Just-in-Time Manufacturing

Suppose that you have a machine that completes a specific process for manufacturing a product. This machine can complete the process of a sub-product within 3 minutes on average and the process completion time is exponentially distributed. Sub-products arrive randomly with a rate of 15 per hour at the station in front of the machine. The inter-arrival times of the sub-products is exponentially distributed.

There should always be an operator next to the machine and an operator's cost is $15 per hour. The machine has a cost of $20 per each hour it is active, i.e., processing a sub-product. Furthermore, the sub-products have a penalty for the time they spend from the moment they arrive at the station until their processes have been completed. Specifically, due to just-in-time requirements; the penalties for sub-products are given by the following schedule:

• If the sub-product takes less than 10 minutes to be processed, the penalty is $1
• If the sub-product takes between 10 and 15 minutes to be processed, the penalty is $2
• If the sub-product takes more than 15 minutes to be processed, the penalty is $3

Please answer the following questions about the above system.

You can use the excel templates in your calculations. However, you need to show how you are using the information given by the excel template in your calculations as excel templates gives your only information for L, Lq, W, Wq, Pn, P(W>t), and P(Wq>t).

a) Describe the above system as a queueing system by defining

• Customers, expected inter-arrival time, and arrival rate
• Server(s), expected service time, and service rate
• Express the queueing model using Kendall's notation

b) Explain if the above queueing system satisfies the following criteria by showing your calculations for each criterion.

• On average, there should not be more than 2 sub-products waiting to start their processes at the station in front of the machine.
• On average, there should be less than or equal to 3 sub-products waiting to start their processes at the station in front of the machine for at least 75% of the time.
• On average, less than 50% of the sub-products should wait more than 5 minutes to start their processes at the station in front of the machine.
• As the machine is a sensitive machine, the time it is active should not be 4 times more than the time it is inactive

c) Calculate the expected hourly cost of the queueing system. Expected hourly cost of the system includes the labor cost per hour plus the expected hourly cost of the machine plus the expected penalties charged per hour. Show your calculations.