5. 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?