Question

Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production when machines are out of service is \(\$ 10\) per hour per machine. What is the average cost rate incurred due to failed machines?

Short answer

The average cost rate incurred due to failed machines is \(\$30\) per hour.

Step-by-step solution

Identify the Failure Rate and Repair Rate

In the problem, we are given the failure rate and repair rate for machines. The failure rate \(\lambda\) is six per hour, and the repair rate \(\mu\) is eight per hour.

Calculate Average Number of Failed Machines

Using the fact that the average number of failed machines can be found using the formula \(\frac{\lambda}{\mu-\lambda}\), we can plug in our failure and repair rates: \[\text{Average Number of Failed Machines} = \frac{6}{8-6} = \frac{6}{2} = 3\]

Calculate the Average Cost Rate

Now that we have the average number of failed machines, we can calculate the average cost rate. Given that the cost incurred per hour per failed machine is \(\$10\), the average cost rate is: \[ \text{Average Cost Rate} = 3 \cdot 10 = \$ 30 \text{ per hour} \] Thus, the average cost rate incurred due to failed machines is \(\$30\) per hour.