Savemore has 4 check-out counters. Prediction says that the duration of customer transaction is exponentially distributed with a mean of 4.9 minutes. In the peak hour, customer arrivals are exponentially distributed with a mean arrival rate of 17.3 customers/hr. Determine the probability that a customer will wait in a queue? Answer in four decimal places.
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- During nearly four decades of business operations, Memphis-based FedEx has earned a reputation for reliable, on-time delivery of packages to homes and offices around the country. Founder Fred Smith originally focused on overnight deliveries, choosing Memphis as the company’s headquarters because the airport rarely closes due to bad weather. With FedEx’s planes departing and arriving on schedule nearly all the time, its express shipments usually remained on schedule, then and now. To reassure customers that delivery will take place when and where promised, the firm offers a money-back guarantee on time-sensitive express shipments, among other services. FedEx has steadily expanded its portfolio of services since the 1970s. Its original overnight express delivery is currently available to U.S. customers in various forms, including “first-overnight” delivery, next-morning delivery, next-afternoon delivery, and budget-pleasing two- or three-day delivery. The company’s services also include cost-effective ground delivery for parcels and extra-speedy same-day delivery for urgent deliveries within 1,800 cities. Over the years, FedEx has widened its delivery network to more than 220 countries. It has purchased more cargo jets and acquired specialized shipping firms, including Tiger International, Roberts Express, RPS, and TNT Express, to support global growth. For international business customers needing products, parts, or raw materials shipped across countries or continents, the company now offers time-saving services such as commercial freight forwarding and cross-border logistical support. To add the convenience of local drop-off and pickup points for U.S. consumers and small businesses, FedEx acquired the Kinko’s office services company in 2004 and later rebranded it as FedEx Office. This acquisition also added printing and copying to the menu of services offered. Then the company arranged for large U.S. retailers such as Walgreens, Albertsons, Kroger, and Safeway to accept packages for shipment and receive package delivery for customer pickup in thousands of store locations. This means people who want to send a package can head to a nearby retailer and ship where they shop, rather than making a separate trip to the FedEx location. It’s also a safe alternative for packages to be picked up by people who don’t want FedEx shipments left by the front door. Another service FedEx offers to small and mid-sized businesses, including retailers, is FedEx Fulfillment. The purpose is to expedite order fulfillment by having each business store its products in a FedEx warehouse. Then, when the business’s customers place orders, FedEx puts the products into boxes bearing the business’s own logo and ships directly to those customers. The business doesn’t need a separate warehouse or staff for fulfillment, and packages are on their way to customers more quickly because the products were in FedEx’s warehouse, ready to be packed and shipped. This service puts FedEx into direct competition with Amazon.com, which offers a similar service to merchants that sell through the online Amazon Marketplace. But it also gives businesses that don’t sell via Amazon a fast and professional fulfillment alternative. FedEx is careful to let customers know, through media and social-media announcements, when it anticipates that extreme weather or other conditions will cause delays or force it to halt pickups and deliveries. For the duration of Hurricane Irma, for example, FedEx said it would suspend deliveries in Florida. Some Florida customers who had ordered generators to be delivered via FedEx were unhappy, because they worried about being without power during and after the storm. But one FedEx employee loaded several generator orders into his car and took them to customers himself. When a customer posted a grateful compliment to FedEx on Facebook, the message generated thousands of likes, shares, and positive comments. The company also received positive comments for its donations of cash and transportation services to areas devastated by Hurricanes Irma, Harvey, and Maria. According to the American Customer Satisfaction Index (ACSI), FedEx often tops the list of U.S. shipping companies as ranked by customers surveyed. Every day, the company delivers 13 million packages—and during the busy year-end holiday season, it delivers many more. By meeting customers’ expectations for on-time deliveries, FedEx has increased annual revenues beyond $60 billion and positioned itself for continued growth in the future. How does FedEx’s money-back guarantee address customers’ concerns about heterogeneity?The VBO grocery store has a checkout counter with a full-time cashier. Customers arrive randomly with an average rate of 20 per hour. The distribution of service time is exponential with an average of 2 minutes. Occasionally, this situation causes a long queue and customer complaints. Since there is no room for another cashier, the manager plans to hire another person to help pack the groceries and reduce the expected service time to 1.5 minutes, still with exponential distribution. The manager wants the percentage of time when there are more than 2 customers at the cash register to be less than 25%. He also wants no more than 5% of customers to have to wait 5 minutes or more before starting service or 7 minutes or more before finishing service. a) Use the formulas from the M/M/1 model to calculate L, Lq, W and Wq, π0, π1 and π2 of the current transaction. What is the probability that there are more than 2 customers at the cash register? (b) Repeat (a) for the helper alternative.At noon time, customers arrive at Metrobank counter managed by a single cashier with mean arrival rate of 6 customers per hour. The cashier attends the customers on first come first served basis at 10 customers per hour. What is the probability association if there are 5 customers in the system?
- BBA Bank (a fictitious one) has a drive-through teller window and observed that 15 customers arrive for service per hour, on an average, and the average service time per customer is 3 minutes. Assume inter-arrival time and service time follow a negative Exponential distribution. The bank hires you as a consultant. You guess that this is an M|M|1 system and you are required to determine the following: Probability (teller is busy) Probability (teller is idle) Probability of 3 customers in the system. Average number of customers waiting for service, that is, number of autos in the line excluding the one at the teller window. Average number of customers in the system, that is, number of autos in the line including the one at the teller window. Average time a customer spends in the system, that is, waiting time plus service time. Average time a customer spends in the waiting line before reaching the teller window.A petrol station in the capital Kingstown has a single pump manned by one attendant.Vehicles arrive at the rate of 20 customers per hour and petrol filling takes 2 minutes on anaverage. Assume the arrival rate is Poisson probability distribution and service rate isexponentially distributed. Arrivals tend to follow a Poisson distribution, and service timestend to be exponential. The attendant is paid $10 per hour, but because of lost goodwill andsales, station loses about $15 per hour of customer time spent waiting for the attendant toservice and order. Part BThe Petrol station is considering adding a second pump with an attendant to servicecustomers. The station would pay that person the same $10 per hour.Using appropriate formula for the multiple channel model, answer the following questions:a. What is the probability that no customers are in the system (Po)?b. What is the average number of customers waiting for service (Lq)? c. What is the average number of customers in the system…A petrol station in the capital Kingstown has a single pump manned by one attendant.Vehicles arrive at the rate of 20 customers per hour and petrol filling takes 2 minutes on anaverage. Assume the arrival rate is Poisson probability distribution and service rate isexponentially distributed. Arrivals tend to follow a Poisson distribution, and service timestend to be exponential. The attendant is paid $10 per hour, but because of lost goodwill andsales, station loses about $15 per hour of customer time spent waiting for the attendant toservice and order. a. What is the probability that no customers are in the system (Po)?b. What is the average number of customers waiting for service ( Lq)? c. What is the average number of customers in the system (L)?d. What is the average time a customer waits for service(Wq)? e. What is the average time in the system (W)?f. What is the probability that a customer will have to wait for service (Pw)?g What is the probability that there is exactly 2…
- The arrival rate of patients at a hospital counter follows Poisson distribution with a mean of 25 per hour. The service rate of the treatment to patients also follows Poisson distribution with a mean of 45 per hour. What is the probability of having 0 patients in the system (P 0 )? What is the probability of having 10 patients in the system (P 10 )? What is the probability of having 15 patients in the system (P 15 )? Find Ls, Lq, Ws and WqA restaurant manager is considering to hire waiters. Arrival rates have been found to follow the Poisson distribution, and the service times follow the negative exponential distribution. The average arrival rate is 12 customers per hour, and the average service time is 15 minutes. The manager hopes that there is on average 1 customer waiting in line, so at least how many waiters does he need to hire?