Introduction

The purpose of the package queuecomputer is to compute, deterministically, the output of a queue network given the arrival and service times for all customers. The most important functions are queue_step, lag_step and wait_step.

Input format

The first argument to the functions queue_step, lag_step and wait_step is a dataframe with a column of unique customer IDs labelled ‘ID’ and a column of arrival times labelled ‘times’. For example:

library(queuecomputer)

arrivals <- data.frame(ID = c(1:100), times = cumsum(rexp(100)))

head(arrivals)

##   ID    times
## 1  1 1.272276
## 2  2 1.661925
## 3  3 3.328307
## 4  4 3.912903
## 5  5 4.732425
## 6  6 6.490986

service <- rexp(100)

departures <- queue_step(arrival_df = arrivals, service = service)

head(departures)

##   ID     times
## 1  1  5.479328
## 2  2  9.850967
## 3  3 11.724356
## 4  4 12.663550
## 5  5 14.345809
## 6  6 14.363243

Resourcing schedule

The resourcing schedule is specified with either a non-zero natural number, a server.stepfun or a server.list object. Use a non-zero natural number when the number of servers does not change over time. The server.stepfun specifies a step function to indicate how many servers are available throughout the day. The computation speed for queue_step() is much faster when using a server.stepfun rather than a server.list input for the servers argument.

We create a server.stepfun object with the as.server.stepfun function.

# Zero servers available before time 10
# One server available between time 10 and time 50
# Three servers available between time 50 and time 100
# One server available from time 100 onwards
resource_schedule <- as.server.stepfun(c(10,50,100), c(0, 1, 3, 1))

resource_schedule

## $x
## [1]  10  50 100
## 
## $y
## [1] 0 1 3 1
## 
## attr(,"class")
## [1] "list"           "server.stepfun"

departures <- queue_step(arrival_df = arrivals, service = service, servers = resource_schedule)

head(departures)

##   ID    times
## 1  1 14.20705
## 2  2 18.57869
## 3  3 20.45208
## 4  4 21.39127
## 5  5 23.07353
## 6  6 23.09097

The server.list object is a list of step functions which represent each server, the range is \(\{0,1\}\), where 0 represents unavailable and 1 represents available and the knots represent the times where availability changes.

The as.server.list function is used to create a server.list object.

# Server 1 is available before time 10.
# Server 2 is available between time 15 and time 30.
# Server 3 is available after time 10. 
as.server.list(list(10, c(15,30), 10), c(1,0,0))

## [[1]]
## Step function
## Call: stats::stepfun(times[[i]], y)
##  x[1:1] =     10
## 2 plateau levels =      1,      0
## 
## [[2]]
## Step function
## Call: stats::stepfun(times[[i]], y)
##  x[1:2] =     15,     30
## 3 plateau levels =      0,      1,      0
## 
## [[3]]
## Step function
## Call: stats::stepfun(times[[i]], y)
##  x[1:1] =     10
## 2 plateau levels =      0,      1
## 
## attr(,"class")
## [1] "list"        "server.list"

Setting up a queue network

It is simple to set up a chain of queueing elements with queuecomputer. Suppose passengers must walk to a queue, then wait for service and then wait for their bags.

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

set.seed(500)

arrivals <- data.frame(ID = c(1:100), times = cumsum(rexp(100)))
service_l <- rexp(100, 0.8)
service_q <- rexp(100, 0.5)
arrivals_b <- data.frame(ID = c(1:100), times = cumsum(rexp(100, 0.8)))

# The queue elements can be computed one by one. 

departures_1 <- lag_step(arrivals, service_l)
departures_2 <- queue_step(departures_1, service = service_q, servers = 2)
departures_3 <- wait_step(departures_2, arrivals_b$times)

departure_df <- data.frame(arrival_times = arrivals$times, departures_1 = departures_1$times, departures_2 = departures_2$times, departures_3 = departures_3$times) %>% reshape2::melt()

## No id variables; using all as measure variables

qplot(value, data = departure_df, colour = variable, geom = "density") + xlab("time")

# The queue elements can be chained together with the %>% operator. 

departures <- lag_step(arrivals, service_l) %>% queue_step(service = service_q, servers = 2) %>% wait_step(arrivals_b$times)

all(departures$times == departures_3$times)

## [1] TRUE