A cafeteria serving line has a coffee urn from which customers serve themselves.
ID: 371576 • Letter: A
Question
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 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 urn? (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 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 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 22 seconds, how long would you expect it to take (in minutes) to get a cup of coffee, including waiting time? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Expected time minute(s)
Explanation / Answer
A)Arrival rate or lambda = 2.5 customers per minute.
Service rate or Mu = 60 minutes per second /22 seconds = 2.727 per minute.
Hence number of customers at the cofee urn = Length of the system
= Lambda / ( Mu - Lambda)
=2.5 / (2.727 - 2.5 )
=11 customers
Note : It can also be computed by formula : lambda ^2/mu *(mu - lambda) + lamda/ Mu
B)Time to get a cup of cofee
= Waiting time in the queue = Lambda / Mu *( Mu - Lambda)
= 2.5 / 2.727*(2.727-2.5)
=4.04 minutes.
Waiting time in system = Waiting time in queue + 1/ Mu.
= 4.04 + 1/2.727
=4.41 minutes.
C)Percentage of time is the utilization ratio .
=lambda/ Mu
= 2.5/ 2.727
=0.916 or 91.6%
D)Probability that 3 or more people are at the cofee urn:
= (lambda / Mu )^3
=(2.5/2.727)^3
=0.7692 or 76.9 %
E)If automatice vendor is installed then service time becomes constant.
Length of the system : Length of customers in queue + lambda/ Mu.
Since service time is constant hence length of customers in queue formula is
= Lambda ^2/2 *Mu *(Mu - lambda)
2.5*2.5/2*2.727*(2.727-2.5)
=5.048 customers
Length of system = Length of queue + lambda/mu
=5.048 + 2.5/2.727
=5.96 customers
F)Waiting time in queue : Length of queue/ lambda
= 5.048/ 2.5
=2.019 or 2.02 minutes.
Waiting time in system : waiting time in queue + 1/ Mu.
= 2.02 + 1/2.727
=2.386 or 2.39 minutes.
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