# Queuing models in R, $\lambda$ Little

It's noted that the number of folks in a stationary system will maintain an average equal to the rate of arrival multiplied by the mean of the service distribution.

The formula $$L = \lambda w$$ is valid for any queueing model in steady-state, where $$L$$ and $$w$$ are long-term steady-state average values respectively, and $$\lambda$$ denotes the arrival rate to the system.

We can add up the total service time in the system as follows:

$$\sum w_{j} = \sum i T_i$$

where $$T_i$$ represents the time units in which $$i$$ entities were in the system.

But things are always more interesting with sample sizing and simulating results, so say in $$5000$$ iterations we estimate a state of a system in 1-min intervals, between the first minute and the last arrival at a determined time.

So suppose we use a random interarrival rate of $$\lambda = 2$$ per minute and the service distribution is $$N(8,1)$$ minutes for the system.

How can I simulate this model in R, using rexp() and rnorm()? I also want to display in ts() to show in time plot.

• While I think the topic does fit into our site, asking for a model in R will probably get more attention on SO (see stackoverflow.com/search?q=%5BR%5D+lambda for some questions there). But let's see what our other users will come up to answer your question. – JakobS Jul 12 '19 at 22:56
• I came up with a portion, perhaps there can be come feedback see below – pulselaserman_quantum Jul 13 '19 at 0:17
• Is your question about OR concepts in the queuing model, e.g., some aspect of Little’s Law, or is it about the implementation of some functions in R? If the latter, I agree with @JakobS that the question might get more response on SO. If the former, I think the question could use some clarification about what exactly you are asking. – LarrySnyder610 Jul 14 '19 at 13:23
• It was more oh do we set this up programmatically, but also a question could be why would a distribution of outcomes (ala simulation) provide and more rigor than any other methods – pulselaserman_quantum Jul 14 '19 at 20:33
• I am happy to expand my answer if you can refine your question appropriately. We are all seeking to keep the answered rate high. – CMichael Jul 15 '19 at 22:05

Not directly answering your question of how to code it manually but for discrete simulation of queues in R I would strongly recommend the simmer package. The minimal code for your example would look like this (adapted from the tutorial).

library(simmer)
library(simmer.plot)
lambda <- 2

queue <- trajectory() %>%
seize("server", amount=1) %>%
timeout(function() {rnorm(1, 8, 1)}) %>%
release("server", amount=1)

env <- simmer() %>%
add_generator("arrival", queue, function() {rnorm(1, lambda)}) %>%
run(until=100000)

resources <- get_mon_resources(env)
arrivals <- get_mon_arrivals(env)

plot(resources, metric = "usage", items = "server")
plot(arrivals, metric = "flow_time")


However, looking at the plot of flow times I am unsure if you conceived a stable queuing system - for mean = 1 in the service time it looks better:

Service Time Mean 8:

Service Time Mean 1:

• Thanks for the suggestion! I’ll share what I came up with! – pulselaserman_quantum Jul 14 '19 at 21:14
• Have fun - it is a powerful yet neat package. I updated my answer to be a bit more illustrative. – CMichael Jul 14 '19 at 21:24

So far this is what I have come up with

lambda <- 2
interarrivals <- rexp(5000,lambda)  ## (2 items per minute)


Provided the $$\mu$$ we expect that the interarrivals is about half a minute

mean(interarrivals) <- 0.516
service.times <- rnorm(5000, mean=8,sd=1)


where the service distribution is $$N(8,1)$$

arrival.times <- cumsum(interarrivals)
departure.times <- arrival.times+service.times


Where I am having issues is determining how many individuals remain in the system at time 1-min to the last.

I am having issues conceptualizing the model - this is what you are supposed to estimate (Little's law), which states that rate of arrival = 2 units per minute × mean of the service distribution = 8.

• my intent is to convert the system state to a time.series ts() so I can plot. – pulselaserman_quantum Jul 13 '19 at 0:28