

library(ggplot2)
library(trstyles)
x <- data.frame(x = 0:5)
rate <- 1:5
rateGroups <- factor(1:5)
e <- ggplot(x, aes(x = x)) +
  labs(y = 'density', title = 'Exponential PDF') +
  theme_tr()
for (i in 1:5) {
e <- e +
  stat_function(fun = dexp, 
                aes_(color = rateGroups[i]),
                args = list(rate = rate[i]),
                size = 1)
}
e +
  scale_color_tr(name = "Rate",
                 breaks = rateGroups)

Deterministic Distributions

Deterministic means that the rate is always the same. It is a constant. The shorthand notation is D.

General Distributions

General probability distributions refer to when you make no assumptions on the shape and form. It can be any probability distribution. The shorthand notation is G. Below are some common distributions:

library(fitur)
library(actuar)
set.seed(438)
x <- rweibull(10000, shape = 5, scale = 1)
dists <- c('gamma', 'lnorm', 'weibull', 'gamma', 'llogis')
fits <- lapply(dists, fit_univariate, x = x)
plot_density(x, fits, nbins = 30) +
  scale_color_tr(name = 'Distribution') +
  theme_tr()