queuecomputer implements a new and computationally efficient method for simulating from a general set of queues. The current most popular method for simulating queues is Discete Event Simulation (DES). The top R package for DES is called simmer and the top Python package is called SimPy. We have validated and benchmarked queuecomputer against both these packages and found that queuecomputer is two orders of magnitude faster than either package.

Simulating arbitrary queues is difficult, however once:

- The arrival times A and service times S are known for all customers and,
- the server resource schedule is specified

then the departure times D for all customers can be computed deterministically.

The focus on this package is:

- fast computation of departure times given arrival and service times, and
- a flexible framework to allow for extensions such as server effects.

It is up to the user to provide arrival and service times, and therefore very complicated distributions can be simulated (by the user) and tested with this package.

For detailed information regarding the algorithm used in this package see our paper:

Ebert A, Wu P, Mengersen K, Ruggeri F (2020). “Computationally
Efficient Simulation of Queues: The R Package queuecomputer.”
*Journal of Statistical Software*, *95*(5), 1-29. doi:
10.18637/jss.v095.i05 (URL: https://doi.org/10.18637/jss.v095.i05).

`install.packages('queuecomputer')`

We demonstrate simulating the first 50 customers from a M/M/2 queue. In queueing theory, M/M/2 refers to a queue with exponential inter-arrival times, exponential service times and two servers.

```
library(queuecomputer)
<- 50
n
<- cumsum(rexp(n, 1.9))
arrivals <- rexp(n)
service
<- queue_step(arrivals = arrivals, service = service, servers = 2) queue_mm2
```

You can see the table of customer arrival, service and departure
times by accessing the `departures_df`

object from
queue_mm2.

```
$departures_df
queue_mm2#> # A tibble: 50 x 6
#> arrivals service departures waiting system_time server
#> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 0.397 0.422 0.820 0 0.422 1
#> 2 1.02 2.18 3.20 0 2.18 2
#> 3 1.10 3.22 4.31 0 3.22 1
#> 4 1.17 0.558 3.76 2.03 2.59 2
#> 5 1.40 0.595 4.35 2.36 2.95 2
#> 6 2.92 0.977 5.29 1.39 2.37 1
#> 7 3.57 0.210 4.56 0.781 0.991 2
#> 8 3.85 0.309 4.87 0.706 1.02 2
#> 9 4.36 1.11 5.98 0.512 1.62 2
#> 10 4.43 0.774 6.07 0.856 1.63 1
#> # ... with 40 more rows
```

You can see visualisations of the queueing system.

```
plot(queue_mm2)
#> [[1]]
```

```
#>
#> [[2]]
```

```
#>
#> [[3]]
```

```
#>
#> [[4]]
```

```
#>
#> [[5]]
```

A summary of the performance of the queueing system can be computed.

```
summary(queue_mm2)
#> Total customers:
#> 50
#> Missed customers:
#> 0
#> Mean waiting time:
#> 2.91
#> Mean response time:
#> 3.99
#> Utilization factor:
#> 0.897026364068736
#> Mean queue length:
#> 4.9
#> Mean number of customers in system:
#> 6.64
```

In this example of a queueing network, customers must pass through two queues. The arrival times to the first queue come in two waves starting at time 100 and time 500. The arrival times to the second queue are the departure times of the first queue plus the time they spent walking to the second queue.

```
library(queuecomputer)
library(ggplot2)
library(dplyr)
#>
#> Vedhæfter pakke: 'dplyr'
#> De følgende objekter er maskerede fra 'package:stats':
#>
#> filter, lag
#> De følgende objekter er maskerede fra 'package:base':
#>
#> intersect, setdiff, setequal, union
set.seed(1)
<- 100
n
<- c(100 + cumsum(rexp(n)), 500 + cumsum(rexp(n)))
arrivals_1 <- rexp(2*n, 1/2.5)
service_1
<- queue_step(arrivals = arrivals_1, service = service_1, servers = 2)
queue_1
<- rexp(2*n, 1/100)
walktimes
<- lag_step(arrivals = queue_1, service = walktimes)
arrivals_2 <- rexp(2*n, 1/3)
service_2
<- queue_step(arrivals = arrivals_2, service = service_2, servers = 1)
queue_2
head(arrivals_1)
#> [1] 100.7552 101.9368 102.0825 102.2223 102.6584 105.5534
head(queue_1$departures_df)
#> # A tibble: 6 x 6
#> arrivals service departures waiting system_time server
#> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 101. 0.189 101. 0 0.189 1
#> 2 102. 2.57 105. 4.44e-15 2.57 2
#> 3 102. 1.69 104. 0 1.69 1
#> 4 102. 2.00 106. 1.55e+ 0 3.55 1
#> 5 103. 0.435 105. 1.84e+ 0 2.28 2
#> 6 106. 1.68 107. 0 1.68 2
head(arrivals_2)
#> [1] 120.3923 105.6711 227.5242 175.9008 339.9853 108.7119
head(queue_2$departures_df)
#> # A tibble: 6 x 6
#> arrivals service departures waiting system_time server
#> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 120. 5.16 126. 0 5.16 1
#> 2 106. 1.58 107. 0 1.58 1
#> 3 228. 0.114 291. 63.2 63.3 1
#> 4 176. 2.35 186. 8.02 10.4 1
#> 5 340. 3.20 404. 61.3 64.5 1
#> 6 109. 1.23 110. 0 1.23 1
summary(queue_1)
#> Total customers:
#> 200
#> Missed customers:
#> 0
#> Mean waiting time:
#> 11.4
#> Mean response time:
#> 13.7
#> Utilization factor:
#> 0.38410206651912
#> Mean queue length:
#> 3.7
#> Mean number of customers in system:
#> 4.46
summary(queue_2)
#> Total customers:
#> 200
#> Missed customers:
#> 0
#> Mean waiting time:
#> 34.1
#> Mean response time:
#> 37.2
#> Utilization factor:
#> 0.519160307775743
#> Mean queue length:
#> 5.71
#> Mean number of customers in system:
#> 6.21
```

I’d like to thank my supervisors Professor Kerrie Mengersen, Dr Paul Wu and Professor Fabrizio Ruggeri.

This work was supported by the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). This work was funded through the ARC Linkage Grant “Improving the Productivity and Efficiency of Australian Airports” (LP140100282).