The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.
The purpose of this vignette is to demonstrate how to use
queuecomputer to understand M/M/k queues. We consider one
M/M/1 queue and two M/M/3 queues. The working follows the theoretical
results for M/M/k queues as shown in Thomopoulos, N (2012).
library(queuecomputer)P_0_func <- function(rho, k){
sum_i <- rep(NA, k)
for(i in 0:I(k-1))
{
sum_i[i+1] <- rho^i / factorial(i)
}
p_0 <- 1/(sum(sum_i) + rho^k/(factorial(k - 1) * (k - rho)))
return(p_0)
}
P_n <- function(rho,n,k){
p_0 <- P_0_func(rho, k)
if(n <= k){
output <- rho^n / factorial(n) * p_0
} else {
output <- rho^n / (factorial(k) * k^(n-k)) * p_0
}
return(output)
}
Lq <- function(rho, k){
p_0 <- P_0_func(rho, k)
output <- p_0 * rho^{k+1} / ( factorial(k-1) * (k - rho)^2)
return(output)
}set.seed(1)
n_customers <- 1e6
lambda_a <- 1/1
lambda_s <- 1/0.8
interarrivals <- rexp(n_customers, lambda_a)
arrivals <- cumsum(interarrivals)
service <- rexp(n_customers, lambda_s)
rho <- (1/lambda_s) / (1/lambda_a)k = 1
p_0 <- P_n(rho, n=0, k)
### System lengths -----------------------
Vectorize(P_n, "n")(rho=rho, n=c(0:30), k = k)## [1] 0.2000000000 0.1600000000 0.1280000000 0.1024000000 0.0819200000
## [6] 0.0655360000 0.0524288000 0.0419430400 0.0335544320 0.0268435456
## [11] 0.0214748365 0.0171798692 0.0137438953 0.0109951163 0.0087960930
## [16] 0.0070368744 0.0056294995 0.0045035996 0.0036028797 0.0028823038
## [21] 0.0023058430 0.0018446744 0.0014757395 0.0011805916 0.0009444733
## [26] 0.0007555786 0.0006044629 0.0004835703 0.0003868563 0.0003094850
## [31] 0.0002475880### Estimated queue length -----------------
LQ <- Lq(rho, k)
LQ## [1] 3.2### Estimated units in system -----------
LQ + rho## [1] 4Ws = 1/lambda_s
Wq = LQ / lambda_a
W = Ws + Wq
Wq # Mean waiting time (time in queue)## [1] 3.2W # Mean response time (time in system)## [1] 4MM1 <- queue_step(arrivals = arrivals, service = service, servers = k)
MM1_summary <- summary(MM1)
MM1_summary$slength_sum## # A tibble: 52 x 2
## queuelength proportion
## <int> <dbl>
## 1 0 0.200
## 2 1 0.160
## 3 2 0.128
## 4 3 0.102
## 5 4 0.0817
## 6 5 0.0655
## 7 6 0.0524
## 8 7 0.0420
## 9 8 0.0337
## 10 9 0.0270
## # ... with 42 more rows# Mean queue length
MM1_summary$qlength_mean## [1] 3.210141# Mean system length (number of customers in system)
MM1_summary$slength_mean## [1] 4.009881MM1_summary$mwt # Mean waiting time## [1] 3.212666MM1_summary$mrt # Mean response time## [1] 4.01304k = 3
p_0 <- P_n(rho, 0, k)
### System lengths -----------------------
Vectorize(P_n, "n")(rho=rho, n=c(0:30), k = k)## [1] 4.471545e-01 3.577236e-01 1.430894e-01 3.815718e-02 1.017525e-02
## [6] 2.713400e-03 7.235732e-04 1.929529e-04 5.145410e-05 1.372109e-05
## [11] 3.658958e-06 9.757221e-07 2.601926e-07 6.938468e-08 1.850258e-08
## [16] 4.934022e-09 1.315739e-09 3.508638e-10 9.356368e-11 2.495031e-11
## [21] 6.653417e-12 1.774245e-12 4.731319e-13 1.261685e-13 3.364493e-14
## [26] 8.971982e-15 2.392529e-15 6.380076e-16 1.701354e-16 4.536943e-17
## [31] 1.209851e-17### Estimated queue length -----------------
LQ <- Lq(rho, k)
LQ## [1] 0.01892092### Estimated units in system -----------
LQ + rho## [1] 0.8189209Ws = 1/lambda_s
Wq = LQ / lambda_a
W = Ws + Wq
Wq # Mean waiting time (time in queue)## [1] 0.01892092W # Mean response time (time in system)## [1] 0.8189209MM3 <- queue_step(arrivals = arrivals, service = service, servers = k)
MM3_summary <- summary(MM3)
MM3_summary$slength_sum## # A tibble: 13 x 2
## queuelength proportion
## <int> <dbl>
## 1 0 0.447
## 2 1 0.358
## 3 2 0.143
## 4 3 0.0380
## 5 4 0.0101
## 6 5 0.00268
## 7 6 0.000704
## 8 7 0.000193
## 9 8 0.0000399
## 10 9 0.0000120
## 11 10 0.00000201
## 12 11 0.00000276
## 13 12 0.000000106# Mean queue length
MM3_summary$qlength_mean## [1] 0.01865591# Mean system length (number of customers in system)
MM3_summary$slength_mean## [1] 0.8184002MM3_summary$mwt # Mean waiting time## [1] 0.01867057MM3_summary$mrt # Mean response time## [1] 0.8190443# Setup ----------
set.seed(2)
n_customers <- 1e6
lambda_a <- 1/1
lambda_s <- 1/2.5
interarrivals <- rexp(n_customers, lambda_a)
arrivals <- cumsum(interarrivals)
service <- rexp(n_customers, lambda_s)
rho <- (1/lambda_s) / (1/lambda_a)rho <- (1/lambda_s) / (1/lambda_a)
# MM3 queue ------------------------------
k = 3
## Theoretical -------------------
p_0 <- P_n(rho, 0, k)
### System lengths -----------------------
Vectorize(P_n, "n")(rho=rho, n=c(0:30), k = k)## [1] 0.0449438202 0.1123595506 0.1404494382 0.1170411985 0.0975343321
## [6] 0.0812786101 0.0677321751 0.0564434792 0.0470362327 0.0391968606
## [11] 0.0326640505 0.0272200421 0.0226833684 0.0189028070 0.0157523392
## [16] 0.0131269493 0.0109391244 0.0091159370 0.0075966142 0.0063305118
## [21] 0.0052754265 0.0043961888 0.0036634906 0.0030529089 0.0025440907
## [26] 0.0021200756 0.0017667297 0.0014722747 0.0012268956 0.0010224130
## [31] 0.0008520108### Estimated queue length -----------------
LQ <- Lq(rho, k)
LQ## [1] 3.511236### Estimated units in system -----------
LQ + rho## [1] 6.011236### Waiting times -----------
Ws = 1/lambda_s
Wq = LQ / lambda_a
W = Ws + Wq
Wq # Mean waiting time (time in queue)## [1] 3.511236W # Mean response time (time in system)## [1] 6.011236MM3_2 <- queue_step(arrivals = arrivals, service = service, servers = k)
MM3_2_summary <- summary(MM3_2)
MM3_2_summary$slength_sum## # A tibble: 53 x 2
## queuelength proportion
## <int> <dbl>
## 1 0 0.0450
## 2 1 0.113
## 3 2 0.140
## 4 3 0.118
## 5 4 0.0980
## 6 5 0.0820
## 7 6 0.0679
## 8 7 0.0568
## 9 8 0.0471
## 10 9 0.0391
## # ... with 43 more rows# Mean queue length
MM3_2_summary$qlength_mean## [1] 3.472223# Mean system length (number of customers in system)
MM3_2_summary$slength_mean## [1] 5.971736MM3_2_summary$mwt # Mean waiting time## [1] 3.46892MM3_2_summary$mrt # Mean response time## [1] 5.966074Thomopoulos, N (2012). Fundamentals of Queuing Systems: Statistical Methods for Analyzing Queuing Models. Springer Science & Business Media
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.