DEAR TUTOR, PLEASE SHOW ALL THE WORKING OUT. SO I CAN UNDERSTAND AND DO SIMILAR QUESTIONS:
The Campus Deli provides an excellent testing ground of queueing theory we learned in
class. There are four checkout lines in the deli, each of which is served by an employee
who collects payments. Lately, we noticed that a checkout line was reserved for
customers who used “dining dollars.” (Customers with cash payments can only join the
remaining three lines.) It is the purpose of this question to understand the impact of the
dining-dollars line on customer service.
First, let’s consider the original setup without the dining-dollars line. During the busy
lunch period from 11:30am to 2pm, customers arrive at the deli at an average rate of 20
customers per minute. Suppose the customers are equally split among the four checkout
lines. (In probability lingo, each customer randomly chooses a line to join, with the
probability of choosing a particular line equal to the probability of choosing any other
line.) For simplicity, assume that each customer, once joins a line, does not switch to
another line (as at McDonald’s.) An employee spends an average of 10 seconds per
customer to determine the payment amount and to collect the payment with proper
changes, if necessary. Assume that customer arrivals follow a Poisson process, and the
service times are exponentially distributed.
a) What is the average time a customer spends in the checkout process (from joining a
checkout line to departing)?
b) What is the average number of customers in the checkout area (including all four
checkout lines, waiting or being served)?
Now suppose one of the four checkout lines is dedicated to customers with dining dollars.
Because collecting dining dollars is simpler than collecting cash (e.g., no changes are
necessary), the average service time for a dining-dollars customer is only 4 seconds.
Suppose 25% of customers use dining dollars. The remaining 75% of customers pay by
cash, and thus must join one of the three cash lines. Again, these customers randomly
pick a cash line to join (with equal probabilities). The average service time for a cash
customer is 12 seconds. Assume that the arrival processes of the two customer classes are
both Poisson, and their service times are exponentially distributed.
c) What is the average time a dining-dollars customer spends in the checkout process
(from joining the line to departing)? What about a cash customer?
d) What suggestions do you have to improve service without increasing staffing levels?
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