To get you started, the function for simulating M/M/1 queues in Mathematica is
QueueingProcess. There are a number of examples in that documentation which might be helpful.
You defined the process as you described with the following code:
myProcess=QueueingProcess[0.1, 0.8, 1, Infinity, 3000]
There are many things you do with myProcess. The first is to ask about its properties:
QueueProperties[myProcess]
This will give you some steady state properties such as:
"MeanSystemSize" - mean number of jobs in the system
"MeanSystemTime" - mean time spent in the system
"MeanQueueSize" - mean number of jobs in the queue
"MeanQueueTime" - mean time spent in the queue
You can simulate it:
data = RandomFunction[myProcess, {0, 100}]
ListLinePlot[data, InterpolationOrder -> 0, Filling -> Axis]
