Lisa Howell asked 6 months ago
Charlie joins a new reading club, from which he receives books to read. Suppose that books arrive as a Poisson process at a rate λ of books per week. For each book, the time it takes for Charlie to read it is exponentially distributed with parameter μ; i.e., on average it takes Charlie 1/μ weeks to finish one book. Assume that the reading times for different books are independent, and also independent from the book arrival process.

The problem with Charlie is that he is easily distracted. If he is reading a book when a new book arrives, he immediately turns to read the new one, and only comes back to the older book when he finishes the new book.

For all of the parts below, give your answers in terms of μ and λ, using standard notation. Enter μ as “mu” and λ as “lambda”.

Hint: When Charlie starts reading a book, the total time he spends on reading it can be viewed as the first arrival from a Poisson process of rate μ, and you can then think about merging or splitting of Poisson processes.

Question 1 –> When Charlie starts a new book, what is the probability that he can finish this book without being interrupted by a new book?

Question 2 –> Given that Charlie receives a new book while reading a book, what is the probability that he can finish both books, the new one and the interrupted one, without further interruption?

Question 3 –> What is the expected reading time of a book given that it is not interrupted? Hint: The answer is not 1/μ.