Package 'queuecomputer' reference manual

Title:	Computationally Efficient Queue Simulation
Description:	Implementation of a computationally efficient method for simulating queues with arbitrary arrival and service times. Please see Ebert, Wu, Mengersen & Ruggeri (2020, <doi:10.18637/jss.v095.i05>) for further details.
Authors:	Anthony Ebert [aut, cre] , Kerrie Mengersen [ths], Paul Wu [ths], Fabrizio Ruggeri [ths]
Maintainer:	Anthony Ebert <[email protected]>
License:	GPL-2 \| file LICENSE
Version:	1.2.0
Built:	2024-11-08 05:29:33 UTC
Source:	https://github.com/anthonyebert/queuecomputer

Creates a `"server.list"` object from a list of times and starting availability.

Description

Creates a "server.list" object from a list of times and starting availability.

Usage

as.server.list(times, init)
as.server.list(times, init)

Arguments

`times`	list of numeric vectors giving change times for each server.
`init`	vector of 1s and 0s with equal length to `times`. It represents whether the server starts in an available `(1)` or unavailable `(0)` state.

Value

an object of class "server.list", which is a list of step functions of range {0, 1}.

Examples

# Create a server.list object with the first server available anytime before time 10,
# and the second server available between time 15 and time 30.
as.server.list(list(10, c(15,30)), c(1,0))
# Create a server.list object with the first server available anytime before time 10,
# and the second server available between time 15 and time 30.
as.server.list(list(10, c(15,30)), c(1,0))

Create a `server.stepfun` object with a roster of times and number of available servers.

Description

Create a server.stepfun object with a roster of times and number of available servers.

Usage

as.server.stepfun(x, y)
as.server.stepfun(x, y)

Arguments

`x`	numeric vector giving the times of changes in number of servers.
`y`	numeric vector one longer than `x` giving the number of servers available between x values.

Details

This function uses the analogy of a step function to specify the number of available servers throughout the day. It is used as input to the queue_step function. Alternatively one may use as.server.list to specify available servers as a list, however queue_step is much faster when as.server.stepfun is used as input rather than as.server.list.

If any of the service times are large compared to any element of diff(x) then the as.server.list function should be used.

Value

A list and server.stepfun object with x and y as elements.

Examples


servers <- as.server.stepfun(c(15,30,50), c(0, 1, 3, 2))
servers

servers <- as.server.stepfun(c(15,30,50), c(0, 1, 3, 2))
servers

Compute time average queue length

Description

Compute time average queue length

Usage

average_queue(times, queuelength)
average_queue(times, queuelength)

Arguments

`times`	numeric vector of times
`queuelength`	numeric vector of queue lengths

Examples

n <- 1e3
arrivals <- cumsum(rexp(n))
service <- rexp(n)
departures <- queue(arrivals, service, 1)

queuedata <- queue_lengths(arrivals, service, departures)
average_queue(queuedata$times, queuedata$queuelength)
n <- 1e3
arrivals <- cumsum(rexp(n))
service <- rexp(n)
departures <- queue(arrivals, service, 1)

queuedata <- queue_lengths(arrivals, service, departures)
average_queue(queuedata$times, queuedata$queuelength)

get departure times from `queue_list` object

Description

get departure times from queue_list object

Usage

depart(x)
depart(x)

Arguments

`x`	an `queue_list` object

Value

departure times

Examples

arrivals <- cumsum(rexp(10))
service <- rexp(10)
queue_obj <- queue_step(arrivals, service)

depart(queue_obj)
queue_obj$departures_df$departures
arrivals <- cumsum(rexp(10))
service <- rexp(10)
queue_obj <- queue_step(arrivals, service)

depart(queue_obj)
queue_obj$departures_df$departures

Add lag to vector of arrival times.

Description

Add lag to vector of arrival times.

Usage

lag_step(arrivals, service)
lag_step(arrivals, service)

Arguments

`arrivals`	Either a numeric vector or an object of class `queue_list`. It represents the arrival times.
`service`	A vector of service times with the same ordering as arrivals

Value

A vector of response times for the input of arrival times and service times.

Examples

# Create arrival times
arrivals <- rlnorm(100, meanlog = 3)

# Create service times
service <- rlnorm(100)
lag_step(arrivals = arrivals, service = service)

# lag_step is equivalent to queue_step with a large number of queues, but it's faster to compute.

cbind(queue(arrivals, service = service, servers = 100),
lag_step(arrivals = arrivals, service = service))
# Create arrival times
arrivals <- rlnorm(100, meanlog = 3)

# Create service times
service <- rlnorm(100)
lag_step(arrivals = arrivals, service = service)

# lag_step is equivalent to queue_step with a large number of queues, but it's faster to compute.

cbind(queue(arrivals, service = service, servers = 100),
lag_step(arrivals = arrivals, service = service))

ggplot2 method for output from queueing model

Description

ggplot2 method for output from queueing model

Usage

## S3 method for class 'queue_list'
plot(x, which = c(2:6), annotated = TRUE, ...)
## S3 method for class 'queue_list'
plot(x, which = c(2:6), annotated = TRUE, ...)

Arguments

`x`	an object of class `queue_list`
`which`	Numeric vector of integers from 1 to 6 which represents which plots are to be created. See examples.
`annotated`	logical, if `TRUE` annotations will be added to the plot.
`...`	other parameters to be passed through to plotting functions.

Examples


## Not run: 

n_customers <- 50
arrival_rate <- 1.8
service_rate <- 1
arrivals <- cumsum(rexp(n_customers, arrival_rate))
service <- rexp(n_customers, service_rate)
queue_obj <- queue_step(arrivals, service, servers = 2)
plot(queue_obj)



library(ggplot2)

## density plots of arrival and departure times
plot(queue_obj, which = 1)

## histograms of arrival and departure times
plot(queue_obj, which = 2)

## density plots of waiting and system times
plot(queue_obj, which = 3)

## step function of queue length
plot(queue_obj, which = 4)

## line range plot of customer and server status
plot(queue_obj, which = 5)

## empirical distribution plot of arrival and departure times
plot(queue_obj, which = 6)


## End(Not run)

## Not run: 

n_customers <- 50
arrival_rate <- 1.8
service_rate <- 1
arrivals <- cumsum(rexp(n_customers, arrival_rate))
service <- rexp(n_customers, service_rate)
queue_obj <- queue_step(arrivals, service, servers = 2)
plot(queue_obj)



library(ggplot2)

## density plots of arrival and departure times
plot(queue_obj, which = 1)

## histograms of arrival and departure times
plot(queue_obj, which = 2)

## density plots of waiting and system times
plot(queue_obj, which = 3)

## step function of queue length
plot(queue_obj, which = 4)

## line range plot of customer and server status
plot(queue_obj, which = 5)

## empirical distribution plot of arrival and departure times
plot(queue_obj, which = 6)


## End(Not run)

Print method for output of `summary.queue_list`.

Description

Print method for output of summary.queue_list.

Usage

## S3 method for class 'summary_queue_list'
print(x, ...)
## S3 method for class 'summary_queue_list'
print(x, ...)

Arguments

`x`	an object of class `summary_queue_list`, the result of a call to `summary.queue_list()`.
`...`	further arguments to be passed to or from other methods.

Value

A list of performance statistics for the queue:

"Total customers": Total customers in simulation,

"Missed customers": Customers who never saw a server,

"Mean waiting time": The mean time each customer had to wait in queue for service,

"Mean response time": The mean time that each customer spends in the system (departure time - arrival time),

"Utilization factor": The ratio of available time for all servers and time all servers were used. It can be greater than one if a customer arrives near the end of a shift and keeps a server busy,

"Mean queue length": Average queue length, and

"Mean number of customers in system": Average number of customers in queue or currently being served.

Examples

n <- 1e3
arrivals <- cumsum(rexp(n, 1.8))
service <- rexp(n)

queue_obj <- queue_step(arrivals, service, servers = 2)
summary(queue_obj)
n <- 1e3
arrivals <- cumsum(rexp(n, 1.8))
service <- rexp(n)

queue_obj <- queue_step(arrivals, service, servers = 2)
summary(queue_obj)

Summarise queue lengths

Description

Summarise queue lengths

Usage

ql_summary(times, queuelength)
ql_summary(times, queuelength)

Arguments

`times`	numeric vector of times
`queuelength`	numeric vector of queue lengths

Examples

n <- 1e3
arrivals <- cumsum(rexp(n))
service <- rexp(n)
departures <- queue(arrivals, service, 1)

queuedata <- queue_lengths(arrivals, service, departures)
ql_summary(queuedata$times, queuedata$queuelength)
n <- 1e3
arrivals <- cumsum(rexp(n))
service <- rexp(n)
departures <- queue(arrivals, service, 1)

queuedata <- queue_lengths(arrivals, service, departures)
ql_summary(queuedata$times, queuedata$queuelength)

Compute the departure times for a set of customers in a queue from their arrival and service times.

Description

queue is a faster version of queue_step but the input returned is much simpler. It is not compatible with the summary.queue_list method or the plot.queue_list method.

Usage

queue(arrivals, service, servers = 1, serveroutput = FALSE)
queue(arrivals, service, servers = 1, serveroutput = FALSE)

Arguments

`arrivals`	numeric vector of non-negative arrival times
`service`	numeric vector of non-negative service times
`servers`	a non-zero natural number, an object of class `server.stepfun` or an object of class `server.list`.
`serveroutput`	boolean whether the server used by each customer should be returned.

Details

If the arrival vector is out of order the function will reorder it. The same reordering will be applied to the service vector, this is so each customer keeps their service time. Once the queue is computed the original order is put back.

Examples

n <- 1e2
arrivals <- cumsum(rexp(n, 1.8))
service <- rexp(n)

departures <- queue(
    arrivals, service, servers = 2)

head(departures)
curve(ecdf(departures)(x) * n,
    from = 0, to = max(departures),
    xlab = "Time", ylab = "Number of customers")
curve(ecdf(arrivals)(x) * n,
    from = 0, to = max(departures),
    col = "red", add = TRUE)

n <- 1e2
arrivals <- cumsum(rexp(n, 1.8))
service <- rexp(n)

departures <- queue(
    arrivals, service, servers = 2)

head(departures)
curve(ecdf(departures)(x) * n,
    from = 0, to = max(departures),
    xlab = "Time", ylab = "Number of customers")
curve(ecdf(arrivals)(x) * n,
    from = 0, to = max(departures),
    col = "red", add = TRUE)

Compute queue lengths from arrival, service and departure data

Description

Compute queue lengths from arrival, service and departure data

Usage

queue_lengths(arrivals, service = 0, departures, epsilon = 1e-10, ...)
queue_lengths(arrivals, service = 0, departures, epsilon = 1e-10, ...)

Arguments

`arrivals`	vector of arrival times
`service`	vector of service times. Leave as zero if you want to compute the number of customers in the system rather than queue length.
`departures`	vector of departure times
`epsilon`	numeric small number added to departures to prevent negative queue lengths
`...`	additional arguments - does nothing, for compatibility

Examples

library(dplyr)
library(queuecomputer)

set.seed(1L)
n_customers <- 100

queueoutput_df <- data.frame(
  arrivals = runif(n_customers, 0, 300),
  service = rexp(n_customers)
)

queueoutput_df <- queueoutput_df %>% mutate(
  departures = queue(arrivals, service, servers = 2)
)

queue_lengths(
  queueoutput_df$arrivals,
  queueoutput_df$service,
  queueoutput_df$departures
)

# The dplyr way
queueoutput_df %>% do(
  queue_lengths(.$arrivals, .$service, .$departures))

n_customers <- 1000

queueoutput_df <- data.frame(
  arrivals = runif(n_customers, 0, 300),
  service = rexp(n_customers),
  route = sample(c("a", "b"), n_customers, TRUE)
)

server_df <- data.frame(
  route = c("a", "b"),
  servers = c(2, 3)
)

output <- queueoutput_df %>%
  left_join(server_df) %>%
  group_by(route) %>%
  mutate(
    departures = queue(arrivals, service, servers = servers[1])
  ) %>%
  do(queue_lengths(.$arrivals, .$service, .$departures))


if(require(ggplot2, quietly = TRUE)){
    ggplot(output) +
      aes(x = times, y = queuelength) + geom_step() +
      facet_grid(~route)
}
library(dplyr)
library(queuecomputer)

set.seed(1L)
n_customers <- 100

queueoutput_df <- data.frame(
  arrivals = runif(n_customers, 0, 300),
  service = rexp(n_customers)
)

queueoutput_df <- queueoutput_df %>% mutate(
  departures = queue(arrivals, service, servers = 2)
)

queue_lengths(
  queueoutput_df$arrivals,
  queueoutput_df$service,
  queueoutput_df$departures
)

# The dplyr way
queueoutput_df %>% do(
  queue_lengths(.$arrivals, .$service, .$departures))

n_customers <- 1000

queueoutput_df <- data.frame(
  arrivals = runif(n_customers, 0, 300),
  service = rexp(n_customers),
  route = sample(c("a", "b"), n_customers, TRUE)
)

server_df <- data.frame(
  route = c("a", "b"),
  servers = c(2, 3)
)

output <- queueoutput_df %>%
  left_join(server_df) %>%
  group_by(route) %>%
  mutate(
    departures = queue(arrivals, service, servers = servers[1])
  ) %>%
  do(queue_lengths(.$arrivals, .$service, .$departures))


if(require(ggplot2, quietly = TRUE)){
    ggplot(output) +
      aes(x = times, y = queuelength) + geom_step() +
      facet_grid(~route)
}

Compute the departure times and queue lengths for a queueing system from arrival and service times.

Description

Compute the departure times and queue lengths for a queueing system from arrival and service times.

Usage

queue_step(arrivals, service, servers = 1, labels = NULL)
queue_step(arrivals, service, servers = 1, labels = NULL)

Arguments

`arrivals`	numeric vector of non-negative arrival times
`service`	numeric vector of service times with the same ordering as arrival_df.
`servers`	a non-zero natural number, an object of class `server.stepfun` or an object of class `server.list`.
`labels`	character vector of customer labels (deprecated).

Details

If only departure times are needed, the queue function is faster.

Value

An list object of class queue_list with the following components:

departures - A vector of response times for the input of arrival times and service times.
server - A vector of server assignments for the input of arrival times and service times.
departures_df - A data frame with arrivals, service, departures, waiting, system time, and server assignments for each customer.
queuelength_df - A data frame describing the evolution of queue length over time
systemlength_df - A data frame describing the evolution of system length over time
servers_input - A copy of the server argument
state - A vector of availability times for the servers

Examples


# With two servers
set.seed(1)
n <- 100

arrivals <- cumsum(rexp(n, 3))
service <- rexp(n)


queue_obj <- queue_step(arrivals,
    service = service, servers = 2)


summary(queue_obj)
plot(queue_obj, which = 5)

# It seems like the customers have a long wait.
# Let's put two more servers on after time 20


server_list <- as.server.stepfun(c(20),c(2,4))

queue_obj2 <- queue_step(arrivals,
    service = service,
    servers = server_list)

summary(queue_obj2)
if(require(ggplot2, quietly = TRUE)){

    plot(queue_obj2, which = 5)

}


# With two servers
set.seed(1)
n <- 100

arrivals <- cumsum(rexp(n, 3))
service <- rexp(n)


queue_obj <- queue_step(arrivals,
    service = service, servers = 2)


summary(queue_obj)
plot(queue_obj, which = 5)

# It seems like the customers have a long wait.
# Let's put two more servers on after time 20


server_list <- as.server.stepfun(c(20),c(2,4))

queue_obj2 <- queue_step(arrivals,
    service = service,
    servers = server_list)

summary(queue_obj2)
if(require(ggplot2, quietly = TRUE)){

    plot(queue_obj2, which = 5)

}

Summary method for queue_list object

Description

Summary method for queue_list object

Usage

## S3 method for class 'queue_list'
summary(object, ...)
## S3 method for class 'queue_list'
summary(object, ...)

Arguments

`object`	an object of class `queue_list`, the result of a call to `queue_step`.
`...`	further arguments to be passed to or from other methods.

Compute maximum time for each row from two vectors of arrival times.

Description

Compute maximum time for each row from two vectors of arrival times.

Usage

wait_step(arrivals, service)
wait_step(arrivals, service)

Arguments

`arrivals`	Either a numeric vector or an object of class `queue_list`. It represents the arrival times.
`service`	A vector of times which represent the arrival times of the second type of customers. The ordering of this vector should have the same ordering as `arrivals`.

Details

A good real-world example of this is finding the departure times for passengers after they pick up their bags from the baggage carousel. The time at which they leave is the maximum of the passenger and bag arrival times.

Value

The maximum time from two vectors of arrival times.

Examples

set.seed(500)
arrivals <- rlnorm(100, meanlog = 4)
service <- rlnorm(100)

#Airport example ------------------------

# Create a number of bags for each of 100 customers
bags <- rpois(100,1)

# Create a bags dataframe, with each bag associated with one customer.
bags.df <- data.frame(BagID = 1:sum(bags),
   ID = rep(1:100, bags), times = rlnorm(sum(bags), meanlog = 2))

# Create a function which will return the maximum time from each customer's set of bags.

reduce_bags <- function(bagdataset, number_of_passengers){
   ID = NULL
   times = NULL

   zerobags <- data.frame(BagID = NA, ID = c(1:number_of_passengers), times = 0)
   reduced_df <- as.data.frame(dplyr::summarise(dplyr::group_by(
   rbind(bagdataset, zerobags), ID), n = max(times, 0)))
   ord <- order(reduced_df$ID)
   reduced_df <- reduced_df[order(ord),]
   names(reduced_df) <- c("ID", "times")
   return(reduced_df)
}


arrivals2 <- reduce_bags(bags.df, 100)$times

# Find the time when customers can leave with their bags.
wait_step(arrivals = arrivals, service = arrivals2)
set.seed(500)
arrivals <- rlnorm(100, meanlog = 4)
service <- rlnorm(100)

#Airport example ------------------------

# Create a number of bags for each of 100 customers
bags <- rpois(100,1)

# Create a bags dataframe, with each bag associated with one customer.
bags.df <- data.frame(BagID = 1:sum(bags),
   ID = rep(1:100, bags), times = rlnorm(sum(bags), meanlog = 2))

# Create a function which will return the maximum time from each customer's set of bags.

reduce_bags <- function(bagdataset, number_of_passengers){
   ID = NULL
   times = NULL

   zerobags <- data.frame(BagID = NA, ID = c(1:number_of_passengers), times = 0)
   reduced_df <- as.data.frame(dplyr::summarise(dplyr::group_by(
   rbind(bagdataset, zerobags), ID), n = max(times, 0)))
   ord <- order(reduced_df$ID)
   reduced_df <- reduced_df[order(ord),]
   names(reduced_df) <- c("ID", "times")
   return(reduced_df)
}


arrivals2 <- reduce_bags(bags.df, 100)$times

# Find the time when customers can leave with their bags.
wait_step(arrivals = arrivals, service = arrivals2)

Package 'queuecomputer'

Help Index

Creates a "server.list" object from a list of times and starting availability.

Description

Usage

Arguments

Value

See Also

Examples

Create a server.stepfun object with a roster of times and number of available servers.

Description

Usage

Arguments

Details

Value

See Also

Examples

Compute time average queue length

Description

Usage

Arguments

Examples

get departure times from queue_list object

Description

Usage

Arguments

Value

Examples

Add lag to vector of arrival times.

Description

Usage

Arguments

Value

See Also

Examples

ggplot2 method for output from queueing model

Description

Usage

Arguments

Examples

Print method for output of summary.queue_list.

Description

Usage

Arguments

Value

Examples

Summarise queue lengths

Description

Usage

Arguments

Examples

Compute the departure times for a set of customers in a queue from their arrival and service times.

Description

Usage

Arguments

Details

See Also

Examples

Compute queue lengths from arrival, service and departure data

Description

Usage

Arguments

Examples

Compute the departure times and queue lengths for a queueing system from arrival and service times.

Description

Usage

Arguments

Details

Value

See Also

Examples

Summary method for queue_list object

Description

Usage

Arguments

Compute maximum time for each row from two vectors of arrival times.

Description

Usage

Arguments

Details

Creates a `"server.list"` object from a list of times and starting availability.

Create a `server.stepfun` object with a roster of times and number of available servers.

get departure times from `queue_list` object

Print method for output of `summary.queue_list`.