# Thread: Number serviceable at given time

1. ## Number serviceable at given time

If I have 50 systems each operating 400 hours per year with an MTBF of 900 hours how do I calculate how many would be serviceable at one time? Systems are repairable and have a repair turn round time of 21 days.

2. ## Re: Number serviceable at given time

It sounds like you want to calculate the Availability. This is often calculated using the MTBF and the MTTR (Mean Time To Repair). Availability = MTBF / (MTBF + MTTR). Be aware that there are several different methods for calculating different Availabilities. However, given the information that you presented, this is the most likely version.

Be careful, like MTBF, MTTR is based on an Exponential distribution, not a Normal distribution, so do not use the arithmetic mean for turnaround time.

3. ## Re: Number serviceable at given time

Looks like a queuing modeling problem. The most common/easiest choice would be assuming an exponential waiting time for each failure (same as Poisson process)

Now you need to clarify the question a little bit:

If I have 50 systems each operating 400 hours per year
how many would be serviceable at one time?
In 1 year there are more than 400 hours. So what is the schedule for each 50 systems - switching from working to idling and vice versa? Are they working in parallel at the same time or some other form? Are they independent?

And the number of working systems, at one time should be a random variable as the failure time is random. The distribution depends on your assumption behind.

4. ## Re: Number serviceable at given time

Originally Posted by Miner
It sounds like you want to calculate the Availability. This is often calculated using the MTBF and the MTTR (Mean Time To Repair). Availability = MTBF / (MTBF + MTTR). Be aware that there are several different methods for calculating different Availabilities. However, given the information that you presented, this is the most likely version.

Be careful, like MTBF, MTTR is based on an Exponential distribution, not a Normal distribution, so do not use the arithmetic mean for turnaround time.
Thanks for the reply. Availability will donate the probability of a system being available at a given point in time, normall expressed as a decimal or percentage. What I am trying to work out is the number of systems that are serviceable at any one time, or the converse i.e. the number of systems that will be in the repair loop at any given time.

5. ## Re: Number serviceable at given time

Originally Posted by BGM
Looks like a queuing modeling problem. The most common/easiest choice would be assuming an exponential waiting time for each failure (same as Poisson process)

Now you need to clarify the question a little bit:

In 1 year there are more than 400 hours. So what is the schedule for each 50 systems - switching from working to idling and vice versa? Are they working in parallel at the same time or some other form? Are they independent?

And the number of working systems, at one time should be a random variable as the failure time is random. The distribution depends on your assumption behind.
Thank you for the reply. Each system is independent of all others. Given that there are 50 systems operating 400 hours each there are 20,000 operating hours per year giving 20000/900 = 22 failures per year. Am I right that based on that and assuming a constant failure rate that there would be 22/52 *3 = 1.3 failures every 3 weeks giving an average of 2 equipments in the 3 week repair loop?

6. ## Re: Number serviceable at given time

I am still not quite understand the schedule of the system.

1. Will all 50 systems work at the same time whenever there are no failure among them?

2. Do all systems have two states only - either working or repairing - so that the sum of the working time and repairing time will always equal to 24 hours in 1 day?

3. If I am understanding correctly, you are asking the long-run average number of working systems.

7. ## Re: Number serviceable at given time

Originally Posted by BGM
I am still not quite understand the schedule of the system.

1. Will all 50 systems work at the same time whenever there are no failure among them?

2. Do all systems have two states only - either working or repairing - so that the sum of the working time and repairing time will always equal to 24 hours in 1 day?

3. If I am understanding correctly, you are asking the long-run average number of working systems.
Thank you for your questions and continuing help

Each system will operate for 400 hours per year (approx 1.1 hours per day)
The systems are either serviceable or unserviceable (in repair which takes 3 weeks before they are serviceable again)
I do not know how long-run average number of working systems is defined but it does sound like what I am after

8. ## Re: Number serviceable at given time

Each system will operate for 400 hours per year (approx 1.1 hours per day)
So either you are not answering my 1st question or I have missed something.

If I change the wordings of question like this:

If there is no failure, and what you say is correct, do you mean you have all the 50 systems will work together in the first 8 hours in a year and then idling for the rest of years? Do you mean as a whole the systems will work for a total of 400 hours whenever possible, and idle afterward?

9. ## Re: Number serviceable at given time

Originally Posted by BGM
So either you are not answering my 1st question or I have missed something.

If I change the wordings of question like this:

If there is no failure, and what you say is correct, do you mean you have all the 50 systems will work together in the first 8 hours in a year and then idling for the rest of years? Do you mean as a whole the systems will work for a total of 400 hours whenever possible, and idle afterward?
Each system will operate for a total of just over 1 hour a month which may not be continuous. When it is not operating it is switched off.

10. ## Re: Number serviceable at given time

Ok, better. Could you describe the schedule of the system in more details? At least now we know that each system have three states instead of two: on, off, repair.

Unless you can provide the exact details of the mechanism of switching on and off, I am afraid that we cannot help further.

Maybe you can provide us an example what will happen to the 50 systems when there is no failure in a given period which is long enough.

11. ## Re: Number serviceable at given time

Originally Posted by BGM
Ok, better. Could you describe the schedule of the system in more details? At least now we know that each system have three states instead of two: on, off, repair.

Unless you can provide the exact details of the mechanism of switching on and off, I am afraid that we cannot help further.

Maybe you can provide us an example what will happen to the 50 systems when there is no failure in a given period which is long enough.
Thank you for your patience in trying to understand the issue.

The 50 systems are used as and when required to perform a task which is on average once a day for about one hour each system. The systems are totally separate from each other and are in different locations. Power is applied by the user and disconnected by the user once the task is complete for that day. Systems will continue to be used this way until they fail. When they fail they are repaired and once repaired are put back into service. So question is how many will be in repair loop at any one time?

12. ## Re: Number serviceable at given time

Assuming the failure only occur at the working state. (without this assumption the modelling will be a little bit more complicated, depends on which fits better to your actual situation)

In the following I assume all system work independently for different tasks (so that the starting working time, ending time and failure time will be all different for each system). If not then the question you ask may not make sense.

Now I describe one possible modelling:

For each system, assume initially started at serviceable and are idling. Each system will wait for a random exponential time with mean 23 hours to receive a task requested by users. Then the system will start to work. Next, it will work for an random exponential time with mean 1 hours to finish the task if there is no failure, and then will be switched off and becomes idle again. At the moment when it start to work, we have a random exponential time with mean 900/24 = 37.5 hours (re-scaled as the failure only occur at work) representing the failure time. If this failure time is less than the working time, then the failure occur and the system goes to repair for a deterministic 21*24 = 504 hours, and switch to idle state after repair. And the cycle continues.

If these descriptions are correct, then we can calculate the long-run fraction of time of a particular system in the repair loop. And in fact, as the systems are independent and have identically distributed random times, the number of systems in repair loop (out of systems) will follows a binomial distribution and thus we just need to multiply the probability by to obtain the expectation.

If the model looks formidable to you, one can always use simulation to estimate the probability first to obtain a rough idea.

13. ## Re: Number serviceable at given time

Originally Posted by BGM
Assuming the failure only occur at the working state. (without this assumption the modelling will be a little bit more complicated, depends on which fits better to your actual situation)

In the following I assume all system work independently for different tasks (so that the starting working time, ending time and failure time will be all different for each system). If not then the question you ask may not make sense.

Now I describe one possible modelling:

For each system, assume initially started at serviceable and are idling. Each system will wait for a random exponential time with mean 23 hours to receive a task requested by users. Then the system will start to work. Next, it will work for an random exponential time with mean 1 hours to finish the task if there is no failure, and then will be switched off and becomes idle again. At the moment when it start to work, we have a random exponential time with mean 900/24 = 37.5 hours (re-scaled as the failure only occur at work) representing the failure time. If this failure time is less than the working time, then the failure occur and the system goes to repair for a deterministic 21*24 = 504 hours, and switch to idle state after repair. And the cycle continues.

If these descriptions are correct, then we can calculate the long-run fraction of time of a particular system in the repair loop. And in fact, as the systems are independent and have identically distributed random times, the number of systems in repair loop (out of systems) will follows a binomial distribution and thus we just need to multiply the probability by to obtain the expectation.

If the model looks formidable to you, one can always use simulation to estimate the probability first to obtain a rough idea.
Thank you for this. I follow your explanation and now understand the approach I need to take.

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