How to generate number of points in a m/m/c/c process with Poisson arrival and exponential service time?

Question

I am considering the Earling-B or m/m/c/c process.

Let the request arrival rate follows Poisson Process with average rate, $\lambda$

Let the service time follows Exponential Distribution, i.e., $\mu$ is the mean of Exponential Distribution.

Let's say, I have a grid of $c \times T$. Here $c=5$, i.e., in the Queueing process, there are 5 servers and 5 requests can be served at most concurrently. Let, $T=100$ is the number of time-slots that I need to simulate. Therefore, each block in the x-axis represents a time-slot and each block in the y-axis is the amount of resource required to serve a request.

For example,

Let the number of requests at the first time-slot is 3. The first and the third requests need service time of 2 times slots, while the second request requires service time of only 1 time-slot.

As a result, in the grid,

The first row will have two grid-element (first and second column) filled

The second row will have one grid-element (first column) filled

The third row will have two grid-element (first and second column) filled

Now, let's say, at the second time-slot, the number of requests is 5. In this case, due to the nature of Earling-B or m/m/c/c Queueing Process, only 3 more requests can be served at this particular time-slot as we already have two requests being served (first and third request arrived at first time-slot). Other requests are blocked or lost.

How do I generate the number of resources being used by the system in different time-slots?

Lets assume, $\lambda=0.2$ per time-slot

$\mu=2$ time-slots

Answer 1

You should be able to get the data you want using QueueProperties and/or RandomFunction (both of which are linked in the docs of QueueingProcess and Queueing Processes):

q = QueueingProcess[\[Lambda], \[Mu], c, c]
Simplify[QueueProperties[q], c > 0 && c \[Element] Integers]

yields:

You can also query e.g. the throughput or utilization factor directly for your values:

In[1]:= QueueProperties[q /. {\[Lambda] -> 0.2, \[Mu] -> 2, c -> 5}, "Throughput"]
Out[1]= 0.2
In[2]:= QueueProperties[q /. {\[Lambda] -> 0.2, \[Mu] -> 2, c -> 5}, "UtilizationFactor"]
Out[2]= 0.02

You can also simulate your process (obviously, you need to insert actual values here):

d = RandomFunction[q /. {\[Lambda] -> 0.2, \[Mu] -> 2, c -> 5}, {0, 100}]
ListStepPlot[d, Filling -> Axis]

May yield:

where the plot shows how many slots are used at each point in time.

Note

If you meant $\mu=\frac{1}{2}$, the results are a bit more interesting (as the queue is not so empty):

In[1]:= QueueProperties[q /. {\[Lambda] -> 0.2, \[Mu] -> 1/2, c -> 5}, "Throughput"]
Out[1]= 0.199989
In[2]:= QueueProperties[q /. {\[Lambda] -> 0.2, \[Mu] -> 1/2, c -> 5}, "UtilizationFactor"]
Out[2]= 0.0799954