4.37. A professor receives requests for letters of recommendation according to a Poisson process with rate...

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4.37. A professor receives requests for letters of recommendation according to a Poisson process with rate 8 per month. When k are on his desk he feels stressed and writes them at rate 2k (per month). (a) Find the stationary distribution of the number of letters to be written. (b) What is the average time a person has to wait before their letter is written. (c) He just received a request for a letter. What is the probability his queue will return to 0 before it hits 4, i.e., what is P(To < T4). 4.37. A professor receives requests for letters of recommendation according to a Poisson process with rate 8 per month. When k are on his desk he feels stressed and writes them at rate 2k (per month). (a) Find the stationary distribution of the number of letters to be written. (b) What is the average time a person has to wait before their letter is written. (c) He just received a request for a letter. What is the probability his queue will return to 0 before it hits 4, i.e., what is P(To < T4).