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---
title: "The Bank Tutorial: Part II"
author: "Duncan Garmonsway"
date: "`r Sys.Date()`"
output:
rmarkdown::html_vignette:
toc: yes
vignette: >
%\VignetteIndexEntry{04. The Bank Tutorial: Part II}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
```{r, cache = FALSE, include=FALSE}
knitr::opts_chunk$set(collapse = T, comment = "#>",
fig.width = 6, fig.height = 4, fig.align = "center")
required <- c("simmer.plot")
if (!all(sapply(required, requireNamespace, quietly = TRUE)))
knitr::opts_chunk$set(eval = FALSE)
```
## Introduction
This tutorial is adapted from a tutorial for the Python 2 package 'SimPy',
[here](https://pythonhosted.org/SimPy/Tutorials/TheBank2.html). Users familiar
with SimPy may find this tutorial helpful for transitioning to `simmer`. Some
very basic material is not covered. Beginners should first read
[_The Bank Tutorial: Part I_](simmer-04-bank-1.html).
## Priority customers
In many situations there is a system of priority service. Those customers with
high priority are served first, those with low priority must wait. In some
cases, preemptive priority will even allow a high-priority customer to interrupt
the service of one with a lower priority.
Simmer implements priority requests with an extra integer priority argument to
`add_generator()`. By default, priority is zero; higher integers have higher
priority. For this to operate, the resource must have been created with
`preemptive = TRUE`.
### Priority customers without preemption
In the first example, we modify the program with random arrivals, one counter,
and a fixed service time (like _One Service counter_ in _The Bank Tutorial: Part
I_) to process a high priority customer.
Here, we give each customer a priority. Since the default is `priority = 0` this
is easy for most of them.
To observe the priority in action, while all other customers have the default
priority of 0, we create and activate one special customer, Guido, with priority
1 who arrives at time 23. This is to ensure that he arrives after Customer2.
Since the activity trace does not produce the waiting time by default, this is
calculated and appended using the `transform` function.
```{r, message = FALSE}
library(simmer)
set.seed(1933)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
set_attribute("start_time", function() {now(bank)}) %>%
log_(function() {
paste("Queue is", get_queue_count(bank, "counter"), "on arrival")
}) %>%
seize("counter") %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
timeout(12) %>%
release("counter") %>%
log_("Completed")
bank <-
simmer("bank") %>%
add_resource("counter") %>%
add_generator("Customer", customer, function() {c(0, rexp(4, 1/10), -1)}) %>%
add_generator("Guido", customer, at(23), priority = 1)
bank %>% run(until = 400)
bank %>%
get_mon_arrivals() %>%
transform(waiting_time = end_time - start_time - activity_time)
```
The output above displays the number of customers in the queue just as each one
arrives. That count does not include any customer in service.
Reading carefully one can see that when Guido arrives Customer0 has been served
and left at 12, Customer1 is in service and two (customers 2 and 3) are
queueing. Guido has priority over those waiting and is served before them at 24.
When Guido leaves at 36, Customer2 starts service.
### Priority customers with preemption
Now we allow Guido to have preemptive priority. He will displace any customer in
service when he arrives. That customer will resume when Guido finishes (unless
higher priority customers intervene). It requires only a change to one line of
the program, adding the argument, `preemptive = TRUE` to the `add_resource`
function call.
```{r, message = FALSE}
library(simmer)
set.seed(1933)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
set_attribute("start_time", function() {now(bank)}) %>%
log_(function() {
paste("Queue is", get_queue_count(bank, "counter"), "on arrival")
}) %>%
seize("counter") %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
timeout(12) %>%
release("counter") %>%
log_("Completed")
bank <-
simmer("bank") %>%
add_resource("counter", preemptive = TRUE) %>%
add_generator("Customer", customer, function() {c(0, rexp(4, 1/10), -1)}) %>%
add_generator("Guido", customer, at(23), priority = 1)
bank %>% run(until = 400)
bank %>%
get_mon_arrivals() %>%
transform(waiting_time = end_time - start_time - activity_time)
```
Though Guido arrives at the same time, 23, he no longer has to wait and
immediately goes into service, displacing the incumbent, Customer1. That
customer had already completed 23 - 12 = 11 minutes of his service. When Guido
finishes at 35, Customer1 resumes service and takes 36 - 35 = 1 minutes to
finish. His total service time is the same as before (12 minutes).
## Balking and reneging customers
Balking occurs when a customer refuses to join a queue if it is too long.
Reneging (or, better, abandonment) occurs if an impatient customer gives up
while still waiting and before being served.
### Balking customers
Another term for a system with balking customers is one where “blocked
customers” are “cleared”, termed by engineers a BCC system. This is very
convenient analytically in queueing theory and formulae developed using this
assumption are used extensively for planning communication systems. The easiest
case is when no queueing is allowed.
As an example let us investigate a BCC system with a single server but the
waiting space is limited. We will estimate the rate of balking when the maximum
number in the queue is set to 1. On arrival into the system the customer must
first check to see if there is room. If there is not enough room, the customer
balks.
To get the balking rate, we first count the number of arrivals that didn't
finish, using the data given by `get_mon_arrivals()`. Then we divide it by the
current model time from `now(bank)`.
```{r}
library(simmer)
timeInBank <- 12 # mean, minutes
ARRint <- 10 # mean, minutes
numServers <- 1 # servers
maxInSystem <- 2 # customers
maxInQueue <- maxInSystem - numServers
maxNumber <- 8
maxTime <- 400 # minutes
set.seed(59098)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
log_("Here I am") %>%
set_attribute("start_time", function() {now(bank)}) %>%
seize("counter",
continue = FALSE,
reject =
trajectory("Balked customer") %>%
log_("BALKING")) %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
timeout(function() {rexp(1, 1/timeInBank)}) %>%
release("counter") %>%
log_("Finished")
bank <-
simmer("bank") %>%
add_resource("counter",
capacity = numServers,
queue_size = maxInQueue) %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(maxNumber - 1, 1 / ARRint)))))
bank %>% run(until = maxTime)
number_balked <- sum(!get_mon_arrivals(bank)$finished)
paste("Balking rate is", number_balked / now(bank), "customers per minute.")
```
When Customer2 arrives, Customer0 is already in service and Customer1 is
waiting. There is no room, so Customer2 balks. By the vagaries of exponential
random numbers, Customer0 takes a very long time to serve (22.7358 minutes) so
the first one to find room is number Customer6 at 25.5339.
### Reneging (or abandoning) customers
Often in practice an impatient customer will leave the queue before being
served. Simmer can model this reneging behaviour using the `renege_in()`
function in a trajectory. This defines the maximum time that a customer will
wait before reneging, as well as an 'out' trajectory for them to follow when
they renege.
If the customer reaches the server before reneging, then their impatience must
be cancelled with the `renege_abort()` function.
```{r, message = FALSE}
library(simmer)
timeInBank <- 15 # mean, minutes
ARRint <- 10 # mean, minutes
numServers <- 1 # servers
maxNumber <- 5
maxTime <- 400 # minutes
maxWaitTime <- 12 # minutes, maximum time to wait before reneging
set.seed(59030)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
log_("Here I am") %>%
set_attribute("start_time", function() {now(bank)}) %>%
renege_in(maxWaitTime,
out = trajectory("Reneging customer") %>%
log_(function() {
paste("Waited", now(bank) - get_attribute(bank, "start_time"), "I am off")
})) %>%
seize("counter") %>%
renege_abort() %>% # Stay if I'm being attended within maxWaitTime
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
timeout(function() {rexp(1, 1/timeInBank)}) %>%
release("counter") %>%
log_("Completed")
bank <-
simmer("bank") %>%
add_resource("counter",
capacity = numServers) %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(maxNumber - 1, 1 / ARRint)))))
bank %>% run(until = maxTime)
```
Customer1 arrives after Customer0 but has only 12 minutes patience. After that
time in the queue (at time 28.5058) he abandons the queue to leave Customer2 to
take his place. Customer2 and Customer3 also renege. Customer4 is served within
12 minutes.
## Interrupting a process
Klaus goes into the bank to talk to the manager. For clarity we ignore the
counters and other customers. During his conversation his cellphone rings. When
he finishes the call he continues the conversation.
In this example, the call is another trajectory, whose only activities are to
send a signal (the ringing of the phone), and to write that event to the log.
In Klaus' trajectory, the `trap` activity causes him to listen for the phone to
ring. Supposing the phone doesn't ring, then his trajectory would continue to
the `timeout` activity, where he would do his banking business for 20 minutes,
and then finish.
Supposing the phone _does_ ring, then Klaus would enter the sub-trajectory
defined within the `trap` function as a 'handler'. In that trajectory, he makes
his excuses, answers the phone, then returns to business. At the end of the
trajectory, he continues the original trajectory at the next step _following_
the original `timeout`, without spending the rest of the 20 minutes on his
banking.
To make Klaus spend a full 20 minutes banking, we add a `timeout` activity to
the end of the 'handler', but first we have to calculate how much time remains
after the interruption. This is done by storing the 'start' time in an
attribute, and calculating how much time is left when the phone rings.
By default, interruptions can themselves be interrupted, as illustrated in this
example by the phone ringing twice. This could be avoided by setting
`interruptible = FALSE` in the `trap` activity.
```{r, message = FALSE}
library(simmer)
timeInBank <- 20
timeOfCall <- 9
onphone <- 3
maxTime <- 100
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
trap("phonecall",
handler = trajectory() %>%
log_("Excuse me") %>%
set_attribute(
"timeleft", function() {
sum(get_attribute(bank, c("timeleft", "start"))) - now(bank)
}) %>%
log_("Hello! I'll call back") %>%
timeout(onphone) %>%
log_("Sorry, where were we?") %>%
set_attribute("start", function() {now(bank)}) %>%
log_(function() {paste("Time left:", get_attribute(bank, "timeleft"))}) %>%
timeout_from_attribute("timeleft")
) %>%
log_("Here I am") %>%
set_attribute("timeleft", timeInBank) %>%
set_attribute("start", function() {now(bank)}) %>%
timeout(timeInBank) %>%
log_("Completed")
phone <-
trajectory("Phone") %>%
log_("Ringgg!") %>%
send("phonecall")
bank <-
simmer("bank") %>%
add_generator("Klaus", customer, at(0)) %>%
add_generator("Phone", phone, at(timeOfCall, timeOfCall + 7))
bank %>% run(until = maxTime)
```
As this has no random numbers the results are reasonably clear: the first
interrupting call occurs at 9. It takes Klaus 3 minutes to listen to the message
and he resumes the conversation with the bank manager at 12. The phone rings
again at 16, he listens for three more minutes, and resumes the conversation at
19, finally finishing at 26. The total time of conversation is 9 + 4 + 7 = 20
minutes, the same as it would have been if the interrupt had not occurred.
## Wait until the bank door opens
Customers arrive at random, some of them getting to the bank before the door is
opened by a doorman. They wait for the door to be opened and then rush in and
queue to be served.
This model defines the door as a resource, just like the counter. The capacity
of the door is defined according to the `schedule` function, so that it has zero
capacity when it is shut, and infinite capacity when it is open. Customers
'seize' the door and must then wait until it has capacity to 'serve' them. Once
it is available, all waiting customers are 'served' immediately (i.e. they pass
through the door). There is no `timeout` between 'seizing' and 'releasing' the
door.
For the sake of announcing in the log that the door has been opened, a doorman
trajectory is defined.
```{r, message = FALSE}
library(simmer)
maxTime = 400
set.seed(393937)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
log_(function()
if (get_capacity(bank, "door") == 0)
"Here I am but the door is shut."
else "Here I am and the door is open."
) %>%
seize("door") %>%
log_("I can go in!") %>%
release("door") %>%
seize("counter") %>%
timeout(function() {rexp(1, 1/10)}) %>%
release("counter")
openTime <- rexp(1, 1/10)
door_schedule <- schedule(c(0, openTime), c(0, Inf))
doorman <-
trajectory() %>%
timeout(openTime) %>%
log_("Ladies and Gentlemen! You may all enter.")
bank <-
simmer("bank") %>%
add_resource("door", capacity = door_schedule) %>%
add_resource("counter") %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(5 - 1, 0.1))))) %>%
add_generator("Doorman", doorman, at(0))
bank %>% run(until = maxTime)
bank %>%
get_mon_arrivals() %>%
transform(waiting_time = end_time - start_time - activity_time)
```
The output above programs shows how the first two customers have to wait until
the door is opened.
# Wait for the doorman to give a signal
Customers arrive at random, some of them getting to the bank before the door is
open. This is controlled by an automatic machine called the doorman which opens
the door only at intervals of 30 minutes (it is a very secure bank). The
customers wait for the door to be opened and all those waiting enter and proceed
to the counter. The door is closed behind them.
There are at least two ways to implement this model. The first example uses a
schedule, and the second uses batching.
The principle behind the schedule is that the door is modelled as a server with
zero capacity for 30 minutes, then infinite capacity for zero minutes, then
repeat the 30-minute cycle. In the moment that it has infinite capacity, all
the customers will pass through the door (i.e. they will be 'served').
For the sake of announcing in the log that the door has been opened, a doorman
trajectory is defined. The doorman has a `rollback` step so that it keeps
opening and shutting the door every 30 minutes for ever.
```{r, message = FALSE}
library(simmer)
maxTime = 150
customer <-
trajectory("Customer's path") %>%
log_("Here I am, but the door is shut.") %>%
set_attribute("start_time", function() {now(bank)}) %>%
seize("door") %>%
log_("The door is open!") %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
release("door") %>%
seize("counter") %>%
timeout(function() {rexp(1, 1/10)}) %>%
release("counter") %>%
log_("Finished.")
door_schedule <- schedule(c(30, 30), c(Inf, 0), period = 30)
doorman <-
trajectory("Doorman") %>%
timeout(30) %>%
log_("You may enter.") %>%
rollback(2, times = Inf)
set.seed(393939)
bank <- simmer("bank")
bank %>%
add_resource("door", capacity = door_schedule) %>%
add_resource("counter") %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(5 - 1, 0.1))))) %>%
add_generator("Doorman", doorman, at(0))
bank %>% run(until = maxTime)
bank %>%
get_mon_arrivals() %>%
transform(waiting_time = end_time - start_time - activity_time)
```
The output run for this program shows how the first two customers have to wait
until the door is opened, and then the next three have to wait.
The second method is batching. Customers can be collected into batches of a
given size, or for a given time, or whichever occurs first. Here, they are
collected for periods of 30, and the number of customers in each batch is
unrestricted.
After the batch is created with `batch`, usually the customers will all be
processed together by a server, before separating with `separate`. In this
example, there is no need for a server -- the door is modelled by the batch
itself -- so the customers are separated immediately after the batch.
```{r, message = FALSE}
library(simmer)
maxTime = 150
customer <-
trajectory("Customer's path") %>%
log_("Here I am, but the door is shut.") %>%
set_attribute("start_time", function() {now(bank)}) %>%
batch(n = Inf, timeout = 30) %>%
separate() %>%
log_("The door is open!") %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
seize("counter") %>%
timeout(function() {rexp(1, 1/10)}) %>%
release("counter") %>%
log_("Finished.")
doorman <-
trajectory("Doorman") %>%
timeout(30) %>%
log_("You may enter.") %>%
rollback(2, times = Inf)
set.seed(393939)
bank <- simmer("bank")
bank %>%
add_resource("door") %>%
add_resource("counter") %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(5 - 1, 0.1))))) %>%
add_generator("Doorman", doorman, at(0))
bank %>% run(until = maxTime)
bank %>%
get_mon_arrivals() %>%
transform(waiting_time = end_time - start_time - activity_time)
```
This second method gives the same output as the first.
## Monitors
Monitors record events in a simulation. Unlike SimPy, Simmer does this by
default, so the records are available after -- in fact, even during -- a
simulation. Data that is collected by monitors is available from the following
functions:
```{r, eval = FALSE}
get_capacity
get_mon_arrivals
get_mon_attributes
get_mon_resources
get_n_activities
get_n_generated
get_queue_count
get_queue_size
get_server_count
```
The `get_mon_*()` functions return a data frame (which is growing during the
simulation, so that, although possible, it is computationally expensive to call
them from inside a trajectory). The others return a numeric value.
### Plotting a histogram of monitor results
A histogram of the amount of time that customers spend in the
bank can be plotted by taking some basic information -- start and end time of
each customer -- from the `get_mon_arrivals()` function, and then calculating
the elapsed time. This example draws the plot with the `ggplot2` package, but
other plotting packages, and base R graphics, could do something similar.
```{r}
library(simmer)
library(simmer.plot)
# library(ggplot2) # (automatically loaded with simmer.plot)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
seize("counter") %>%
timeout(12) %>%
release("counter")
set.seed(393939)
bank <-
simmer("bank") %>%
add_resource("counter") %>%
add_generator("Customer",
customer,
at(c(0, cumsum(rexp(20 - 1, 0.1)))))
bank %>% run(400)
bank %>%
get_mon_arrivals %>%
ggplot(aes(end_time - start_time)) +
geom_histogram() +
xlab("Time spent in the system") +
ylab("Number of customers")
```
### Monitoring a resource
Now consider observing the number of customers waiting or active in a Resource.
`get_mon_resources()` returns a table of states and times for the counter.
Whenever a customer enters/leaves a queue/counter, a new row is created
recording the number of customers in the queue, the counter and the system as a
whole. We call this the 'state'. The amount of time that the state lasts is
the difference in time between one state and the next, which we calculate with
the `diff()` function. Finally, we multiply the number of customers in each
state by the duration of the state, and divide by the duration of the simulation
(the time at the end) to get the average.
```{r, message = FALSE}
library(simmer)
set.seed(1234)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
log_("Arrived") %>%
set_attribute("start_time", function() {now(bank)}) %>%
seize("counter") %>%
log_("Got counter") %>%
log_(function() {paste("Waited", now(bank) - get_attribute(bank, "start_time"))}) %>%
timeout(12) %>%
release("counter") %>%
log_("Finished")
bank <-
simmer("bank") %>%
add_resource("counter") %>%
add_generator("Customer", customer, function() {c(0, rexp(4, 1/10), -1)})
bank %>% run(until = 400)
customer_monitor <-
get_mon_arrivals(bank) %>%
transform(wait = end_time - start_time - activity_time)
mean_waiting_time <- mean(customer_monitor$wait)
resource_monitor <- get_mon_resources(bank)
queue_state <- head(resource_monitor$queue, -1)
server_state <- head(resource_monitor$server, -1)
time_state_lasted <- diff(resource_monitor$time)
time_at_end <- max(resource_monitor$time)
mean_active_customers <- sum(server_state * time_state_lasted) / time_at_end
mean_waiting_customers <- sum(queue_state * time_state_lasted) / time_at_end
cat(" Average waiting = ", mean_waiting_customers, "\n",
"Average active = ", mean_active_customers, "\n")
```
### Plotting from resource monitors
All the `get_mon_*()` return information that enables us to graph the output.
Alternative plotting packages can be used; here we use the `simmer.plot`
package just to graph the number of customers waiting for the counter.
The `simmer.plot` package is imported at line 2. The function
`get_mon_resources()` is not called because the function `plot()` calles
`get_mon_resources()` itself. The `plot()` function has arguments to specifiy
what to plot.
```{r, message = FALSE}
library(simmer)
library(simmer.plot)
timeInBank <- 12 # mean, minutes
set.seed(1234)
bank <- simmer()
customer <-
trajectory("Customer's path") %>%
set_attribute("start_time", function() {now(bank)}) %>%
seize("counter") %>%
timeout(function() {rexp(1, 1/timeInBank)}) %>%
release("counter")
bank <-
simmer("bank") %>%
add_resource("counter") %>%
add_generator("Customer", customer, function() {c(0, rexp(19, 1/10), -1)})
bank %>% run(until = 400)
plot(bank,
what = "resources",
metric = "usage",
names = "counter",
items = "system",
steps = TRUE)
```