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1.The site The site Stackoverflow.com is a popular question/answer site for programmers. Assume that the total number of questions asked on the site follows a homogenous Poisson process and that on average the site receives 288 questions per hour.
(a) Once a question is asked – How long would you have to wait (on average) before a new question is asked? Answer in terms of seconds.
(b) At the time this is being written there are 4839500 questions on StackOverflow.
i. What is the distribution of the amount of time it will take until the 5000000th question is asked (starting from when there are 4839500 questions).
ii. How long should I expect to wait until the 5000000th question is asked?
(c) If I watch the stackoverflow home page for new questions for a single minute then what is the probability that I don’t observe any new questions being asked? What is the probability that I observe 10 or more new questions get asked?
(d) Identify a problem with modeling this as a homogenous Poisson process.
2. A fast food restaurant oers a drive through. They aren’t a major chain though and there is only one window where you both place your order and receive your food. Assume that we can model the drive through as a M/M/1 queue with an arrival rate of 8 customers per hour and on average it takes a customer 5 minutes to place and receive their order
(a)What is the probability that when a customer pulls up that the drive through is empty?
(b)On average how long will it take between when a customer pulls into the drive through and is able to leave with their food?
(c)On average how many cars are in the drive through?
(d) On average how many cars are waiting in the drive through but are not at the service window?
(e)On average how long does it take between when a car pulls into the drive through and when they arrive at the service window?
(f)I pull into the drive through at 12:00pm. There are 3 cars in front of me in the drive-through. I have a meeting I need to get to and I’ll be on time if I get through the drive through before 12:30pm. What is the probability that I am late to the meeting?
(g) What is the probability that there are 4 cars in the drive through?
3. Identify a queueing system in the real world. You are being asked to collect data from this system. You need to watch the queue for a minimum of 10 minutes and observe a minimum of 5 arrivals into the queue.
(a) Describe the queue you observed. This doesn’t need to belong – for example: The customers waiting in line at the Caribou on campus – there is only one person taking orders.
(b) Provide the data you collected, which should include at a minimum the times whenever a person entered or exited the system and the queue length at those times. Ideally we would track each person in the queue and know when they entered, when they entered the service center, and when they were nished. However, in a fast queue that would be a lot of information to keep track of so noting the length of the queue at the beginning and when anybody either entered or exited the queue (along with whether it was an arrival or an departure). Note that I am mentioning people entering/exiting the queue but you are allowed to observe a queue that isn’t comprised of people.
(c) Graph the number of items in your queueing system over time
(d) What is the most appropriate model for the data you observed?
(e) Estimate the parameters for your model. You can assume the data is M/M/1 even if that isn’t the model you identied in the previous problem if you would like (which would mean identifying and).
(f) Estimate the average amount of time it takes to go through the entire process in the system you observed.
(g) Estimate the average number of items in the queueing system (including those receiving service).
(h) What is the probability that queueing system is empty?