DEAR TUTOR, PLEASE SHOW ALL THE WORKING OUT. SO I CAN UNDERSTAND AND DO SIMILAR QUESTIONS:

Problem 2

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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