A small business has recently purchased a specialised printing and binding machine that can read a PDF document and automatically print and bind the document. The business has designed an online interface where customers can upload documents, and then the specialised machine will automatically print and bind the document as soon as it is ready. Immediately after a customer uploads a document to the website, the document is added to a queue for processing and status updates will be made available for the customer. The status updates given to the customer are Waiting, In Progress and Ready. Upon being uploaded, the status will display Waiting until the specialised machine begins printing the document. Once it starts being printed, it will display In Progress, and when it is ready for the customer to pick up from the store, it will show Ready. The store advertises a pickup time of 3 hours, suggesting to customers that, on average, they will be able to pick up their document approximately 3 hours after uploading it. A customer will be satisfied if they can pick up their document exactly after the advertised pickup time, and dissatisfied otherwise. Assume that the machine runs every day, non-stop, without any breaks. Based on historical data, an average of 37 documents are uploaded per day, and the machine takes an average time of 29 minutes to print and bind the document. To analyse the system, the company has decided to model the service as an M/M/1 queue using these parameters. For Part 1 of the project, you may only use facts and results that have been given in the subject materials, unless otherwise stated. 1. State the arrival rate, service rate and traffic intensity for this queue. Include appropriate units. 2. Determine the average time taken for the status to change from Waiting to In Progress. 3. Determine the probability that a document takes 16 minutes to finish being printed and bound after it starts being printed. 4. Let N denote the number of documents in the queue. State the probability distribution that N follows, giving your answer in the form N Then use this to determine, on average, how many hours per day the machine is inactive. 5. Is the store's advertised pickup time reasonable? Explain. You are now given the following general fact that may be used for the remaining questions: for an M/M/1 queue with p < 1, if the random variable T denotes the total time an individual spends in the queue, then T~ Exp(-X). 6. Explain how the given fact results in the formula for the average time spent in the queue. 7. Using the given fact, determine the probability that a customer will be satisfied. 8. Assume that the store wants at least 95% of customers to be satisfied based on the advertised pickup time. Should they change their advertisement? Explain. If so, suggest a suitable change. 9. Suppose that exactly 40 customers will upload a document tomorrow. Let C denote the number of tomorrow's satisfied customers (using the currently advertised pickup time). What probability distribution does C follow? Give your answer in the form C 10. The store would like to understand the effects of increased demand on the system. Provide some insight to the store that would help them understand the level of demand that would make the system unmanagable. You must give the result of any calculations to at least 3 decimal places.