Lognormal Distribution Setup

[Lloyd Cross]

I have some actual failure data where a machine state is sampled every second over the course of 24 hours. These state flags in the data allow me to calculate the time between failures (TBF) and time to recover (TTR) for each unplanned stoppage. The TBF follows a 2 parameter Weibull distribution, and the TTR can be approximated nicely with a Lognormal distribution.

In Jaamsim, the Weibull distribution is straightforward, and when I use an expression logger to capture the simulated breakdowns, an analysis of these data yields Weibull parameters that fall nicely between the confidence intervals for the estimates of the real data - so far so good.

Where I'm having trouble is with the Lognormal distribution in Jaamsim. In my data, the Scale (alpha) parameter is -0.53, but Jaamsim will not accept a number less than zero here. A negative scale is a valid input for a lognormal distribution (and when I transform the TTR data to ln(TTR) I get a normal distribution with a mean of -0.53, sigma of 0.858.

The Shape (sigma) parameter does seem to be working, as the simulated output reproduces this value within the confidence limits - this occurs when I enter the Shape Parameter from the real data's PDF (in this case, 0.858) into the NormalStandardDeviation field in Jaamsim.

I tried entering -0.53 for the NormalMean field - but that resulted in an output of the simulated TTR data with a Scale Parameter of -4.64, so I'm interpreting that field wrong.

I've attached an image of the Lognormal PDF from the actual data. I'm using SAS JMP to fit the distributions. How would this be properly mapped to the Jammsim Lognormal distribution input editor?

Thanks!

Lloyd

[Harry King]

Llyod,

The labels 'scale' and 'shape' given by SAS are a bit confusing. The Wikipedia article gives the usual terminology which is consistent with JaamSim. Note that the Scale input used by JaamSim is NOT the scale value used by SAS. It is a factor that gives the distribution the correct units, in this case minutes.

The input data should be as follows:

• Scale = 1 min
• NormalMean = -0.530652
• NormalStandardDeviation = 0.8578484

Check the definitions for the SAS parameters. Normally, the variable mu is the mean of the underlying normal distribution and sigma is the standard deviation of the normal distribution.

Harry

[Lloyd Cross]

Harry,

Thanks for the quick response! To have the someone who developed the software writing you back within a few hours is unheard of anywhere else, especially for a tool as complex and capable as Jaamsim. That's amazing - thanks.

Sure enough, I had left the Scale field blank, so my original results were off by a factor of 60x Once that was fixed, my test model did a remarkable job reproducing the actual live data, both in terms of TBF and TTR. Given there were 86,400 opportunities (seconds in a day), the model predicted the actual number of failures within a few percent.

To clarify, if my data were in seconds, the Scale Field would have been 1 s, in hours 1 h, and such?

The NIST's online statistics handbook is the reference I use quite a bit, which describes mu and sigma as scale and shape for the lognormal PDF, as do others. If I'm correct about the Jaamsim "scale" field used as a time unit of measure scaler - perhaps some clarification in the fly-over help would prevent confusion?

Thanks so much for the help!

Lloyd

[Harry King]

Llyod,

Yes, the Scale input should be 1 s if the data used for the lognormal distribution is in seconds.

Harry