This week, you are visiting a GSO grocery store in a rural town of Quebec to see if you could increase their customer satisfaction. During the three days you have been here, you have noticed that customers seem to be waiting a long time to pay for their groceries. It is a small local grocery store with only one cashier. The store is open daily from 8 a.m. to 6 p.m. On average, it takes the cashier 6 minutes to serve a client (the service time follows an exponential distribution). In the three days you have been in the store, you have counted an average of 80 clients 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 add a second cashier. If the average service time is still 6 minutes for each cashier and the clients form a single line waiting to go to the first cashier to be free, calculate the same parameters as in a), b) and c).

e) However, adding a second cashier would require considerable investment and take up some of the much-needed space in this store. Another idea would be to add a small self service checkout machine. You estimate that this would help reduce the average number of clients per day for the cashier to about 50 clients. If the opening hours and service time remain unchanged and we stay with only one cashier, calculate the same parameters as in a), b) and c). Note: we exclude the self-service machine and the clients who use the machine from our calculations at this point. We assume they are an investment needed to reduce the arrival rate for our single server.

f) You are also quite certain that by improving the cashier's process, you could reduce the average service time to 5 minutes/client. If the opening hours remain unchanged, they still have only one cashier and the average number of customers per day is still 80, 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 cashiers currently working at this GSO have a cumulative salary of $60,000 per 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).

Adding a second cashier would entail an added salary of $25,000 per year. It would also require equipment which you estimate at an annualized cost of $5,000.

Adding a self-service checkout would have an annualized cost estimated at $8,000.

You estimate that analysing the cashier's process to reduce the service time to 5 min/client will cost about $2,000.

Which alternative would you recommend? Why