---
title: "Queueing Systems"
author: "Iñaki Ucar"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{06. Queueing Systems}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, cache = FALSE, include=FALSE}
knitr::opts_chunk$set(collapse = T, comment = "#>", 
                      fig.width = 6, fig.height = 4, fig.align = "center")

required <- c("simmer.plot", "dplyr")

if (!all(sapply(required, requireNamespace, quietly = TRUE)))
  knitr::opts_chunk$set(eval = FALSE)
```

```{r, message=FALSE}
library(simmer)
library(simmer.plot)
set.seed(1234)
```

## M/M/c/k systems

In Kendall's notation, an M/M/c/k system has exponential arrivals (__M__/M/c/k), a $c$ servers (M/M/__c__/k) with exponential service time (M/__M__/c/k) and an $k-c$ positions in queue (M/M/c/__k__). For instance, a router may have several processors to handle packets, and the in/out queues are necessarily finite.

This is the simulation of an M/M/2/3 system (2 server, 1 position in queue). Note that the trajectory is identical to an M/M/1 system (one server, infinite queue).

```{r}
lambda <- 3
mu <- 4

mm23.trajectory <- trajectory() %>%
  seize("server", amount=1) %>%
  timeout(function() rexp(1, mu)) %>%
  release("server", amount=1)

mm23.env <- simmer() %>%
  add_resource("server", capacity=2, queue_size=1) %>%
  add_generator("arrival", mm23.trajectory, function() rexp(1, lambda)) %>%
  run(until=2000)
```

There are rejections when the queue is full.

```{r}
mm23.arrivals <- get_mon_arrivals(mm23.env)

rejection_rate <- mm23.arrivals %>%
  dplyr::summarise(rejection_rate = sum(!finished)/length(finished)) %>%
  dplyr::pull(rejection_rate)
rejection_rate
```

By solving the balance equations for this system, we obtain the following:

$$
\begin{aligned}
p_0 &= \left[ \sum_{n=0}^{c-1}\frac{r^n}{n!} + \sum_{n=c}^{k}\frac{r^n}{c!c^{n-c}} \right]^{-1}\\
N &= \sum_{n=0}^{\infty}nP_n = p_0\left(\sum_{n=0}^{c-n}\frac{nr^n}{n!} + \sum_{n=c}^{k}\frac{nr^n}{c!c^{n-c}}\right)
\end{aligned}
$$

where $r=\lambda/\mu$. Finally, we can see how the simulation quickly converges to the theoretical average number of customers in the system $N$:

```{r}
# Theoretical value
rho <- lambda/mu
div <- 1 / c(1, 1, factorial(2) * 2^(2:3-2))
mm23.N <- sum(0:3 * rho^(0:3) * div) / sum(rho^(0:3) * div)

# Evolution of the average number of customers in the system
plot(mm23.env, "resources", "usage", "server", items="system") +
  geom_hline(yintercept=mm23.N)
```

## Queueing Networks

Let us consider the following network of queues (example taken from [Grotto Networking](http://www.grotto-networking.com/DiscreteEventPython.html#NetQueues)):

![A network of queues.](includes/queueing-network.svg)

There are three exponential generators ($\lambda_1$, $\lambda_2$, $\lambda_4$) that inject messages with exponential size of mean 100 bytes. There are four M/D/1 queues with a deterministic rate equal to 220 bytes/s. There is a 25\% probability that messages from $\lambda_1$ are dropped before the second queue, and a fork 35/65\% for the output of the second queue to reach queues 3 and 4. We are interested, for instance, in the accumulated waiting time for messages entering queue 1 and exiting queue 3 or 4.

The original author of this example, Greg M. Bernstein, simulated this problem with [this Python code](http://www.grotto-networking.com/files/DESPython/QueueNet2.py). We'll start by setting up the main constants and a couple of functions to set the message size and seize an M/D/1 queue:

```{r}
mean_pkt_size <- 100        # bytes
lambda1 <- 2                # pkts/s
lambda3 <- 0.5              # pkts/s
lambda4 <- 0.6              # pkts/s
rate <- 2.2 * mean_pkt_size # bytes/s

# set an exponential message size of mean mean_pkt_size
set_msg_size <- function(.)
  set_attribute(., "size", function() rexp(1, 1/mean_pkt_size))

# seize an M/D/1 queue by id; the timeout is function of the message size
md1 <- function(., id)
  seize(., paste0("md1_", id), 1) %>%
  timeout(function() get_attribute(env, "size") / rate) %>%
  release(paste0("md1_", id), 1)
```

The next step is to set up the three points of attachment for our generators:

```{r}
to_queue_1 <- trajectory() %>%
  set_msg_size() %>%
  md1(1) %>%
  leave(0.25) %>%
  md1(2) %>%
  branch(
    function() (runif(1) > 0.65) + 1, continue=c(F, F),
    trajectory() %>% md1(3),
    trajectory() %>% md1(4)
  )

to_queue_3 <- trajectory() %>%
  set_msg_size() %>%
  md1(3)

to_queue_4 <- trajectory() %>%
  set_msg_size() %>%
  md1(4)
```

Finally, we prepare and run the simulation environment with the resources and generators required:

```{r}
env <- simmer()
for (i in 1:4) env %>% 
  add_resource(paste0("md1_", i))
env %>%
  add_generator("arrival1_", to_queue_1, function() rexp(1, lambda1), mon=2) %>%
  add_generator("arrival3_", to_queue_3, function() rexp(1, lambda3), mon=2) %>%
  add_generator("arrival4_", to_queue_4, function() rexp(1, lambda4), mon=2) %>%
  run(4000)
```

In the analysis, we filter arrivals from generator 1 that reach queues 3 and 4, and we examine the average waiting time and the total number of messages:

```{r}
res <- get_mon_arrivals(env, per_resource = TRUE) %>%
  dplyr::select(name, resource) %>%
  dplyr::filter(resource %in% c("md1_3", "md1_4"))
arr <- get_mon_arrivals(env) %>%
  dplyr::mutate(waiting_time = end_time - (start_time + activity_time),
                generator = regmatches(name, regexpr("arrival[[:digit:]]", name))) %>%
  dplyr::left_join(res) %>%
  dplyr::group_by(generator, resource)

dplyr::summarise(arr, average = sum(waiting_time) / n())
get_n_generated(env, "arrival1_") + get_n_generated(env, "arrival4_")
dplyr::count(arr)
```
