C.7 Additional Distribution Modeling Concepts
This section wraps up the discussion of input modeling by covering some additional topics that often come up during the modeling process.
Throughout the service time example, a continuous random variable was being modeled. But what do you do if you are modeling a discrete random variable? The basic complicating factor is that the only discrete distribution available within the Input Analyzer is the Poisson distribution and this option will only become active if the data input file only has integer values. The steps in the modeling process are essentially the same except that you cannot rely on the Input Analyzer. Commercial software will have options for fitting some of the common discrete distributions. The fitting process for a discrete distribution is simplified in one way because the bins for the frequency diagram are naturally determined by the range of the random variable. For example, if you are fitting a geometric distribution, you need only tabulate the frequency of occurrence for each of the possible values of the random variable 1, 2, 3, 4, etc.. Occasionally, you may have to group bins together to get an appropriate number of observations per bin. The fitting process for the discrete case primarily centers on a straightforward application of the Chi-Squared goodness of fit test, which was outlined in this chapter, and is also covered in many introductory probability and statistics textbooks.
If you consider the data set as a finite set of values, then why can’t you just reuse the data? In other words, why should you go through all the trouble of fitting a theoretical distribution when you can simply reuse the observed data, say for example by reading in the data from a file. There are a number of problems with this approach. The first difficulty is that the observations in the data set are a sample. That is, they do not necessarily cover all the possible values associated with the random variable. For example, suppose that only a sample of 10 observations was available for the service time problem. Furthermore, assume that the sample was as follows:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| 36.84 | 38.5 | 46.23 | 46.23 | 46.23 | 48.3 | 52.2 | 53.46 | 53.46 | 56.11 |
Clearly, this sample does not contain the high service times that were in the 100 sample case. The point is, if you resample from only these values, you will never get any values less than 36.84 or bigger than 56.11. The second difficulty is that it will be more difficult to experimentally vary this distribution within the simulation model. If you fit a theoretical distribution to the data, you can vary the parameters of the theoretical distribution with relative ease in any experiments that you perform. Thus, it is worthwhile to attempt to fit and use a reasonable input distribution.
But what if you cannot find a reasonable input model either because you have very limited data or because no model fits the data very well? In this situation, it is useful to try to use the data in the form of the empirical distribution. Essentially, you treat each observation in the sample as equally likely and randomly draw observations from the sample. In many situations, there are repeated observations within the sample (as above) and you can form a discrete empirical distribution over the values. If this is done for the sample of 10 data points, a discrete empirical distribution can be formed as shown in Table C.10. Again, this limits us to only the values observed in the sample.
| \(X\) | PMF | CDF |
|---|---|---|
| 36.84 | 0.1 | 0.1 |
| 38.5 | 0.1 | 0.2 |
| 46.23 | 0.3 | 0.5 |
| 48.3 | 0.1 | 0.6 |
| 53.46 | 0.2 | 0.8 |
| 55.33 | 0.1 | 0.9 |
| 56.11 | 0.1 | 1 |
One can also use the continuous empirical distribution, which interpolates between the distribution values.
What do you do if the analysis indicates that the data is dependent or that the data is non-stationary? Either of these situations can invalidate the basic assumptions behind the standard distribution fitting process. First, suppose that the data shows some correlation. The first thing that you should do is to verify that the data was correctly collected. The sampling plan for the data collection effort should attempt to ensure the collection of a random sample. If the data was from an automatic collection procedure then it is quite likely that there may be correlation in the observations. This is one of the hazards of using automatic data collection mechanisms. You then need to decide whether modeling the correlation is important or not to the study at hand. Thus, one alternative is to simply ignore the correlation and to continue with the model fitting process. This can be problematic for two reasons. First, the statistical tests within the model fitting process will be suspect, and second, the correlation may be an important part of the input modeling. For example, it has been shown that correlated arrivals and correlated service times in a simple queueing model can have significant effects on the values of the queue’s performance measures. If you have a large enough data set, a basic approach is to form a random sample from the data set itself in order to break up the correlation. Then, you can proceed with fitting a distribution to the random sample; however, you should still model the dependence by trying to incorporate it into the random generation process. There are some techniques for incorporating correlation into the random variable generation process. An introduction to the topic is provided in (Banks et al. 2005).
If the data show non-stationary behavior, then you can attempt to model the dependence on time using time series models or other non-stationary models. Suffice to say, that these advanced techniques are beyond the scope of this text; however, the next section will discuss the modeling of a special non-stationary model, the non-homogeneous Poisson process, which is very useful for modeling time dependent arrival processes. For additional information on these methods, the interested reader is referred to (Law 2007) and (L. M. Leemis and Park 2006) or the references therein.
Finally, all of the above assumes that you have data from which you can perform an analysis. In many situations, you might have no data whatsoever either because it is too costly to collect or because the system that you are modeling does not exist. In the latter case, you can look at similar systems and see how their inputs were modeled, perhaps adopting some of those input models for the current situation. In either case, you might also rely on expert opinion. In this situation, you can ask an expert in the process to describe the characteristics of a probability distribution that might model the situation. This is where the uniform and the triangular distributions can be very useful, since it is relatively easy to get an expert to indicate a minimum possible value, a maximum possible value, and even a most likely value.
Alternatively, you can ask the expert to assist in making an empirical distribution based on providing the chance that the random variable falls within various intervals. The breakpoints near the extremes are especially important to get. Table 1.8 presents a distribution for the service times based on this method.
| \(X\) | PMF | CDF |
|---|---|---|
| (36, 100] | 0.1 | 0.1 |
| (100, 200] | 0.1 | 0.2 |
| (200 - 400] | 0.3 | 0.6 |
| (400, 600] | 0.2 | 0.8 |
| (600, \(\infty\)) | 0.2 | 1.0 |
Whether you have lots of data, little data, or no data, the key final step in the input modeling process is sensitivity analysis. Your ultimate goal is to use the input models to drive your larger simulation model of the system under study. You can spend significant time and energy collecting and analyzing data for an input model that has no significant effect on the output measures of interest to your study. You should start out with simple input models and incorporate them into your simulation model. Then, you can vary the parameters and characteristics of those models in an experimental design to assess how sensitive your output is to the changes in the inputs. If you find that the output is very sensitive to particular input models, then you can plan, collect, and develop better models for those situations. The amount of sensitivity is entirely modeler dependent. Remember that in this whole process, you are in charge, not the software. The software is only there to support your decision making process. Use the software to justify your art.