Over the last few years a change has swept across our supermarkets. The regular single server first-come first-served queues that we are used to are being replaced by banks of self service stations supervised by one, or a few, store employees The time has come for yet another store to update their checkout procedure The store owners want to work out the savings they stand to make by replacing two of their three checkout counters with one bank of 10 self service stations supervised by just one employee. The store owners are particularly interested in the peak time between 5 pm and 8 pm when the number of people in the store is roughly constant, and the rates given in the following have been observed to govern the queues in the Store This is the current situation, before the implementation of Self Service: Customers arrive at each of the checkout counters according to a Poisson process with rate 0.3 per minute The time that it takes to process each customer is exponentially distributed, and the highly skilled checkout operators take an average of 2 minutes to process each customer. Employing each operator costs the store $10 per hour This is for the situation after the change Find, in the Appendix, the output from the Worksheet for memory less (M M/c/n) queue for the projected situation after the change It is assumed that the total arrival rate of customers to the checkout area of the store. as well as the rate of customers going to the one remaining old style checkout counter will remain the same after the change State the Kendall notation, as well as service and arrival rates, for an infidel checkout counter The time of the customers is valued the same before and after the change to the checkout system Find, based on the input in the spreadsheet, how much an hour of customer waiting time is assumed to cost the store Calculate the cost per hour for one checkout counter before the change. Include the cost of the server as well as the cost of customer time found in part (b) Give the total hourly cost before the change for the parts of the system that are to be replaced The self-checkout stations are leased on an hourly basis from the company Cyber dyne Payment Systems. Find, using the spreadsheet output, how much each of the stations costs the store per hour Find the total cost per hour for the whole bank of self-service stations, including the cost of the supervising store employee who cost as much per unit time as a checkout operator Comment on the savings that the store makes by making this change. In particular at least comment on the following three questions: How does the change impact the time that the users spend waiting and in service? How much money does the store save by this change? Where do the savings come from?

## Expert Answer

Service and arrival rates – abbreviations for the service and arrival processes

M-poisson process

D-deterministic process

G-general process

M/M/1 queue notation for when there is 1server system.

M – arrival process is Poisson

M – service/departure process is Poisson

1 – 1 server system is denoted when there is “1”

infinite queue capacity ;no arriving client will be rejected.

Infinite population size ,arrival process will be unaffected by the number of clients alredy in the system

FIFO – first in first out service.

g)

2)This change impacted the time ,with the poisonn processs with the rate of 0.3per minute. The time this takes to process each customer is distributed,and the highly skilled checkout operator take an average of 2 minute to process each customer. Employing an opetator each costs the store $10per hour.

3) They save money by replacing 2 of their three checkout counters with one bank of 10self service stations supervised by just one employee. So, the cost per server gets saved and this profit is gained.