A supermarket you work part-time at has one express lane open from 5 to 6 PM...

A supermarket you work part-time at has one express lane open from 5 to 6 PM on weekdays (Monday through Friday). This time of the day is usually the busiest since people tend to stop on their way home from work to buy groceries. The number of items allowed in the express lane is limited to 10 so that the average time to process an order is fairly constant at about 1 minute. The manager of the supermarket notices that there is frequently a long line of people waiting and hears customers grumbling about the wait. To improve the situation he decides to open additional express lanes during this time period. If he does, however, he will have to "pull" workers from other jobs around the store to serve as cashiers. Hence, he is reluctant to open more lanes than necessary.
Knowing that you are a college student studying probability, your manager asks you to help him decide how many express lanes to open. His requirement is that there should be no more than one person waiting in line 95% of the time.
With the task at hand, you set out to study the problem first. You start by counting the number of customer arrival in the express lane on a Monday from 5 to 6pm. There are a total of 81 arrivals. You repeat the experiment on the following four days (Tuesday through Friday) and note the total arrivals of 68, 72, 61 and 66 customers, respectively.

1) What is the average number of customer arrivals at the express lane from 5 to 6pm on weekdays?

2) Assume the customer arrivals at the express lane from 5 to 6pm on weekdays can be modeled by a Poisson random variable, what is the PMF for the number of customers arrived during a one-minute interval in this period?

3) What is the probability of two or fewer customers arriving at the one express lane during a oneminute interval in this period? Does it satisfy the manager’s requirement of no more than one person waiting in line 95% of the time?

4) If your answer to the previous question is no, how many express lanes should the manager open in order to satisfy his requirement? You can assume that the arriving customer is equally likely to join any of the express lane if there are more than one express lanes. Also you can assume the lanes are independent, but all lanes must satisfy the manager’s requirement.

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Manojponduru answered 2 years ago

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