6.1.6 Easy Path
Leanna Perr
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@Megan, What behavior are you expecting in different browser? ? No the browser (whatever is in use) decides what symbol how to be treated. It's Dropbox server responsibility! And... there many symbols (including tilde) are treated as words separators and not as a part of the text to be looked for. I'm not aware of any way such symbols to be tagged (so included in the search).
Yeah... I agree. Even more: to avoid "a massing PITA", the current search mode may stay as default one, so the people that don't get interested wont ever noticed the addition. It's easy to be added iconified dropdown menu at the searchline beginning (where a magnifying glass icon resides now).
There can be added command line path template option - representing a template feature for looking like POSIX shell does (for example: * - zero or more arbitrary symbols, ? - exactly one arbitrary symbol). It's simple and easy to use and let you represent more precisely what you're looking for.
It's a good idea, I think, to be added and more advanced search options, where the search template can describe even better the thing to be searched (usable not only for file or folder search, based on path). One such option can be regular expression (RE) usage - either basic RE (BRE) or extended RE (ERE) or maybe both. Despite of more difficult for usage, it's very powerful tool to search for something. Even more I cannot think of some platform not providing some kind of RE by default, so I'm ready to bet that Dropbox system core has it available already. So, it cannot be actually completely new feature, but rather providing access to existing feature, currently usable internally only (or it might have completely forgotten).
Final result: Instead of one magnifying glass icon at the searchline beginning, interchangeable icon set (the magnifying glass lefts the default), representing different search modes, providing lot of choices.
So, there are several reasons why fitting a probability distribution, and then generating random variates from it to drive the simulation, is often preferred over using the observed real-world data directly. One specific situation where driving the simulation directly from the observed real-world data might be a good idea is in model validation, to assess whether your simulation model accurately reproduces the performance of the real-world system. Doing this, though, would require that you have the output data from the real-world system against which to match up your simulation outputs, which might not be the case if your data-collection efforts were directed at specifying input distributions. However, some recent work ((Song, Nelson, and Pegden 2014), (Song and Nelson 2015)) has shown that there are cases where sampling from sets of observed data is the preferred approach to fitting a probability distribution. We will briefly discuss this topic and the Simio method for sampling from observed data sets in Section 6.5. Our initial job, then, is to figure out which probability distribution best represents our observed data.
It should match whether the input quantity is inherently discrete or continuous. A batch size should not be modeled as a continuous RV (unless perhaps you generate it and then round it to the nearest integer), and a time duration should generally not be modeled as a discrete RV.
So getting discrete vs. continuous right, and the range right, is a first step and will narrow things down.But still, there could be quite a few distributions remaining from which to choose, so now you need to look for one whose shape (PMF for discrete or PDF for continuous) resembles, at least roughly, the shape of a histogram of your observed real-world data. The reason for this is that the histogram is an empirical graphical estimate of the true underlying PMF or PDF of the data. Figure 6.3 shows a histogram (made with part of the Stat::Fit\(^(R)\) distribution-fitting software, discussed more below) of the 47 observed service times from Figure 6.1 and the file Data_06_01.xls. Since this is a service time, we should consider a continuous distribution, and one with a finite left tail (to avoid generating negative values), and perhaps an infinite right tail, in the absence of information placing an upper bound on how large service times can possibly be. Browsing through a list of PDF plots of distributions (such as the Simio Reference Guide cited above and in Simio via the F1 key or the ? icon near the upper right corner), possibilities might be Erlang, gamma (a generalization of Erlang), lognormal, Pearson VI, or Weibull. But each of these has parameters (like shape and scale, for the gamma and Weibull) that we need to estimate; we also need to test whether such distributions, with their parameters estimated from the data, provide an acceptable fit, i.e., an adequate representation of our data. This estimating and goodness-of-fit testing is what we mean by fitting a distribution to observed data.
To view PMFs and PDFs of the supported distributions, follow the menu path Utilities \(\rightarrow\) Distribution Viewer and then select among them via the pull-down field in the upper right of that window. Note that the free textbook version of Stat::Fit includes only seven distributions (binomial and Poisson for discrete; exponential, lognormal, normal, triangular, and uniform for continuous) and is limited to 100 data points, but the full commercial version supports 33 distributions and allows up to 8000 observed data points.
The Calculations tab (Figure 6.6) has choices about the method to form the Estimates (MLE, for Maximum Likelihood Estimation, is recommended), and the Lower Bound for the distributions (which you can select as unknown and allow Stat::Fit to choose one that best fits your data, or specify a fixed lower bound if you want to force that). The Calculations tab also allows you to pick the kinds of goodness-of-fit Tests to be done: Chi Squared, Kolmogorov-Smirnov, or Anderson-Darling, and, if you selected the Chi Squared test, the kind of intervals that will be used (Equal Probability is usually best). See Section 6.1.5 for a bit more on assessing goodness of fit; for more detail see (Banks et al. 2005) or (Law 2015).
To fit the four distributions you chose to represent the service-time data, follow the menu path Fit \(\rightarrow\) Goodness of Fit, or click the Fit Data button (labeled just FIT) in the toolbar, to produce a detailed window of results, the first part of which is in Figure 6.7.
A brief summary is at the top, with the test statistics for all three tests applied to each distribution; for all of these tests, a smaller value of the test statistic indicates a better fit (the values in parentheses after the Chi Squared test statistics are the degrees of freedom). The lognormal is clearly the best fit, followed by exponential, then triangular, and uniform is the worst (largest test statistics).
The final step is to translate the results into the proper syntax for copying and direct pasting into Simio expression fields. This is on File \(\rightarrow\) Export \(\rightarrow\) Export fit, or the Export toolbar button, and brings up the EXPORT FIT dialog shown in Figure 6.10.
The chi-squared goodness-of-fit test can also be applied to fitting a discrete distribution to a data set whose values must be discrete (like batch sizes). In the foregoing discussion, \(p_j\) is just replaced by the sum of the fitted PMF values within the \(j\)th interval, and the procedure is the same. Note that for discrete distributions it will generally not be possible to attain exact equiprobability on the choice of the intervals.
In Stat::Fit, P-P plots are available via the menu path Fit \(\rightarrow\) Result Graphs \(\rightarrow\) PP Plot, to produce Figure 6.12 for our 47-point data set and our four trial fitted distributions. As in Figures 6.9 and 6.11, you can add fitted distributions by clicking them in the upper right box, and remove them by clicking them in the lower right box. The P-P plot for the lognormal fit appears quite close to the diagonal line, signaling a good fit, and the P-P plots for the triangular and uniform fits are very far from the diagonal, once again signaling miserable fits for those distributions. The exponential fit appears not unreasonable, though not as good as the lognormal.
So it appears that the independence assumption for our 47 service times is at least plausible, having found no strong evidence to the contrary. Finding that your data are clearly not independent mandates some alternative course of action on your part, either in terms of modeling or further data collection. One important special case of this in simulation is a time-varying arrival pattern, which we discuss below in Section 6.2.3, and modeling this is well-supported by Simio. But in general, modeling non-independent input process in simulation is difficult, and beyond the scope of this book; the interested reader might peruse the time-series literature (such as (Box, Jenkins, and Reinsel 1994)), for examples of methods for fitting various kinds of autoregressive processes, but generating them is generally not supported in simulation software.
In this case, Stat::Fit can produce an empirical distribution, which is basically just a version of the histogram, and variates can be generated from it in the Simio simulation. On the Stat::Fit menus, do File \(\rightarrow\) Export \(\rightarrow\) Export Empirical, and select the Cumulative radio button (rather than the default Density button) for compatibility with Simio. This will copy onto the Windows clipboard a sequence of pairs \(v_i, c_i\) where \(v_i\) is the \(i\)th smallest of your data values, and \(c_i\) is the cumulative probability of generating a variate that is less than or equal to the corresponding \(v_i\). Exactly what happens after you copy this information into Simio depends on whether you want a discrete or continuous distribution. In the Simio Reference Guide (F1 or ? icon from within Simio), follow the Contents-tab path Modeling in Simio \(\rightarrow\) Expression Editor, Functions and Distributions \(\rightarrow\) Distributions, then select either Discrete or Continuous according to which you want:
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