Charlie joins a new reading club, from which he receives books to read. Suppose that books arrive as a Poisson process with rateλbooks per week. For each book, suppose the time it takes for Charlie to finish reading is exponentially distributed with parameterμ, i.e. on average it takes Charlie 1/μweeks to finish one book. Assume that the reading time for different books is independently distributed. 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. In the following, please not only give an answer but carefully reason your results. Hint: when Charlie starts reading a book, the finishing time can be viewed as the first arrival from a Poisson process of rateμ, and you can apply merge and split of this Poisson process with other processes.

a.(4 points) When Charlie starts a new book, what is the probability that he can finish this book without being interrupted? 

b. 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? 

c. What is the average reading time of a book given that it is not interrupted?  

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