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# Waiting lines problems

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I have some Quantitative Analysis questions I need help understanding.

Waiting lines and queuing Theory models

1. The New Providence shopping mall is considering setting up an information desk manned by one employee. Because of the complex design of the mall, it is expected that people will arrive at the desk at about twice the rate for most malls. The expected rate is 25 per hour. It will also take longer to answer their questions - approximately four minutes per person on average.

(a) Find the probability that the employee is idle.
(b) Find the proportion of the time that the employee is busy.
(c) Find the average number of people receiving and waiting to receive some information.
(d) Find the average number of people waiting in line to get some information.
(e) Find the average time a person seeking information spends waiting and at the desk.
(f) Find the expected time a person spends just waiting in line to have his question answered.

2. The new Providence shopping mall is considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately four minutes for the Information Desk employee to help the average person. By utilizing a new map and special guide signs, the manager believes that the required service time can be reduced to an average of two minutes per person. Assuming that he implements the new map and guide signs:

(a) Find the probability that the Information Desk employee is idle.
(b) Find the proportion of the time that the Information Desk employee is busy.
(c) Find the average number of people receiving and waiting to receive some information.
(d) Find the average number of people waiting in line to get some information.
(e) Find the average time a person seeking information spends waiting and at the desk.
(f) Find the expected time a person spends just waiting in line to have his question answered.

3. The new Providence shopping mall has been considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person. By utilizing a new map and special guide signs, he believes that the required service time can be reduced to an average of two minutes per patron. The manager has also noticed that the people seeking help at the information desk may come from one of two groups: (a) mall patrons, and (b) mall staff, or delivery persons. Therefore, the manager has decided that he wants to consider another option: establishing two information desks - one desk to help mall patrons, the other to help staff and delivery persons. The manager believes that he can expect patrons to arrive at the rate of 20 per hour, and mall staff or delivery persons to arrive at the rate of 5 per hour. It is likely to take an average of two minutes to answer the questions of a patron, and an average of ten minutes to answer those of a staff or delivery person. Assuming that the two-desk concept is implemented,

(a) find the probability that both Information Desk employees are idle.
(b) determine whether the patron or the staff person is likely to have the longer wait.

4. The new Providence shopping mall has been considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person. He has now come to realize that employing only a single person at the information desk would lead to a very lengthy line - theoretically, an infinite line! He has decided, therefore, to employ two staff members at the information desk.

(a) Find the proportion of the time that the employees are busy.
(b) Find the average number of people waiting in line to get some information.
(c) Find the expected time a person spends just waiting in line to have his question answered.
(d) If the manager has a goal that, for the average patron, the time spent having one's question answered is less than half the time spent waiting, has he met that goal?
(e) Assuming that the manager also has the goal that the combined idle time for the two workers does not exceed 45 minutes in an 8 hour day has he met that goal?
(f) What would the arrival rate have to be for the manager to meet the 45 minute idle time goal?