Question

Each time a machine is repaired it remains up for an exponentially distributed time with rate \(\lambda\). It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with rate \(\mu_{1}\); if it is a type 2 failure, then the repair time is exponential with rate \(\mu_{2} .\) Each failure is, independently of the time it took the machine to fail, a type 1 failure with probability \(p\) and a type 2 failure with probability \(1-p\). What proportion of time is the machine down due to a type 1 failure? What proportion of time is it down due to a type 2 failure? What proportion of time is it up?

Short answer

The proportion of time the machine is down due to a type 1 failure is given by \(p_{1} = \frac{p\cdot\frac{1}{\mu_{1}}}{\frac{1}{\lambda} + p\cdot \frac{1}{\mu_{1}} + (1-p)\cdot \frac{1}{\mu_{2}}}\). The proportion of time the machine is down due to a type 2 failure is given by \(p_{2} = \frac{(1-p)\cdot\frac{1}{\mu_{2}}}{\frac{1}{\lambda} + p\cdot \frac{1}{\mu_{1}} + (1-p)\cdot \frac{1}{\mu_{2}}}\). The proportion of time the machine is up and running is given by \(p_{3} = \frac{\frac{1}{\lambda}}{\frac{1}{\lambda} + p\cdot \frac{1}{\mu_{1}} + (1-p)\cdot \frac{1}{\mu_{2}}}\).

Step-by-step solution

Identify The Variables

In this problem, we have the following variables: 1. Exponential rate λ for uptime. 2. Exponential rates µ₁ and µ₂ for repair times for type 1 and 2 failures, respectively. 3. The probability of type 1 failure is given by p, and probability of type 2 failure is given by 1 - p.

Calculate The Average Uptime And Downtime Durations

We'll first find the average time the machine spends in each state. 1. The average uptime duration is given by the inverse of the uptime rate, which is \(\frac{1}{\lambda}\) 2. The average downtime due to a type 1 failure is given by the inverse of the repair rate for type 1 failure, which is \(\frac{1}{\mu_{1}}\) 3. The average downtime due to a type 2 failure is given by the inverse of the repair rate for type 2 failure, which is \(\frac{1}{\mu_{2}}\)

Calculate The Total Expected Cycle Time

The total cycle time is the combination of uptime and downtime. Considering the probabilities of each failure type, the expected downtime for any given cycle can be calculated as follows: \(E[Downtime] = p\cdot \frac{1}{\mu_{1}} + (1-p)\cdot \frac{1}{\mu_{2}}\) The total expected cycle time (T) is given by the sum of expected uptime and expected downtime: \(T = \frac{1}{\lambda} + E[Downtime]\)

Calculate The Proportion Of Time Spent In Each State

Now, we'll find the proportion of time spent in each state by dividing the duration of that state by the total cycle time. 1. The proportion of time spent in type 1 failure downtime (p₁) is given by: \(p_{1} = \frac{p\cdot\frac{1}{\mu_{1}}}{T}\) 2. The proportion of time spent in type 2 failure downtime (p₂) is given by: \(p_{2} = \frac{(1-p)\cdot\frac{1}{\mu_{2}}}{T}\) 3. The proportion of time spent in uptime (p₃) is given by: \(p_{3} = \frac{\frac{1}{\lambda}}{T}\) These proportions represent the fraction of time the machine is down due to each type of failure and the fraction of time it is up. Note that \(p_{1}+p_{2}+p_{3}=1\), indicating that the proportions cover 100% of the time.

Key concepts

Failure Rate and Repair Time

Understanding how often a machine fails and how long it takes to repair are crucial aspects of reliability engineering. Let's take an industrial machine as an example. It stays operational for periods that can be described by an exponential distribution with a certain failure rate, denoted as \(\textstyle\frac{1}{\text{MTTF}}\), where MTTF stands for Mean Time To Failure. This rate, represented by \(\textstyle\frac{1}{\text{\textstyle\frac{1}{\text{MTTF}}}}\), defines the average duration the machine operates before a failure occurs.

When a failure does happen, the machine goes through a downtime period for repairs. If this repair time also follows an exponential distribution, we can use the rates \(\textstyle\frac{1}{\text{MTTR}}\) for type 1 failure and \(\textstyle\frac{1}{\text{MTTR}}\) for type 2 failure, with MTTR representing Mean Time To Repair. By determining these rates, we can predict the average time the machine will be unavailable due to repairs of each failure type. This information is crucial for planning maintenance schedules and optimizing machine availability.

Probability of Failure Types

In reliability applications, it's essential to distinguish between different types of failures because they may have varying impacts and require different responses. Our example includes two failure types; type 1 and type 2, each with its respective probabilities, \(p\) and \(1-p\).

The probability of each type implies how often that kind of failure is expected to occur over the long run. If \(p\) is high, type 1 failures are more common; conversely, if \(1-p\) is high, we expect more type 2 failures. By understanding these probabilities, a maintenance team can prepare resources and prioritize repair strategies effectively to reduce downtime.

Proportion of Uptime and Downtime

The proportion of uptime and downtime is a way to measure the reliability of a machine. It represents the fractions of the machine's lifecycle it is expected to be operational (uptime) or undergoing repairs (downtime).

In our context, calculating these proportions shows how the machine's time is distributed between being active and being repaired. For example, a high uptime proportion means the machine is reliably in service most of the time, which is usually the goal in a production setting. On the other hand, the proportion of downtime, further broken down into type 1 and type 2 failures, indicates the time lost and possibly the costs involved in repairing the machine. These metrics are crucial for understanding and improving the operational efficiency of equipment.

Expected Cycle Time in Reliability

The expected cycle time is the sum of the average duration of all states a machine can be in during its operation, including active and repair periods. In the context of reliability, this cycle time provides insight into the length of time from when a machine starts operating to the time when it is ready to begin a new cycle after being repaired.

To calculate this in our example, we consider the exponential distributions for uptime and for both types of downtime. The expected cycle time helps us understand the full scope of machine availability and plan accordingly. For instance, if the expected cycle time is long due to lengthy repairs, this could indicate the need for preventative maintenance or the evaluation of the repair processes. It's a valued figure in reliability management as it encapsulates the overall performance and availability of the machine.