## UQ-Maintenance is developing a Machine Health Monitoring System to help in maintain- ing the many machines in various parts of UQ

UQ-Maintenance is developing a Machine Health Monitoring System to help in maintain- ing the many machines in various parts of UQ. For efficiency, the machines’ maintenance record are kept at the school level.

For resource allocation purposes, UQ-Maintenance models the arrival of new machines in each school as independent Bernoulli Process, though they may not be identical.

UQ models the health of a machine as a Markov Chain, where the states represent the health levels of the machines at the end of each month and a single transition step represents the changes in the machine’s health over the duration of one month. The transition from one state to another is modeled as independent Binomial distributions. This means, assuming Xi is the random variable that represents the state of the Markov Chain at time-step i, P (Xt+1|Xt = s) is a Binomial distribution, for any state s in the state space of the Markov Chain ; the binomial distribution for different values of Xt are independent, but they might have different parameters.

Of course, different types of machines may be modelled as different Markov Chain. How- ever, each Markov Chain will have one state that represents “break down and must be repaired” and one state that represents “pristine condition”. Furthermore, for simplicity, UQ-Maintenance models all machines in the same school with the same Markov Chain.

[40 points] I: Programming

Please write a Matlab program (.m file) with 2 outputs:

• A single vector, where each element is the average number of machines at each of UQ school that are in state “break down and must be repaired” at the end of month-1, month-2, … , month-12, assuming at the end of month-0, all machines are in state “pristine condition”.
• A single vector, where each element is the average number of steps for the machines in each school to reach the “break down and must be repaired” state from “pristine condition”

The inputs is a sequence of numbers and vectors/matrices in the following order:

1. The number of schools in Let’s denote this number as n.
2. A vector of n elements, where each element indicates the parameter for the arrival process of the machines in each
3. A number representing the number of states in the Markov Chain that represents the health of the machines in school-1.
4. A transition matrix for the Markov Chain that represents the health of the machines in school-1.
5. A number representing the number of states in the Markov Chain that represents the health of the machines in school-2.
6. A transition matrix for the Markov Chain that represents the health of the machines in school-2. :

:

1. A number representing the number of states in the Markov Chain that represents the health of the machines in school-n.
2. A transition matrix for the Markov Chain that represents the health of the machines in school-n.

[60 points] II: Analysis Recall: Strong explanations are backed by evidence, either theo- retical or experimental.

1. [5 points] Please describe the ideas of how you compute the
3. [20 points] Please compute the complexity of your