exponential distribution

[nikvyas]

Good morning. I was wondering if somebody could help me out with, what should be, a very simple question - but for some reason I've got a mental block.

How do I apply the exponential distribution within Vensim in the following case:

Time step = 1 hour
Final Time = 100 hours
MTBF (mean time between failures) = 5

Could somebody also please explain what the parameters 'stretch' and 'shift' mean within the distribution equation.

Thanks

Nik

[bob@vensim.com]

If you have a MTBF you almost certainly want a poisson distribution for failures. I would recommend reading up on Poisson failure processes.

[nikvyas]

Thanks Bob. However, all of the literature I have read indicates an exponential distribution is the most effective and, unfortunately, my boss also wants me to use an exponential distribution.

Any assistance grayefully received.

[LAUJJL]

the poisson (with peaks or not) distribution is currently used to model the new incomers in a queue or the failures of a set of approxiamtely anything, and the exponential distribution is used to model the time of repair or servicing the customer in the queue, at least in the litterature.
Regards.
JJ

[LAUJJL]

Hi
the exponential distribution is effectively used to model the
number of failures during a certain time among a certain number of machines mong other distributions.
Regards.
JJ

[LAUJJL]

the exponential distribution is valid only if the probibility of failure is independant of the age of the equipment and is not valid if it increases when the equipment grows older.
The normal, log normal are more valid in this case.
JJ

[nikvyas]

Thanks very much for your response jj.

The failure rate will be constant over time, which is why I think the exponential distribution is the more valid. Up to this point, my models have assumed a constant MTBF (of 5) so it behaves like a metronome - that is, every 5 hours of usage, the device will experience a failure. However, this is obviously not the case and I want to introduce a degree of randomness as in real life and everyone I talk to want the distribution to be exponential. I haven't found anything within the Vensim literature that explains how to use the Random Exponential function - could you please explain how i use it?

Thanks

Nik

[nikvyas]

Hello again.

What I need in addition (sorry for wanting so much) is to know how I apply this distribution to the time axis. ie., if the finish time is 100, I need a number of failure pulses along the time axis - and these failure pulses come from the exponential distribution.

I just need to explain at this point, that the organisation I work for (which has 80,000 employees) is just starting to use Vensim - and the team within which I work is 'testing' it and if it does what we need it to do, it will be rolled out across the whole organisation. So any help will be of great benefit.

Thanks.

[LAUJJL]

Hi

joined a simplistic model that has exponential failures along the time axis.
Note that the exponential distribution is continuous and it is necessary if the amount of failure is small to convert the continuous value into a discreet one, using the integer value.
Regards.
JJ

[nikvyas]

Thanks again JJ.

However, what I actually wanted was as follows:

Say the Mean Time Between Failures is 5.
Using the exponential function might give us a failure at the following times (for instance):

3, 9, 11, 17, 18 etc etc.

So whereas you have shown the number of failures per day, what I need is a series of pulses (the failure pulses) and in between no pulse (no failure).

Sorry to be a pain JJ.

[LAUJJL]

Hi
It is always the same kind of problem: continuous versus discreet.
In continuous environment where the time is sliced and the
data aggregated during the time step, it is diffucult to represent an event.
In fact you should consider a sufficiently small time step and in this time to simulate the realization or not realization of one failure.
It may be probably done in Vensim, using may be a binomial distribution where the parameters is the number of elements that can fail and the probability of failure of one element in the interval of time considered.
the interval of time must be very small, so that the probability of two failures occuring in the same interval is very small, to simulate a discreet event.
So you will need a very small time step with a very small probability of failure to approximate a discreet event in a
quasi continuous time.
But you will need a high level of time steps to do that, and I think that a pure discreet software will generate discreet events in a completely continuous time more easily.
But be careful as classical discreet software rarely consider
feed back loops.
If you want to mix discreet and continuous simulation, you
have one software: anylogic that apparently mixes discreet and continuous.
But visiting their web sites, I did not see any applications that mixes both.
Whether it works correctly, I do not know!
In fact continous simulation uses not continuous time and
discreet simulation uses continuous time, which looks a bit strange at first time.
But I think that somebody like Bob or another administrator will be more competent in this domain than me.
Ask them.
Regards.
JJ

[bob@vensim.com]

I believe you are confusing the distribution of times between failures which the arrival rate of failures. An exponential arrival rate really makes no sense. To do the time between failures (assuming 1 failure stops additional failures use something like)

next failure time = INTEG(IF THEN ELSE(want new failure time,((Time + random var for fail time)-next failure time)/TIME STEP,0),
INITIAL TIME + random var for fail time)

want new failure = IF THEN ELSE(have failure,1,0)
have failure = IF THEN ELSE(Time >= next failure time,1,0)

- I believe this is the same as using Poisson to get the number of failures.

[nikvyas]

Thanks JJ and Bob.

I've come up with the following:

distribution = RANDOM EXPONENTIAL( 5e-005 , 20000 , 0 , 0.2 , 5555 )
batch = DELAY BATCH(distribution, 1, 0 , 0 , 0 , 0 )
Time period is 1 hour

So, if the MTBF is 5, it means that the average per hour is 0.2. The batch function waits until 'distribution' reaches the value of 1 (which equates to an MTBF of 5) and then pulses. We then have an exponential pattern against time.

All comments on the above approach welcome.

Thanks

Nik

[bob@vensim.com]

Hi Nik,


This probably won't work - once you pick a new failure time you have to wait till that failure occurs to generate yet another failure time.

[nikvyas]

Thanks Bob - I fear you're right.

btw - my company has just placed an order for the first few copies of Vensim!