Exercise 4.1. In a bookshop in the city center there is only one cashier because the other two are off with influenza. Normally the cashier is able to cash up an aver- age of 70 customers per hour with a service rate being described by an exponential distribution. On average there are only 55 customers per hour in the bookshop who want to buy a book. Customer arrivals to the counter are described using a Poisson distribution. Next to cashing up customers the cashier also has to unpack deliveries. We assume that the customers are patient while waiting in the queue. Answer the following questions. 1. What is the amount of time between two consecutive customers wanting to buy a book, on average? 2. Compute how much time the cashier can spend on unpacking deliveries on average. 3. On average, how many people are waiting in the queue? How many people are present in the whole system? 4. A marketing campaign showed that if a customer has to wait more than 5 minutes before being served, he will not come back to the shop in the future. Should the bookshop be worried about that? 5. What is the probability that a customer will not have to wait to buy his book? We assume that the cashier serves a customer as soon as he arrives to the counter even if he was unpacking deliveries. 6. Give two pieces of advice to the bookshop in order to improve the efficiency of the waiting lines and explain them briefly.