Question

There are three machines, all of which are needed for a system to work. Machine \(i\) functions for an exponential time with rate \(\lambda_{i}\) before it fails, \(i=1,2,3 .\) When a machine fails, the system is shut down and repair begins on the failed machine. The time to fix machine 1 is exponential with rate \(5 ;\) the time to fix machine 2 is uniform on \((0,4) ;\) and the time to fix machine 3 is a gamma random variable with parameters \(n=3\) and \(\lambda=2 .\) Once a failed machine is repaired, it is as good as new and all machines are restarted. (a) What proportion of time is the system working? (b) What proportion of time is machine 1 being repaired? (c) What proportion of time is machine 2 in a state of suspended animation (that is, neither working nor being repaired)?

Short answer

In conclusion: (a) The proportion of time the system is working is given by \(P_w\). (b) The proportion of time machine 1 is being repaired is given by \(P_{r1}\). (c) The proportion of time machine 2 is in a state of suspended animation is given by \(P_{s2}\).

Step-by-step solution

Analyzing the System

As all three machines must function for the system to work, the system works if and only if all of them are functioning. When any machine fails, the system shuts down and repair begins. Therefore, we can analyze the system as a whole by finding the proportion of time at least two machines are working simultaneously.

Find the Probability that the System is Working

Since all three machines function for an exponential time with rate \(\lambda_i,\) we should first find the probability that all of them are functioning. Let \(T_i\) be the continuous random variable representing the time taken for machine \(i\) to fail. The joint probability density function (pdf) of \((T_1, T_2, T_3)\) is given by: \[f_{T_1, T_2, T_3}(t_1, t_2, t_3) = \lambda_1 e^{-\lambda_1 t_1} \cdot \lambda_2 e^{-\lambda_2 t_2} \cdot \lambda_3 e^{-\lambda_3 t_3}\] To find the probability that the system is working, we look for the probability that all three machines are functioning. We can represent this as the probability that \(T_1 > t, T_2 > t, T_3 > t\) for some time \(t\). This is given by the integral, \[P(\text{System working}) = \int_{0}^{\infty} \int_{t}^{\infty} \int_{t}^{\infty} f_{T_1, T_2, T_3}(t_1, t_2, t_3) \, dt_1 \, dt_2 \, dt_3\]

Find the Proportion of Time the System is Working

After solving the integral in Step 2, we find the probability that the system is working at time \(t\). To find the proportion of time the system is working, we divide this probability by the sum of the probabilities of the system working and not working. \[P_w = \frac{P(\text{System working})}{P(\text{System working}) + P(\text{System not working})}\] The proportion of time the system is working is \(P_w\).

Find the Proportion of Time Machine 1 is Being Repaired

Let \(T_r\) be the time taken to repair machine 1, which is an exponential random variable with rate \(5\). The proportion of time machine 1 is being repaired is given by the probability that it is being repaired, divided by the total time (either working or under repair): \[P_{r1} = \frac{P(T_r)}{P(T_1) + P(T_r)}\] The proportion of time machine 1 is being repaired is \(P_{r1}\).

Find the Proportion of Time Machine 2 is in a State of Suspended Animation

Suspended animation occurs when the machine is neither working nor being repaired. Let \(T_s\) be the duration of suspended animation. As the time that the machine is working is given by the probability \(P_w\), the proportion of time machine 2 is in suspended animation can be found by subtracting the working time and machine 1's repair time from the total time: \[P_{s2} = 1 - P_w - P_{r1}\] The proportion of time machine 2 is in a state of suspended animation is \(P_{s2}\). In conclusion: (a) The proportion of time the system is working is given by \(P_w\). (b) The proportion of time machine 1 is being repaired is given by \(P_{r1}\). (c) The proportion of time machine 2 is in a state of suspended animation is given by \(P_{s2}\).