Random processesTopics related to the following question: Markov chain Continuous-time Markov chain Queuing theory Invariant measures Please, provide the full answer and explain all parts step-by-step of the question

2. Assume that bacterial cells in a petri dish abide to the following rules: 0 As long as they are alive. each cell gets activated in average every 3 minutes. I When a cell activates two possibilities occur: With probability g all bacteria in the petri dish gets duplicated With probability %, when there are at least 2 cells, half of the cells in the petri dish die, whereas when there is only one. nothing happens. 0 There is initially one cell. (a) Assuming that all activation times are independent and memoryless, give a continuous time Markov chain modelling the number of cells alive. To simplify the presentation, aerjustijying it, you may nd useful to assume that the latter number is a power of 2. and to focus on its binary logarithm. (b) Determine the transition matrix of the jump chain. ( c ) Show that this Markov chain is irreducible and transient. You may use the last point of question I. of W53. (d) Show that there is an invariant probability measure it with 7r(x) > 0 for all possible states x of the chain. (e) Is it explosive? Give your reasons. (1') The experience period is the minimal interval of time needed for the cell population to be duplicated at least once and to return to its initial conguration or to become innite. Starting with one cell. what is in average the amount of time during the experience when there are exactly 2 cells?2