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A university cafeteria line in the student union hall is a self-serve facility in which students select
the food items they want and then form a single line to pay the cashier. Students arrive at a rate
of about 4 per minute according to the Poisson distribution. The single cashier ringing up sales
takes about 12 seconds per customer, following an exponential distribution.
a) Assess the performance of the queue and write a report on your findings.

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Answer:

The answer to this question is a= µ=60/12=5 students/min

Explanation:

Solution

Given that:

λ=4 students / min

The Waiting time in Queue= λ /µ(µ- λ )==4/(5*(5-4))=0.8 min

The Number of students in the line L(q)= λ *W(q)= 4*.8= 3.2 students

TheNumber of students in the system L(q)= λ /(µ- λ )=4/(5-40=4 students

Then,

The Probability of system to be empty= P0= 1-P= 1-0.8= 0.2

Now,

If the management decides to add one more cashier with the same efficiency then  we have

µ= 6 sec/student= 10 students/min.

so,

P= λ /µ =4/10=0.4

Now,

The probability that cafeteria is empty= P0= 1-0.4= 0.6

If we look at the above  system traits, it is clear that the line is not  empty and the students have to standby for 0.8  in the queue waiting to place their order and have it, also on an average there are 3.2 students in the queue and in the entry cafeteria there are 4 students who are waiting to be served.

If the management decides to hire one more cashier with the same work rate or ability, then  the probability of the cafeteria being free moves higher from 0.2 to 0.6 so it suggests that the management must hire one additional cashier.

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