Question

A system consists of two independent machines that each function for an exponential time with rate \(\lambda .\) There is a single repairperson. If the repairperson is idle when a machine fails, then repair immediately begins on that machine; if the repairperson is busy when a machine fails, then that machine must wait until the other machine has been repaired. All repair times are independent with distribution function \(G\) and, once repaired, a machine is as good as new. What proportion of time is the repairperson idle?

Short answer

The proportion of time the repairperson is idle is given by \(X = 1 - \lambda E[R]\), where E[R] represents the mean repair time, and λ is the rate of the exponential function for the machines' functioning time. To find the exact value of X, we need the details of the repair time distribution function G and the exponential rate λ.

Step-by-step solution

Define the Variables

Let's denote the proportion of time the repairperson is idle as X. We can also define the following variables: 1. \(Y\): Proportion of the time a particular machine is up and running 2. \(W\): Waiting time for a machine to be repaired

Calculate the Average Up-Time of a Machine

We know that each machine functions for an exponential time with rate λ. So, the mean up-time of a machine (both running and waiting for repair) is given by: \(E[U] = \frac{1}{\lambda}\) (since the mean of an exponential distribution is the reciprocal of the rate)

Calculate the Mean Repair Time

Let the mean repair time be E[R]. Using the repair time distribution function G, we can calculate the mean repair time. It depends on the specific distribution of G. For example, if it is an exponential distribution with rate μ, then the mean repair time would be \(\frac{1}{\mu}\).

Set Up the Proportion Calculation

Now our task is to find the proportion of time that the repairperson is idle, X. Considering all the possible states (either repairperson is idle, or machine is up and running, or waiting for repair), we can set up the following proportion: \(X = \frac{E[U] - E[R]}{E[U]}\)

Solve for the Proportion of Time the Repairperson is Idle

Substitute the values of the mean up-time (E[U]) and mean repair time (E[R]) in the equation for the proportion of time the repairperson is idle (X): \(X = \frac{\frac{1}{\lambda} - E[R]}{\frac{1}{\lambda}}\) Now solve for X: \(X = 1 - \lambda E[R]\) The proportion of time the repairperson is idle is given by \(X = 1 - \lambda E[R]\). To find the exact value of X, we need the details of the distribution function G (repair time distribution) and the rate λ (exponential rate). We can plug in these values as per the given problem to determine X.

Key concepts

Exponential Distribution

Understanding the exponential distribution is crucial for solving problems related to the time between events, such as the functioning time of a machine before it needs repair. The exponential distribution is often used to model the time until some specific event occurs, and it has a constant hazard rate. This is why it’s frequently associated with life spans (of machines or living organisms).

The main feature of the exponential distribution that is used in this exercise is its memoryless property. This means that the probability of failure at any given moment is the same, regardless of how long the machine has been running. The distribution is characterized by a single parameter, \( \lambda \), the rate of the event. The mean, or average time to the event, for an exponential distribution is the reciprocal of the rate (\( \frac{1}{\lambda} \) in this case).

When dealing with repair times and machine functionality, we assume that after each repair, the machine is 'as good as new,' which aligns with the memoryless nature of the exponential distribution. This property simplifies calculations and models since it does not require tracking the entire history of the machine’s operation.

Mean Repair Time

The mean repair time is a measure of how long, on average, it takes to repair a machine. When we’re looking at a system where machines are continually operated and can fail at random times, knowing the mean repair time is essential for assessing the efficiency of the repair process.

Calculating the mean repair time can involve various statistical distributions, depending on the complexity of the repair process and the nature of the failures. In our exercise, the distribution function \(G\) represents the repair times. If \(G\) is an exponential distribution with rate \(\mu\), then the mean repair time would be \(\frac{1}{\mu}\). However, \(G\) could represent any other distribution, in which case the method of calculating the mean repair time would differ accordingly.

Having an accurate estimate of the mean repair time is critical for managing downtime in industrial settings. It aids in planning and allocating resources efficiently, ensuring that repair personnel are not idle unnecessarily and that machines have minimal downtime.

Distribution Function G

In the context of the exercise, the distribution function \(G\) is used to describe the probability distribution of repair times for a broken machine. A probability distribution function gives us a complete description of the likelihood of various outcomes. More specifically, for a continuous random variable, the distribution function \(G(x)\) at a value \(x\) gives the probability that the repair time will be less than or equal to \(x\).

Understanding the nature of \(G\) is key to predicting repair times and scheduling appropriately. Different distributions might be chosen based on empirical data or known characteristics of the repair process. For example, if the repair times are known to have a fixed average with variation around this average, a normal distribution might be used. However, if the repair times are more unpredictable and are only bounded by a minimum value (zero), but could theoretically be very long, then an exponential distribution would be more appropriate due to its long tail.

Once the distribution function \(G\) is known, it can be integrated into calculations such as expected values, variances, and other statistical measures that are vital for decision-making processes in operations management.