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] "server.stepfun" "list"
departures <- queue_step(arrivals = arrivals, service = service, servers = resource_schedule)
str(departures,1)
## List of 7
## $ departures : num [1:100] 11.3 11.7 12.4 13.2 14 ...
## $ server : int [1:100] 1 1 1 1 1 1 1 1 1 1 ...
## $ departures_df : tibble [100 x 6] (S3: tbl_df/tbl/data.frame)
## $ queuelength_df :'data.frame': 201 obs. of 2 variables:
## $ systemlength_df:'data.frame': 201 obs. of 2 variables:
## $ servers_input :List of 2
## ..- attr(*, "class")= chr [1:2] "server.stepfun" "list"
## $ state : num [1:3] 93.8 91.6 90.8
## - attr(*, "class")= chr [1:2] "queue_list" "list"
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(queuecomputer)
library(dplyr)
set.seed(500)
n <- 100
arrivals <- cumsum(rexp(n))
service_l <- rexp(n, 0.8)
service_q <- rexp(n, 0.5)
arrivals_b <- cumsum(rexp(n, 0.8))
# The queue elements can be computed one by one.
departures_1 <- lag_step(arrivals, service_l)
departures_2 <- queue(departures_1, service = service_q, servers = 2)
departures_3 <- wait_step(departures_2, arrivals_b)
# Or 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)
all(departures == departures_3)
## [1] TRUE
# Plot densities for this tandem queueing network
colours <- rainbow(4)
plot(density(arrivals, from = 0),
col = colours[1], xlim = c(0, 220), ylim = c(0, 0.015),
main = "Density plot")
lines(density(departures_1, from = 0), col = colours[2])
lines(density(departures_2, from = 0), col = colours[3])
lines(density(departures_3, from = 0), col = colours[4])
legend(150,0.012, legend = c("Start walk",
"Finish walk",
"Finish service",
"Pick up bag"),
col = colours, lwd = 1, cex = 0.8
)
