17.6-10. The Friendly Neighbor Grocery Store has a single checkout stand with a full-time cashier. Customers arrive randomly at the stand at a mean rate of 20 per hour. The service-time distribution is exponential, with a mean of 2 minutes. This situation has resulted in occasional long lines and complaints from customers. Therefore, because there is no room for a second checkout stand, the manager is considering the alternative of hiring another person to help the cashier by bagging the groceries. This help would reduce the expected time required to process a customer to 1.5 minutes, but the distribution still would be exponential. The manager would like to have the percentage of time that there are more than two customers at the checkout stand down below 25 percent. She also would like to have no more than 5 percent of the customers needing to wait at least 5 minutes before beginning service, or at least 7 minutes before finishing service. (a) Use the formulas for the M/M/1 model to calculate L, W, W_q, L_q, P_0, P_1, and P_2 for the current mode of operation. What is the probability of having more than two customers at the checkout stand? (b) Use the Excel template for this model to check your answers in part (a). Also find the probability that the waiting time before beginning service exceeds 5 minutes, and the probability that the waiting time before finishing service exceeds 7 minutes. (c) Repeat part (a) for the alternative being considered by the manager. (d) Repeat part (b) for this alternative. (e) Which approach should the manager use to satisfy her criteria as closely as possible?
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1 hours = 6 minutes - Wq (average time a customer spends waiting in line) = Ī» / (Ī¼ * (Ī¼ - Ī»)) = 20 / (30 * (30 - 20)) = 0.067 hours = 4 minutes - P0 (probability the system is empty) = 1 - Ī» / Ī¼ = 1 - 20 / 30 = 0.33 or 33% - P1 (probability there is one customer in Show moreā¦
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Pete's Market is a small local grocery store with only one checkout counter. Assume that shoppers arrive at the checkout lane according to a Poisson probability distribution, with an arrival rate of 18 customers per hour. The checkout service times follow an exponential probability distribution, with a service rate of 20 customers per hour. The manager's service goal is to limit the waiting time prior to beginning the checkout process to no more than five minutes. Also, the manager of Pete's Market wants to consider one of the following alternatives for improving service. Calculate the value of Wq for each alternative. a. Hire a second person to bag the groceries while the cash register operator is entering the cost data and collecting money from the customer. With this improved single-server operation, the service rate could be increased to 28 customers per hour. Round your answer to three decimal places. Do not round intermediate calculations. Wq = _______ minutes b. Hire a second person to operate a second checkout counter. The two-server operation would have a service rate of 20 customers per hour for each server. Note: Use P0 values from Table 11.4 to answer this question. Round your answer to three decimal places. Do not round intermediate calculations. Wq = ____________ minutes
The manager of a home improvement store finds that between 10 A.M. and 11 A.M., customers enter the store at the average rate of 45 customers per hour. The following function gives the probability that a customer will arrive within t minutes of 10 A.M. (Note: A probability of 0.6 means there is a $60 \%$ chance that a customer will arrive during a given time period.) $$ P(t)=1-e^{-0.75 t} $$ a. Find the probability, to the nearest hundredth, that a customer will arrive within 1 minute of 10 A.M. b. Find the probability, to the nearest hundredth, that a customer will arrive within 3 minutes of 10 A.M. c. Use a graph of $P(t)$ to determine how many minutes, to the nearest tenth of a minute, it takes for $P(t)$ to equal $98 \%$ d. Write a sentence that explains the meaning of the answer in part c.
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