Problem 1) The Sandwich Shop (Queueing / Simulation) [65 marks]

A sandwich shop that offers toasted sandwiches is experiencing long lines during its lunch hour, and the owner would like you to investigate. When customers arrive in the store they go to the first station where they choose their sandwich bread and toppings and watch the sandwich get prepared. Then, whether or not the customer wants their sandwich toasted, they wait at the second station in front of the toaster (all customers stay in the order in which they arrived). All customers then proceed to the third station to receive and pay for their sandwich.

There are currently 3 employees: one who works the first station, one who works the toaster at the second station, and one who works the cash register at the third station. The time to prepare a sandwich at the first station is normally distributed, with a mean of 25 seconds and a standard deviation of 10 seconds. The time to toast a sandwich is (continuous) uniformly distributed between 40 seconds and 55 seconds. The time it takes to pay is exponentially distributed with mean 30 seconds.

Customers arrive to the shop at a rate of 100 per hour. Assume inter-arrival times are exponentially distributed. Only 70% of the customers ask for their sandwich to be toasted. [Remember that they stay in order of arrival, whether or not they ask for a toasted sandwich.]

If a customer enters the store and the employee at the first station is busy, they stand in line.

When a customer finishes at the first station, they and their sandwich immediately proceed to the second station, while the employee who was helping them at the first station immediately helps the next customer in line.

If a customer asks for their sandwich to be toasted, they must wait for the toaster to be free. Once the sandwich is toasted, or if they didnt ask for it to be toasted, they proceed to the third station, where they may have to wait for the cashier to be free. Keep in mind the order of the customers never changes, they must always wait for the person in front of them to finish at their station.

Once a customer has paid at the third station, they leave the store.

Q1) Model the current situation by simulating 200 customer arrivals starting at 12pm on 1000 different days. NOTE: You may use Excel add-in ASPE, YASAI, or MCSim to automate the task of running 1000 simulations and summarizing the results. Save both the model and simulation results as separate sheets in one workbook.

a) Describe your modelling approach briefly: if you used a simulation table, what does one row represent? How do you handle some of the key logic in the model (be descriptive, dont just present your formulas)? [21 marks]

b) What is the average time customers spend in the shop (over the lunch hour)? How did you calculate this? [3+3 marks]

c) What is the average time spent waiting before being served at the first station? How did you calculate this? [3+3 marks]

d) What is the utilization rate of the toaster? How did you calculate this? [3+3 marks]