A university, installed a single telephone booth in the student affair office, to provide students with no mobile phone or unwilling to use their mobile phone, the capacity to make a call whenever needed. The arrivals at the telephone booth are considered to be Poisson at an average time of 8 min between our arrival and the next. The length of the phone call is distributed exponentially, with a mean of 4 min. The student affair office stays open 10 hours per day.

Determine for this scenario 1:

1.The arrival rate per hour

2.The service rate per hour

3.Expected in percentage the fraction of the day that the phone will be in use.

4.Expected time in hours the phone will be used per day.

5.Expected number of students in the queue

6.Expected waiting time in the queue.

7.Expected number of students in the system.

8.Expected time spent in the system

9.What is the probability that an arrival will have to wait in queue for service?

10.What is the probability that there are 3 or more persons in the system?

11.What is the probability that more than 5 persons are in system?

12.What is the probability that an arrival will directly enter for service?

With time, the number of phone users increased. The Dean of student is claiming that it is time to install a second telephone booth, as students are waiting for at least 6 min in queue for phone.

In this respect and given the service rate remains the same, determine for this scenario 2:

13.What should be the arrival rate in order to justify the installation of a second booth?

14.What would be the P₀ value?

15.What would be the expected number of students in the system?

16.What would be the expected time spent in the system?

17.What would be the expected number of students in the queue?

18.What would be the expected waiting time in the queue?

19.What is the probability that an arrival will have to wait in queue for service?

The university's operation manager is claiming that considering the variation in the arrival rate to justify the installation of a second telephone booth, (as stated by the Dean of students) is wrong. In his point of view, the parameter that should be taken into consideration is not the arrival rate but the service rate. Therefore, he is suggesting to install a second telephone booth because the the average length of the phone call becomes 4.5 minutes.

In this respect and given the arrival rate is the same than the one you have calculated in scenario 1, determine for this scenario 3:

20.The service rate per hour

21.What would be the P₀ value?

22.What would be the expected number of students in the system?

23.What would be the expected time spent in the system?

24.What would be the expected number of students in the queue?

25.What would be the expected waiting time in the queue?

26.What would be in percentage the expected fraction of the day that the phone will be in use.

27.What is the probability that an arrival will directly enter for service?

To solve the problem resulting from the two different perspectives (in scenario 2 and scenario 3), the university council, asked the team of experts from the telephone company working on the project, to provide their solution. The report of the team stated the following:

Given the standards applied in such environment,

Given the rental fees paid by the university per telephone booth of 2 USD daily,

Given a waiting cost per person of 0.70 USD per minute,

A second booth should be installed when the arrival rate becomes 60 persons per day and the service rate becomes 120 calls per day.

In this respect, determine for this scenario 4:

28.The arrival rate per hour

29.The service rate per hour

30.What would be the P₀ value?

31.What would be the expected number of students in the system?

32.What would be the expected time spent in the system?

33.What would be the expected number of students in the queue?

34.What would be the expected waiting time in the queue?

35.What would be in percentage the expected fraction of the day that the phone will be in use.

Now that all of the queuing indicators (with respect to all scenarios) are calculated,

36.What would be the total cost of the queuing system in the case of the first scenario?

37.What would be the total cost of the queuing system in the case of the second scenario?

38.What would be the total cost of the queuing system in the case of the third scenario?

39.What would be the total cost of the queuing system in the case of the fourth scenario?

40.Which of the scenario is the best?