Step 4: Obtaining the Measures of Interest

Various measures of interest can be calculated with TANGRAM-II, including measures obtained from functions of state variables, and conditional probabilities. For instance, in our M/M/1/k model, we can compute the expected number of packets in the system (queue + server) and the probability mass function of the number of the packets in the queue (pmf).

To obtain a measure of interest, click on the Measures of Interest button in the Modeling Environment interface. The graphical interface is shown in Figure .

Figure: The Measures of Interest Module.

As an example, choose ``PMF of one or more state variables'' . The user must specify the name of the measure of interest, e.g pmf_queue, that will be used to identify the file where the selected measure will be stored. Then the user must specify the file that contains the state probability vector. In our example that file has the form $<$name of the model$>$.SS.gth (in this example, it is the file MM1k.SS.gth).

All state variables of the M/M/1/k model appear in the ``Choose Variables'' box. Our example has only one state variable, so the only element in the box is Server_Queue.queue. Add the Server_Queue.queue variable into the right box. After clicking on the Evaluate button, the measure is evaluated provided that no mistakes were made. The message ``Measures of interest generated'' will pop up as soon as the calculations are concluded.

To plot the results, click on the Plot button and choose the file name that you specified to contain the Measures of Interest (in this case, MM1k.IM.pmf_queue). We can see that the expected value is shown, as well as the pmf of the number of packets in the queue (GNUPlot button). This plot is shown in Figure .

Figure: The plot generated by the PMF Module.

We can perform more experiments. For example, we can change the value of the service rate to 50. Then we generate the state transition probability matrix, the steady-state solution, and compute the same measure of interest for this service rate. Figure shows the result.

Figure: The plot generated by the PMF Module.

We can also evaluate other measures. For example, the utilization of the queue, the expected number of customers in the queue (in this case $E[queue] = (n-1)\pi(n)$, where $\pi(n)$ is the probability that the system has $n$ customers), the average time in the queue using Little's Law, etc.

Assume the measure of interest is the probability mass function of the queue size, conditioned on that the system has more than $50$ customers. We encourage the user to check the Conditional box in Figure and enter the proper condition. The result is shown in .

Figure: The plot generated by the PMF Module.

The user can also obtain a variety of measures of interest from reward models. Examples will be given in the following section.