**PLEASE ANSWER ALL PARTS AND PROVIDE AN EXPLAINATION; IF POSSIBLE** A cafeteria
ID: 371388 • Letter: #
Question
**PLEASE ANSWER ALL PARTS AND PROVIDE AN EXPLAINATION; IF POSSIBLE**
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 2.5 per minute. In serving themselves, customers take about 22 seconds exponentially distributed. a. How many customers would you expect to see on the average at the coffee un? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Average no of customers ·How long would you expect it to take to get a cup of coffee? (Round your answer to 2 decimal places.) Expected time minute(s) c. What percentage of time is the urn being used? (Do not round intermediate calculations. Round your answer to 1 decimal place.) Percentage of time % d. What is the probability that three or more people are at the coffee urn? (Do not round intermediate calculations. Round your answer to 1 decimal place.) Probability e. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 22 seconds, how many customers would you expect to see at the coffee urn (waiting and/or pouring coffee)? (Do not round ntermediate calculations. Round your answer to 2 decimal places.) Average no of customers f. If the cafeteria installs an automatic vendor that ntermediate calculations. Round your answer to 2 decimal places.) a cup of coffee at a constant time of 22 seconds, how long would you expect it to take (in minutes) to get a cup of coffee, including waiting time? (Do not round Expected time minute(s)Explanation / Answer
Arrival rate (A) = 2.5 per minute or 60/2.5 = 24 per hour
Service rate (S) = 22 seconds = 60*60/22 = 163.64 per hour
a) Lq = A^2/(S*(S-A)) = 24^2/(163.64*(163.64-24)) = 0.03
b) Expected time = Lq/A + 1/S = (0.03/24+1/163.64)*60 = 0.44 minutes
c) % Utilization = A/S*100 = 24/163.64*100 = 14.7
d) Probability of 3 or more people = 1 - Probability of less than 3 people = (1 - (1 - (A/S)^3))*100 = (A/S)^3*100 = (24/163.64)^3*100 = 0.3
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