Using R Question 4 A local coffee shop in its busy period gets about 10 customers per minute. A typical server working at the counter can help customers at a rate of 2 per minute. 4(a). What is the minimum number of servers the shop needs to have to make this a stable system, i.e. rho < 1? 4(b). We will assume that the server’s rate of service stays constant. However, the rate at which customers come in might fluctuate. Consider customer arrival rates ranging from 8 per minute to 12 per minute. Conduct a sensitivity analysis on the number of servers the shop should have during its busy period. Range the number of servers from your answer in 4(a) up to 5 more (that is, if you said x in (a), let c = x, x+1,…,x+5) Create simultaneous line plots similar to what was done in the M/M/c practice for the values of R0. Plot the c values on the x-axis and create a separate line plot for each value of lambda(customer arrival rate). Give the appropriate titles and labels and include a legend. Make sure each graph is clear and easy to read. 4(c). Comment on the behavior in the graph and the change in R0 vs the number of servers. Question 5 From Problem 4, suppose that the coffee shop is small and can only hold 15 people inside. Suppose customers come in at a rate of 10 per minute. (a). The shop currently uses only 1 server during its busy period. Suppose the server can help customers at a rate of 2 per minute. To get customers in and out faster, it is considering using more servers. How many less customers will be in the shop on average if they use one more server. (b). Suppose each server makes $8 per hour. The cost of adding another server will require purchasing another cash register at a cost of $150. Suppose that the average profit made from each customer is $3. If the busy period lasts for two hours, will hiring one more server be profitable for the coffee shop? What if it lasts for three hours? (hint: when we find the mean number of customers in the store, this would be the mean number in any given minute.) (c). From (a), how much less time will customers will be in the shop on average if they use one more server?
Using R
Question 4
A local coffee shop in its busy period gets about 10 customers per minute. A typical server
working at the counter can help customers at a rate of 2 per minute.
4(a). What is the minimum number of servers the shop needs to have to make this a
stable system, i.e. rho < 1?
4(b). We will assume that the server’s rate of service stays constant. However, the
rate at which customers come in might fluctuate. Consider customer arrival rates ranging
from 8 per minute to 12 per minute. Conduct a sensitivity analysis on the number of
servers the shop should have during its busy period.
your answer in 4(a) up to 5 more (that is, if you said x in (a), let c = x, x+1,…,x+5)
Create simultaneous line plots similar to what was done in the M/M/c practice for the
values of R0. Plot the c values on the x-axis and create a separate line plot for each value of
lambda(customer arrival rate). Give the appropriate titles and labels and include a legend.
Make sure each graph is clear and easy to read.
4(c). Comment on the behavior in the graph and the change in R0 vs the number of
servers.
Question 5
From Problem 4, suppose that the coffee shop is small and can only hold 15 people inside.
Suppose customers come in at a rate of 10 per minute.
(a). The shop currently uses only 1 server during its busy period. Suppose the server
can help customers at a rate of 2 per minute. To get customers in and out faster, it is
considering using more servers. How many less customers will be in the shop on average if
they use one more server.
(b). Suppose each server makes $8 per hour. The cost of adding another server will
require purchasing another cash register at a cost of $150. Suppose that the average profit
made from each customer is $3. If the busy period lasts for two hours, will hiring one more
server be profitable for the coffee shop? What if it lasts for three hours? (hint: when we find
the mean number of customers in the store, this would be the mean number in any given
minute.)
(c). From (a), how much less time will customers will be in the shop on average if
they use one more server?
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