IOE 316 Winter 2023 - Homework 5 Due 10:30 am on April 6, 2023 1. (15 points) You decide to enroll in two classes at a community college over the summer: a machine learning course and a humanities course. Each course has assignments given on average every 7 days. It takes you on average 1/2 a day to complete an assignment from the humanities course and 1 day to complete an assignment from the machine learning course. Assume you can only work on 1 assignment at a time, that the times between assignments and the time it takes you to complete assignments are exponentially distributed, and that if you are completing an assignment for a given course you will not receive any more assignments for that course until the assignment is completed (you may, however, receive assignments for the other course). Assignments are completed in the order in which they are received. (a) (5 points) Can we analyze this as a birth and death process? Briefly explain your answer. (b) (10 points) Model this as a continuous time Markov chain (CTMC). Clearly define all the states and draw the state transition diagram. 2. (15 points) Every time we travel we bring toys for the children to play with in the car. Assume that children play with each toy for an average of 30 minutes and then not again in that trip. Also assume this can be modelled as a Poisson process. We are going to Windsor and expect the ride to take 1.5 hours. What is the smallest number of toys we should bring if we wish the probability of running out of toys to be less than 15%? 3. (25 points) Juliet is opening up a new restaurant in Ann Arbor, and she is keeping track of how many customers are planning on attending opening night. Part of getting new customers to come to opening night is done by word-of-mouth. Assume each existing customer recruits new customers to come to opening night of the restaurant at a rate of . Unfortunately, each existing customer decides not to attend opening night at an exponential rate of . If the total number of customers planning to attend opening night is less than N, then Juliet will post advertisements in the local newspaper to find more new customers, and new customers will decide to attend opening night at an exponential rate of 0 due to the advertisements. If the total number of customers planning on attending opening night is greater than or equal to N, then Juliet will stop advertising, and new customers will only be recruited through word-of-mouth from other customers. Assume there is no maximum amount of customers that can attend the opening night. (a) (10 points) Set this up as a birth and death model. That is, clearly define the states and transition rates. Draw rate diagram. 1 (b) (10 points) Set up balance equations to be solved to find Pj's for j 0. Do not solve them. (c) (5 points) Let N = 40. Express the proportion of time that Juliet will spend advertising in the local newspaper in terms of Pj's. (Do not need to solve) 4. (20 points) A walk-in vet clinic with a single doctor has space for at most 4 customers. Assume that if a customer arrives and sees 4 other customers waiting, the customer leaves and goes to a different clinic. Customers arrive at a Poisson rate of 1 per hour, and the successive service times are independent exponential random variables with mean 30 minutes. (a) (5 points) What is the average number of customers in the system? Draw the rate diagram, clearly defining the states and label all transition rates. (b) (5 points) How many customers enter the system per hour? (c) (5 points) If the doctor could work twice as fast, how much more business would the clinic get? (That is, on average, how many more customers would be able to enter the system per hour with respect to (b)?) Draw the rate diagram, clearly defining the states and labeling all transition rates. (d) (5 points) Now, if the doctor was working at the original rate of 30 minutes per cus- tomer, but she could hire an additional doctor with the same service rate, how much more business would the clinic get? (That is, on average, how many more customers would be able to enter the system per hour with respect to (b)?) Draw the rate diagram, clearly defining the states and labeling all transition rates. 5. (25 points) A traffic light alternates between having a green light, a yellow light, or a red light. The amount of time that the traffic light displays each color is exponentially distributed. If the light is green, it will always switch to yellow after an average of 15 minutes. When the light is yellow, it displays yellow for an average of 10 minutes, and then switches to green with probability 0.4 or to red with probability 0.6. When the light is red, it is red for an average of 20 minutes. After it is red, the light will switch to green or yellow with equal probability. (a) (5 points) Draw the CTMC rate diagram, clearly label your states and transition rates (qij's) (b) (5 points) Find the transition rates out of the states (vi's) and transition probabilities (Pij's). (c) (5 points) Find the limiting probabilities (make sure to write out the balance equations). (d) (5 points) In the long run, what is the probability that the light is not red? (e) (5 points) A policeman has been notified that there is quite a bit of speeding happening a block from the intersection where the light is located. When there is a green light, 5% of the time, cars are driving over the speed limit. However when there is a yellow light, the policeman observes that cars drive over the speed limit 70% of the time. When there is a red light ahead, cars do not drive over the speed limit. What fraction of the time do cars drive over the speed limit? 2