Exercise 1: In a given country, 40% of the people choose to get vaccinated against an infectious disease. Among vaccinated people, it is estimated that 80% develop immunity, 5% get infected, and 15% remain susceptible. As for the unvaccinated people, 25% become infected and 75% remain susceptible. 90% of infected people recover and develop immunity whereas 10% recover and return to being susceptible. Due to the fact that antibodies fade over time, 30% of immune people lose immunity and become susceptible.

1. Model the problem as a Markov chain.
2. Herd immunity is achieved when more than 50% of the population develops immunity. Given the statistics above, will the population achieve herd immunity?
3. In case herd immunity is not achieved, discuss on what can be done and propose a complete solution.

Exercise 2: A hardware system has two processors P1 and P2 that are identical. Time in the system is divided into slots. The system is governed by the following set of rules:

• With a probability p_a, a maximum of one job arrives each time slot in an independent way.
• The job starts immediately on a free processor (idle) if any. If both are free the job starts at P1.
• Jobs are not buffered. If P1 and P2 are busy, arriving jobs are lost.
• A job is ended by a processor in a slot with probability p_e. Since job completion is independent, the possibility that both processors end their jobs at the same time slot is p_e².
• A processor can end a job in a time slot and starts another.

1. Represent the problem as a Markov chain.

Hint: Consider the following states: State 0: No jobs, State 1: one job in the system (in P1) and State 2: 2 jobs in the system.

Hint: you need to think about all possibilities when looking for the transition probability.