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ASSIGNMENT
PROBLEM:
Consider a popular coffee chain that has decided to introduce warm sandwiches and pastries to its menu. The current system can be described as follows. Customers join a FCFS queue to place their order and then proceed to the next counter to collect their completed order. Coffee customers can be divided into two types:
1) Those that order a Regular coffee. These customers have an arrival pattern that follows an exponential distribution with a mean of 2 minutes and the processing time for their order follows a uniform distribution with parameter values equal to 0.5 and 1 minute.
2) Those that order a Specialty coffee. Their arrival pattern follows an exponential distribution with amean of 4 minutes and the processing time for their order follows a triangular distribution with parameter values equal to 2, 3, and 4 minutes.
3) The proposed system would allow customers to also purchase a sandwich/pastry with their coffee. The process involves the employeeselecting the item the customer wants then placing it in the oven to heat it up. Management estimates that the additional time to process a sandwich/pastry order follows a triangular distribution with parameter values equal to 3, 5, and 9 minutes. They expect the arrival pattern of customers wanting a sandwich/pastry will follow a lognormal distribution with mean 5 minutes and a standard deviation of 3. Since this is one of the best coffee places in the world, management expects that every customer who purchases a food item will also purchase a regular or specialty coffee.
A few additional issues that management would like you to consider are:
1) In the fashion typical of these types of establishments, management feels that one employee at the ordering station and two for preparing food/coffee is sufficient. In this setup, each food preparation employee has their own workstation: Order Prep WS1 and Order Prep WS2. The café has a sophisticated computer system that directs each order (and customer) to the employee with the shortest queue. If the two queues are the same, the computer will direct the order/customer to Order Prep WS1.
2) Management has done some research and studied past data and feels that customers will decide to forgo their coffee or grande double soy mocha lattes and/or foccacia smoked ham with aioli sandwiches if their waiting time exceeds three minutes (which is typical for this type of industry). They feel this translates into a customer balking when the sum of the order preparation queues for WS1 and WS2 exceeds 12 people.
3) They expect that many of the customers will already be familiar with their drinks menu so the ordering time for drinks only will follow a uniform distribution with parameter values [1,1.5]. This includes payment. On the other hand, with their sandwiches/pastries, they expect the same type of customer behavior they experienced when they first opened their establishment many years ago (i.e., a somewhat catatonic-like state while perusing the many options on the menu). In any case, they expect the coffee and food ordering process to follow a lognormal distribution with a mean of 2 minutes and standard deviation equal to 0.5.