5:46 LTE A cafeteria serving line has a coffee um from which customers serve the

Question

5:46 LTE A cafeteria serving line has a coffee um from which customers serve themselves Arrivals at the un follow a Poisson distribution at the rate of 40 per minute. In serving a. How many oustomers would you expect to see on the average at the cofflee um? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Average no of customers b. How long would you expect it to take to get a cup of coffee? (Round your answer to 2 decimal places.) Expected timeminutels) c. What percentage of time is the um 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 um? (Do not round intermediate calculations. Round your answer to 1 decimal place.) Probabity % 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 the coffee urn (waiting and/or pouring coffee)7 (Do not round intermediate calculations. Round your answer to 2 decimal places.) Average no of customers 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 coflee, including waiting time? (Do not round intermediate calculations Round your answer to 2 decimal places.) Expected timeminues) eler

Explanation / Answer

a. a = 1/4 = 0.25, p = 11/60 = 0.1833, m = 1, u = 0.1833/0.25 = 0.733

CVa = 1 and CVp = 1

Tq = (1^2+1^2/2)*(0.733/(1-0.733))*0.1833 = 0.5042

Tsys = Tq+p = 0.5042+0.1833 = 0.6875

Thus average no. of customers = Tsys/a = 0.6875/0.25 = 2.75 customers

b. Expected time = Tsys = Tq+p = 0.5042+0.1833 = 0.69 minutes

c. % of time urn is being used = u = 0.7333 or 73.3%

d. Probability = u^3 = 0.7333^3 = 39.4%

e. Tq = (1^2+0^2/2)*(0.733/(1-0.733))*0.1833 = 0.2521

Tsys = Tq+p = 0.2521+0.1833 = 0.4354 minutes or 0.44 minutes

Isys = Tsys/a = 0.4354/0.25 = 1.74

Thus average no. of customers = 1.74

f. Expected time = 0.44 minutes (as computed in "e")

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