Data centers alternate between "working" and "down." There are many reasons why data centers can be down, but for the purpose of this problem we mention only two reasons: (i) a backhoe accidentally dug up some cable, or (ii) a software bug crashed the machines. Suppose that a data center that is working today will be down tomorrow due to backhoe reasons with probability 1/6, but will be down tomorrow due to a software bug with probability 1/4. A data center that is down today due to backhoe reasons will be up tomorrow with probability 1. A data center that is down today due to a software bug will be up tomorrow with probability 3/4. (a) Draw a DTMC for this problem. (b) Is your DTMC ergodic? Why or why not? (c) Is your DTMC time-reversible? Why or why not? (d) What fraction of time is the data center working? (e) What is the expected number of days between backhoe failures? Data centers alternate between "working" and "down." There are many reasons why data centers can be down, but for the purpose of this problem we mention only two reasons: (i) a backhoe accidentally dug up some cable, or (ii) a software bug crashed the machines. Suppose that a data center that is working today will be down tomorrow due to backhoe reasons with probability 1/6, but will be down tomorrow due to a software bug with probability 1/4. A data center that is down today due to backhoe reasons will be up tomorrow with probability 1. A data center that is down today due to a software bug will be up tomorrow with probability 3/4. (a) Draw a DTMC for this problem. (b) Is your DTMC ergodic? Why or why not? (c) Is your DTMC time-reversible? Why or why not? (d) What fraction of time is the data center working? (e) What is the expected number of days between backhoe failures