Discrete event simulation refers to the simulation of systems that have abrupt, i.e. discrete, changes. In a queuing system, for instance, when a new job arrives, the queue length abruptly increases by 1. Simulation of a weather system, on the other hand, would not fit this definition, as quantities such as temperature vary continuously.
This document will give a quick introduction to the subject of DES, using our R package DES as a running example.
The R DES package here, DES, is not the fastest. If speed is an issue, I recommend the excellent simmer package in R, or in Python, SimPy, on which simmer is based. On the other hand, DES is much easier to learn (good for teaching, for instance), and gives the programmer more control, thus making simulation of more complex systems easier to program.
The DES package uses the event-oriented approach, which means the programmer codes how the system reacts to any specific event. To see how event-oriented simulation works, consider a simple machine-repair model.
We have m machines and r repairpersons. Each machine occasionaly breaks down. If at the time of a breakdown there is at least one idle repairperson, that machine's repair is begun; otherwise, it joins a queue for the repairperson pool. We assume up times and repair times are exponentially distributed, with all times being independent and so on.
The file inst/examples/MachRep.R in the package simulates this system. The functions in that file are thus application-specific, and we will refer to them as user-supplied. They call DES functions, which we will refer to as package functions. The user-supplied wrapper that runs the simulation is named mrp here.
For instance, say we wish to simulate a system in which machines have mean up and repair times of 10.0 and 1.0, respectively, with 2 machines and one repairperson, for 10000.0 units of simulated time. The call would be
mrp(10,1,10000,2,1)Here there are two kinds of events in this application, breakdown and repair. Think about how the system reacts to a breakdown at a machine:
- If some repairperson is free, start service for this machine.
Otherwise, add the job to the queue.
What about reaction to a repair completion at a machine?
-
Start the next up time for this machine.
-
If the queue is nonempty, remove the head and start repair for that job.
The general information about the simulation is contained in the sim list, which mrp not surprisingly has named simlist. Here are the first few lines:
mrp <- function(meanup,meanrep,timelim,m,r,dbg=FALSE) {
# set up structures
simlist <- newsim(timelim,m,
appcols=c('startqtime','startuptime'),dbg=dbg)
simlist$reactevent <- mrpreact
simlist$uprate <- 1.0 / meanup
simlist$reprate <- 1.0 / meanrep
simlist$nmach <- m
simlist$nrepairpersons <- r
simlist$queue <- newqueue(simlist) # queue for the repairpersons
simlist$nup <- m # all machines up initially
simlist$nrepbusy <- 0 # number of busy repairpersons
simlist$breakevnt <- 1 # breakdown
simlist$repairevnt <- 2 # good as new!
simlist$nrepairs <- 0
simlist$totqtime <- 0.0
simlist$totuptime <- 0.0The package function newsim initializes the simulation, particularly the non-application specific parts of the sim list, such as simlist$currtime, which will hold the current simulated time.
The lines we see above are setting application-specific information in the sim list, such as the up time and repair rates. Note too the initialization of the "bookkeeping" variables simlist$nrepairs, which keeps track of the number of repairs done so far, needed so we can determine the mean queuing time later on.
A major DES structure is the event set, a matrix that contains all pending events, say three breakdowns and one repair. It is initialized by newsim as a component of the sim list named evnts. User-supplied code adds events to the event set by calling the package function schedevnt.
There will be one row in the event set for each pending event. The row will contain the simulated time at which the event is to occur, and the event type (in the machine-repair example, breakdown or repair completion). The row will also contain optional application-specific information, which in our call to newsim we have specified as the start of the current queueing time of the machine, if any, and the time at which the current up time for the machine began, if any. The argument appcols in newsim gives the names of these quantities (the names of their columns in the event set matrix), and the appdata argument in schedevnt gives the particular values of this data at the time of the call.
Our user-supplied code, mrp here, must "get the ball rolling" by creating the initial events. Since our simulation will assume that all machines start in the up mode -- it doesn't really matter in the long run, but we need a start -- mrp creates breakdown events for all of them:
for (i in 1:m) {
whenbreak <- rexp(1,simlist$uprate)
schedevnt(simlist,whenbreak,simlist$breakevnt,appdata=c(NA,0))
}The call to schedevnt schedules a breakdown event at time whenbreak. (Note that simulated time begins at 0.0.)
The user must supply a reaction function, which codes how the system reacts to the various events. In the mrp code above, you can see that we have named that function mrpreact, and recorded it as a component of the sim list. We'll look at that function shortly too.
The core package function is mainloop, which works as follows:
while simulated time < time limit do
remove earliest event from the event set
call the user-supplied reaction function with this event
So, let's look at our user-supplied reaction function in this example, mrpreact. The first few lines are
mrpreact <- function(evnt,simlist) {
etype <- evnt['evnttype']
if (etype == simlist$breakevnt) { # machine has gone down
# record this up time
simlist$totuptime <-
simlist$totuptime + simlist$currtime - evnt[4]
# is there is a free repairperson?
nrepb <- simlist$nrepbusy
if (nrepb < simlist$nrepairpersons) {
# start repair, no queuing
simlist$nrepbusy <- nrepb + 1
repduration <- rexp(1,simlist$reprate)
schedevnt(simlist,simlist$currtime+repduration,simlist$repairevnt,
appdata=c(NA,NA))
} else { # no repairpersons free, add job to queue
# record start queue wait
evnt[3] <- simlist$currtime
appendfcfs(simlist$queue,evnt)
}
} else { # etype = simlist$repairevnt
...These are the lines that handle breakdown events. Recall that our user-supplied reaction function is called by the package function mainloop, which has provided us the just-occurred event, evnt. We see that the above code checks evnt for event type, and if it is a breakdown event, executes the code that follows.
First there is bookkeeping to be done. Since an up time has just ended, we need to calculate how long this time lasted, and add that to our running total in simlist$totuptime.
Next we must check whether any repairpersons are free. If so, we call schedevnt to schedule a repair event for the machine that just went down. Otherwise, we'll need to add the machine to the queue, using the package function appendfcfs. Note that in that latter case, we must set the startqtime field in the event vector to the current time, so that later when the repair time begins, we can calculate the queue residence time for this event.
The code for the case of a repair event is similar:
} else { # etype = simlist$repairevnt
# bookkeeping
simlist$nrepairs <- simlist$nrepairs + 1
# start next up time for this machine
uptime <- rexp(1,simlist$uprate)
schedevnt(simlist,simlist$currtime+uptime,simlist$breakevnt,
appdata=c(NA,simlist$currtime))
# repairperson now free
simlist$nrepbusy <- simlist$nrepbusy - 1
# check queue for waiting jobs
if (nrow(simlist$queue$m) > 0) { # nonempty queue
qhead <- delfcfs(simlist$queue)
# bookkeeping
simlist$totqtime <- simlist$totqtime + simlist$currtime - qhead[3]
# start job service
# this repairperson now busy
simlist$nrepbusy <- simlist$nrepbusy + 1
srvduration <- rexp(1,simlist$reprate)
schedevnt(simlist,simlist$currtime+srvduration,simlist$repairevnt,
qhead[3:4])
}
}We schedule an up time for the newly-repaired machine, and now that this repairperson is free, we check whether there are any machines waiting in the queue that this repairperson can work on. If so, we remove the head of the queue and schedule a repair event for this machine. Also, since the queue wait for that machine has ended, we calculate its queue residence time and add it to our running total.
The interarrival and service time distributions are assumed exponential ("M" for "memoryless"), and there is a single server ("1"). The file inst/examples/MM1.R in the package simulates this system, with mm1 containing the overall code, and mm1react being the user-supplied reaction function. The latter handles two kinds of events, job arrival and job service completion.
Most of the code is similar to that of our first example. The main new part is use of the package function exparrivals, which we use to pre-calculate all the arrivals. This makes the code look a little more like process-oriented simulation, with arrivals handled separately. Without this, in the reaction function, each arrival would spawn the next.
In the initialization, the arrivals are pre-calculated with
exparrivals(simlist,meaninterarrv)As explained in the Technical Details section below, these are actually stored in a special auxiliary event set, but that need not concern the application programmer other than a special argument aevntset in newsim that signals use of the auxiliary event set internally.
simlist <-
newsim(timelim,3,appcols=c('arrvtime','jobnum'),aevntset=TRUE,dbg)Note that there are two application-specific fields in appcols: arrvtime is recorded so that when a job finally completes service, we can calculate how long it was in the system, thus enabling the computation of mean wait; jobnum is simply a job ID, 1,2,3,..., not directly used here, but usually a good idea, e.g. to help in debugging.
The rest of mm1 and mm1react are similar to the machine-repair example above.
Setting dbg to TRUE in the call to newsim specifies debug mode. This is then used by mainloop, which pauses after each event occurrence. The event and the new event list are printed out, and R browser mode is entered, enabling single-stepping and querying of variables.
An important function is cancelevnt, which does exactly what its name implies.
Suppose we are simulating a computer system that has some timeout period tmo. We would define a timeout event, and schedule an instance of that event for tmo time later. But if some action turns out to occur before then, we would need to cancel the timeout.
(This material requires advanced knowledge of programming.)
The heart of any DES library is the code that manages the event set processing. Typically that is done via a priority queue, with the word priority being interpreted in DES as earliest event time. This can implemented as a straight queue structure, with all pending events arranged in time order, or as a heap, with the earliest event always at the top.
However, this approach is slow if coded in R, as it does not take advantage of operations in which R is efficient, such as matrix multiplication. This is why for instance simmer's code for this portion of the package is written in C++. In order to stay purely in R, the DES package takes a different approach.
As mentioned earlier, DES implements its event set as a matrix. The rows of the matrix are not ordered, but the earliest event can be obtained efficiently using R's which.min function on the time column of the matrix.
However, even this would be slow for very large event sets. In DES, we provide the option of pre-calculating arrivals, and this would make the event sets large. On the other hand, arrivals are inherently ordered, so we store them separately. Then the earliest event time is determined as the smaller of the first arrival time and the result of applying which.min to the non-arrivals event set.
All this is how event-oriented systems work. Process-oriented systems for DES are generally considered to be clearer. A well-kown example is the Python library SimPy, on which simmer is based.
Process-oriented code is similar to threads programming, and may be implemented using a threads library. In our above examples, process-oriented code would be similar in many respects, but with the difference that we would have a thread for each entity. In the machine repair model, for instance, there may be a thread for each machine and a thread for each repairperson. The code for machine threads would look something like
repeat
simulate up time
wake a repairperson or join queue
simulate down time
The code for a repairperson might be something like
repeat
deactivate
upon receiving wakeup signal, simulate repair time
By focusing on each individual actors, e.g. individual machines, it is hoped that the code is clearer.
If implemented using a threads library, each simulation of a period of time, e.g. up time above, is handled by the thread relinquishing its timeslice, e.g. via a call to pthread_suspend in the pthreads library. A manager thread would then add the relinquishing thread to the event set, and activate whichever thread has the earliest event time.
Though Python does have a threads capability, SimPy instead takes advantage of Python's generator feature, implementing what amounts to a non-preemptive threads system.
I have an online course on DES, using SimPy as the example system. The course is in the PDF files in http://heather.cs.ucdavis.edu/~matloff/156/PLN, with the first unit of the SimPy tutorial being in the file http://heather.cs.ucdavis.edu/~matloff/156/PLN/DESimIntro.pdf See the above directory of PDF files for the remainder of the tutorial.