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This week, you are visiting the FSA's regional branch in Valleyfield and during the three days you have been here,

you have noticed that customers waiting for services with the bank teller seem to be waiting a long time. The branch is open daily from 8 to 12 in the morning and 1 to 5 in the afternoon, with a one-hour lunch break. There is only one bank teller and it takes her an average of about 6 minutes to serve a client (the service time follows an exponential distribution). An average of 64 clients come to the bank each day and their arrival rate follows a Poisson distribution. 


a) What are the arrival rate and service rate in the situation described above? 


b) What is the average number of clients in the waiting line and the average time waiting in line?


c) What is the average total time spent in the system? Is this answer coherent with those given in a)? Justify. You know there are many ways to manage a waiting line and decide to compare different alternatives which should help reduce the waiting time.


d) An obvious solution to reducing customer waiting time would be to hire a second bank teller. If the average service time is still 6 minutes for each bank teller, calculate the same parameters as in a), b) and c).


e) However, another idea would be to hire someone part time to train clients to use the FSA's online services. You estimate that this would help reduce the average number of clients per day to about 50 clients. If the opening hours and service time remain unchanged and we stay with only one bank teller, calculate the same parameters as in a), b) and c).


f) You are also quite certain that by improving the bank teller's processes, you could reduce the average service time to 5 minutes/client. If the opening hours remain unchanged, they still have only one bank teller and the average number of customers per day is still 64, calculate the same parameters as in a), b) and c).


g) Compare the costs of the current situation with the three improvement alternatives according to the following costs:


• The bank tellers are paid 40 000 $/year. • Knowing that clients hate waiting in line, you propose to associate a cost to it. Consider that each client in the waiting line generates a loss (i.e. cost) of 2600$/year (i.e. for example, 2600$/year if there is an average of 1 client in the waiting line or 3900$/year if there is an average of 1,5 clients in the line). • The salary for a part-time worker to train clients for online services is estimated at 20 000$/year. • You estimate that analysing the bank teller's processes to reduce the service time to 5 min/client will cost about 10 000$. 


Which alternative would you choose? Why?

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