The Security & Trust Bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 2 per minute. However, business is growing and management projects that the mean arrival rate will be 3 per minute a year from now. The transaction time between the teller and customer has an exponential distribution with a mean of 1 minute. Management has established the following guidelines for a satisfactory level of service to customers. The average number of customers waiting in line to begin service should not exceed 1. At least 90 percent of the time, the number of customers waiting in line should not exceed 3. a)Use the Excel sheet to determine how many tellers will be needed a year from now to completely satisfy these guidelines.
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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?Answer the given question with a proper explanation and step-by-step solution. Use the information. A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn follow a Poison distribution at the rate of 2.5 per minute. In serving themselves, customers take about 22 seconds, exponentially distributed.Use formulas to solve, no excel. Show all work. a. How many customers would you expect to see, on average, at the coffee urn? b.How long would you expect it to take to get a cup of coffee? c.What percentage of time is the urn being used? d. What is the probability that three or more people are in the cafeteria?During the early morning hours, customers arrive at a branch post office at an average rate of 45 per hour (Poisson), while clerks can handle transactions in an average time (exponential) of 4 minutes each. Find: (a) The average number of customers waiting for service if 5 clerks are used. (b) The minimum number of clerks needed to keep the average waiting time in the system to under 5 minutes. (c) If clerk cost is $40 per hour and customer waiting time represents a “cost” of $30 per hour, how many clerks can be justified on a cost basis? Please show work.
- At a one man barber shop, customers arrive according to poison distribution with a mean arrival rate of 5 per hour and hair cutting time was exponentially distributed with an average hair cutting time was exponentially distributed with an average hair cut taking 19 minutes. It is assumed that because of excellent reputation, customers were always willing to wait. Calculate the following a. Average number of customers in the shop and average numbers waiting for a haircut b .Percentage of time arrival can walk in right without having to wait c. The percentage of customers who have to wait before getting into the barber’s chairCurrently, Commercial Banks have 1 unit of ATM machines to meet customer needs. It is known that there is an average of 12+x customers in the system. Then it is also known that the average time spent by someone in the system is 10 minutes. Assume that the system follows M/M/1. The Commercial Bank operational manager is considering whether it is necessary to add 1 unit of ATM machine. What are your considerations? To strengthen the consideration, compare the queue length, queue length, and utilities before and after adding an ATM machine? Note: x denoted by the last number of my ID Number.Example, if my ID number is 2602296545, then the number used are “5”A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the um follow a Poisson distribution at the rate of three per minute. In serving themselves, customers take about 15 seconds, exponentially distributed. a. How many customers would you expect to see on the average at the coffee um? b. How long would you expect it to take to get a cup of coffee? c. What percentage of time is the urn being used?
- A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn follow a Poisson distribution at the rate of 4.0 per minute. In serving themselves, customers take about 11 seconds, exponentially distributed. a. How many customers would you expect to see on the average at the coffee urn? b. How long would you expect it to take to get a cup of coffee? c. What percentage of time is the urn being used? d. What is the probability that three or more people are in the cafeteria? e. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 11 seconds, how many customers would you expect to see at the coffee urn (waiting and/or pouring coffee)? f.If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 11 seconds, how long would you expect it to take (in minutes) to get a cup of coffee, including waiting time?The following data has been collected on the number of customers seen to arriveto a museum in a succession of 5-minute intervals: 5, 1, 3, 7, 5, 5, 6, 7, 5, 7, 4, 8, 1,5, 2, 3, and 5. Estimate the squared coefficient of variation of the arrival process. Ifthis data was known to come from a Poisson process, what would be your estimateof l, the rate of customer arrivals?