The Neighborhood Grocery store has a single check-out stand with a full-time cashier. Customers arrive at the stand according to a Poisson process with a mean rate of 30 per hour. The service time has an exponential distribution with a mean of 1.5 minutes. The current situation has resulted in occasional long lines and complaints from customers. Therefore, since there is no room for a second check-out stand, the owner is thinking of hiring another person to help the cashier by bagging the groceries. This step would reduce the expected time required to process a customer to 1 minute, but the distribution still would be exponential. The owner has hired you as a consultant to verify the effectiveness of this new alternative. In particular, she has the following three criteria:.
1. She would like to have the percentage of time that there are more than two customers at the check-out stand down below 25 percent.
2. She would like to have no more than 10 percent of customers needing to wait more than 5 minutes before beginning service.
3. She would also like to have no more than 10 percent of customers needing to wait more than 7 minutes for receiving the complete service. Them, answer the following question.
(a) In the current situation, which model would you apply for analyzing this queuing system?
(b) For the current situation, calculate average number of customers at the stand, average number of customers in the queue and average time a customer spends at the stand.
(c) Which of the three criteria mentioned above are satisfied in the current situation?
(d) Repeat part (a)-(c) for the proposed alternative.