EXCEL MINITAB TI-84 PLUS TECHNOLOGY Technology 231 Using Poisson Distributions as Queuing Models Extended solutions are given in the technology manuals that accompany this text. Technical instruction is provided for Minitab, Excel, and the TI-84 Plus. Queuing means waiting in line to be served. There are many examples of queuing in everyday life: waiting at a traffic light, waiting in line at a 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 exercises below, you are asked to use Poisson distributions to analyze the queues at a grocery store checkout counter. In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute. 1. The mean number of customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with m = 4 for x = 0 to 20. Compare your results with the histogram shown at the upper right. 2. Minitab was used to generate 20 random numbers with a Poisson distribution for m = 4. Let the random number represent the number of arrivals at the checkout counter each minute for 20 minutes. 3 3 3 3 5 5 6 7 3 6 3 5 6 3 4 6 2 2 4 1 During each of the first four minutes, only three customers arrived. These customers could all be processed, so there were no customers waiting after four minutes. (a) How many customers were waiting after 5 minutes? 6 minutes? 7 minutes? 8 minutes? (b) Create a table that shows the number of customers waiting at the end of 1 through 20 minutes. 3. Generate a list of 20 random numbers with a Poisson distribution for m = 4. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes. 4. The mean increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for m = 5. Then create a table that shows the number of customers waiting at the end of 20 minutes. 5. The mean number of arrivals per minute is five. What is the probability that 10 customers will arrive during the first minute? 6. The mean number of arrivals per minute is four. Find the probability that (a) three, four, or five customers will arrive during the third minute. (b) more than four customers will arrive during the first minute. (c) more than four customers will arrive during each of the first four minutes. 7. The mean number of arrivals per minute is four. 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. (d) no customers are waiting in line after two minutes. EXERCISES MINITAB
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