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.

1. Brighton surfing company has a very high number of surfboards to rent. The owner makes the following estimates. A new individual customer enters the shop in average every 5 minutes. Then, each customer decides to rent a surfboard with probability }. If they make this choice, they use it in average for 30 minutes before returning it to the shop. a) Give an irreducible continuous-time Markov chain describing the number of surf- boards in use, assuming the number of surfboards of the shop is infinite. Justify your answer. b) Give the transition matrix of the jump chain. c) Is this chain recurrent? Detail your reasons. d) Is it explosive? e) Estimate the proportion of time over a long period when no surfboard is in use. f) The owner call busy period a maximal interval of time when at least one surfboard is in use. How long is a busy period in average. g) Is this chain reversible? h) In the long run, what is the probability that two customers return their surfboard within a time slot of 20 minutes? Give carefully your reasons