Exponential Distribution Question - Service Time Distribution

[coolness]

service times of a device has been given. How can we determine whether the service time distribution is memoryless?
My approach was to prove the services time distribution follows poisson/exponential distribution which inturn is memoryless. But how to prove a distribution follows exponential or poisson ? Assume more than 12000 service times are given.

 

[Miner]

service times of a device has been given. How can we determine whether the service time distribution is memoryless?
My approach was to prove the services time distribution follows poisson/exponential distribution which inturn is memoryless. But how to prove a distribution follows exponential or poisson ? Assume more than 12000 service times are given.

Why is memorylessness a concern?

This sounds like a reliability issue. In reliability, you can fit many types of distributions (e.g., log-normal, normal, weibull, etc.) and are not limited to one that is memoryless.

Do you have access to software such as Minitab? Many statistical packages have distribution fitting routines that aid in identifying the distribution.

 

[Bev D]

by "memoryless" do you mean that the service time for any event is NOT dependent or affected by the previous event? (either in it's duration or some other activity that might result in a longer or shorter time for the subsequent event(s)) In statistical parlance: memoryless = independence...

a bit more information about the specifics would allow us to be more helpful. Frequently qustions involving statistics aren't complicated by the math but by the physics of the specific situation.

It's also important to remember that - if I have correctly interpreted memoryless - although the Poisson distribution was developed for independent events, it does not necessarily follow that a process that fits the Poisson in shape and statistical calculations is itself comprised of independent events...

 

[coolness]

Miner,
The question is to prove the distribution is memoryless.

BevD ,
Yes. you are right. Memoryless = independence. how would you prove a service times follows poissonian or exponential? are they any softwares that help?

 

[Bev D]

Miner,
The question is to prove the distribution is memoryless.

BevD ,
Yes. you are right. Memoryless = independence. how would you prove a service times follows poissonian or exponential? are they any softwares that help?

you have the cart before the horse.

first you must determine if the results are "memoryless" or independent.
Then you can determine which distribution has the best fit for your data.
(I say this becuase you are saying that you want to prove if the 'distribution' is memoryless. teh correct way of saying this is: if the process is memoryless. Distribution shapes themselves are independent of the independence of the data...

You have time based data. A Poisson distribution may fit in terms of the mathematical shape, average, standard deviation but that won't mean that the data are independent.

The best checks for independence:
FIRST plot you data in time sequence. The time sequence should be based on the START of the event and not it's completion. This goes on the X axis. The Y axis is the duration of the event. This is the most intuitive check for most people. you should see no obvious trends or cycles.

A really simple second check (statistically more rigorous, but not always obvious to managers) is to plot the duration of each event vs the duration of the event immediately preceding it.
X axis = event 1, Y axis = event 2
X axis = event 2, Y axis = event 3.

You can also post your data or results for us to look at.

There are a lot of software applications for distribution fitting that are available for a little money or a lot. A simple google search will help you find what fits your needs and budget. Other here can help you with their experiences using a particular software once you have narrowed down your selection.

 

[Tim Folkerts]

As I understand the question and the exponential distribution, "memoryless" is equivalent to "exponential distribution". If the time to failure is indeed independent of the time already elapsed, then you have an exponential distribution. A prime example is radioactive decay. Many electronic parts also come close to this sort of behavior. http://en.wikipedia.org/wiki/Exponential_distribution discusses "memorylessness" and some other properties of the distribution.

A simple first step would be to plot the data and visually compare the number in service to the predicted number is service. The predicted number would be
N(remaining) = N(initial) * exp[- λt]
where 1/λ = mean life time. This should give you a pretty good idea if data comes close to following the exponential distribution.

To be more mathematical, you could do a chi-square test.

Or you could use a standard software package like Minitab to test the distribution. (In fact, if you post the data as an attachment, someone like me should be able to run a quick test. All that would be needed is the time in service for as many devices as you have data for.)


Tim F

 

[Bev D]

As I understand the question and the exponential distribution, "memoryless" is equivalent to "exponential distribution". If the time to failure is indeed independent of the time already elapsed, then you have an exponential distribution. A prime example is radioactive decay.

very true. but the opposite is not true. A distribution (a pile of data arranged by the frequency of occurence of values without regard to their time sequence) that fits the Exponential (or Poisson) mathematically and shape-wise is not necessarrily from a memoryless process.

That - appears - to be the heart of the OP's question. Do I have a memoryless process. To determine that he must test for memory or dependence of the data.

I have seen many processes whose cycle time distributions fit the exponential jsut fine. BUT soem are dependent and some are not. A single lane of flow will result in a backup (just like traffic) when the lead unit gets slowed down. the units behind it are also slowed down. In the trend you see the jam and release, but in the time-less distribution all you see is an exponential shaps that passes the goodness of fit test. Then I have processes that have multiple lanes and when one unit gets slowed down it doesn't affect the others but hte 'slow down' events occur randomly across all lanes and have varying durations of effective time. The trend shows random independent slow and fast times. the process is independent and the distributio looks exponeential and passes teh goodness of fit test.

So what really matters here is what the OP's actual question is? Is he looking for independence or is he looking for which distritbution fits the distribtuion of his data?

If all he cares about is which distribution provides the best statistical description of his distribution (not his trend) then he can proceed directly to goodness of fit tests for distributions.