(Poisson processes and queues) Consider a situation involving a server, e.g., a cashier at a fast-food restaurant, an automatic bank teller machine, a telephone exchange, etc. Units typically arrive for service in a random fashion and form a queue when the server is busy. It is often the case that the number of arrivals at the server, for some specific unit of time t can be modeled by a Poisson(λt)distribution and is such that the number of arrivals in nonoverlapping periods are independent. In Chapter 3, we will show that λ t is the average number of arrivals during a time period of length tand so D is the rate of arrivals per unit of time Suppose telephone calls arrive at a help line at the rate of two per minute. A Poisson process provides a good model.

(a) What is the probability that five calls arrive in the next 2 minutes?

(b) What is the probability that five calls arrive in the next 2 minutes and then five more calls arrive in the following 2 minutes?

(c) What is the probability that no calls will arrive during a 10-minute period?