Problem 6 (35 pts) Using Poisson Distributions as Queuing Models QEUWE means waiting in line to be served. There are many examples of queuing in everyday life: "lms at trafc light, waiting in line at aa grocery checkout counter, waiting for an elevator, holding for a telephone call and so on. Poisson distributions are used to model and predict the number of people (calls, computer programs. vehicles) arriving at the line. In the questions below, you are asked to use Poisson distributions to anal'Ize the queues at a grocery store checkout counter. 1. The mean number of customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with ln = 4 for x = 0 to 20. Compare your results with the histaBram- 0.0 0 2 4 6 8 101214161320 Number of arrivals _- r minute 2. Generate a list of 20 random numbers with a Poisson distribution ,u = 4. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes. a. How many customers were waiting after 5 minutes, 6 minutes, 7 minutes and 8 minutes? b. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes. 3. The mean increases to S arrivals per minute, but the store can still process only four per minute. Generate another list of 20 random numbers with a Poisson distribution for _u = 5. Then create a table that shows the number of customers waiting at the end of 20 minutes as in part 2. 4. The mean number of arrivals is 5 per minute. What is the probability that 10 customers will arrive during the rst minute? 5. The mean number of arrivals per minute is 5 per minute. Find the probability that a. No customers are waiting in line after one minute. b. One customer is waiting in line after one minute. c. One customer is waiting in line after one minute and no customers are waiting in line after the second minute. 6. No customers are waiting in line after two minutes