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

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

• Two things: A poisson distribution won't work for QueueingProcess. According to the documentation, you need a continuous one. Secondly, did you look at the page on Queuing Processes in the docs? – Lukas Lang Sep 5 '17 at 8:28
• @Mathe172, Yes, I have looked at it. But I have Poisson distribution for the request arrivals. How can I solve my problem? I have edited the title of my question. – George Farnandez Sep 5 '17 at 8:40
• Don't you mean that you have a PoissonProcess for the arrivals, rather than a PoissonDistribution? I think that's the confusion here. A PoissonProcess should work in a QueueingProcess. – Sjoerd Smit Sep 5 '17 at 8:49
• Also: note that having a Poisson process for the arrivals is equivalent to having an exponential distribution for the inter-arrival times. – Sjoerd Smit Sep 5 '17 at 8:56
• @SjoerdSmit, Yes, it is a Poisson process with average rate [Lambda]. How do I use it to generate the number of resources used at each time-slot? – George Farnandez Sep 5 '17 at 9:07

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


• Thanks a lot for your answer. It almost answers my question. Where does it say that \[Lambda] is the average rate of the Poisson process and \[Mu] is the mean of Exponential distribution ? If it is not the case then how do I guarantee this? – George Farnandez Sep 6 '17 at 1:46