A spacecraft is sent to transmit photographs of an outer planet. There are three critical systems involved: the camera, the power system, and the antenna. These systems fail independently of each...

The University of Western Australia School of Mathematics & Statistics STAT3361 RANDOM PROCESSES & THEIR APPLICATIONS Practice Exercises 5/Assignment 2 (2012) There is a three-part question for the second (and last) assignment on page 2. Submit your answers by the end of semester. Computational problems. 1. A spacecraft is sent to transmit photographs of an outer planet. There are three critical systems involved: the camera, the power system, and the antenna. These systems fail independently of each other, and once failed they cannot be restored. The mean system lives are: 12 years for the camera, 6 years for power, and 10 years for the antenna. Assume all lifetimes have an exponential distribution. If it takes 3 years for the spacecraft to reach its destination, what is the probability of mission failure? 2. A car is to be used for a long distance trip. The four tyres on the car each have a mean driving distance to failure (puncture, blowout etc.) of 5000 Km. The spare tyre is in poor condition; its mean distance to failure is 1000 Km. Assume the distances to failure are independent exponentially distributed random variables. If two failures occur then the car is stuck. Compute the mean distance which can be driven until the car is stuck. 3. (a) Compute P(0.2) for the Markov process with S = {1, 2, 3, 4, 5} and Q = 2666664 -8 4 4 0 0 5 -15 5 5 0 5 5 -18 4 4 0 5 5 -14 4 0 0 5 5 -10 3777775. (b) Compute P(0.1) if S = {1, 2, 3, 4, 5, 6} and Q = 266666664 -20 6 6 8 0 0 0 -14 6 8 0 0 0 0 -6 6 0 0 0 6 8 -14 0 0 8 0 0 0 -14 6 6 0 0 0 6 -12 377777775 . 4. (a) Compute the limit law for the Markov process in Question 3 (a). (b) Determine the smallest closed subset C of the state-space for the Markov process in Question 3 (b). Compute the limits limt!1 pij(t) for all i, j 2 C. 15. Consider the call centre, Example 7c.4. Suppose the arrival rate is 60 per hour, that the mean service time is 6 minutes, there are 8 servers and 4 holding places, i.e. the system capacity is 12....