Consider a manufacturing process which utilizes CNC routers. There are four routers in the facility. During a shift, the automatic tool changing mechanism on the router may break down. Let the probability that any given router breaks down during a shift be 0.3. At the end of each shift a crew disassembles the mechanism of any broken tool changers and delivers them to a repair shop. The crew also reassembles any repaired mechanisms. The repair shop repairs any broken tool changers it has during the same shift as production. Let Y be the number of repairs in a shift. P[Y=y]=pY(y)=0.20.60.2y=0y=1y=2 You can think of this as a 2 shift operation with production and repair occurring during first shift and disassembly and reassembly occurring during second shift. 1. Compute the probability of k(k=0,1,2,3,4) breakdowns during a shift given 0 operating routers. 2. Do the same given 1,2,3, and 4 operating routers. 3. Compute the probability of k(k=0,1,2,3,4) repairs during a shift given 0 broken tool changers. 4. Do the same given 1,2,3, and 4 broken tool changers. 5. What is the relationship between the number of operating routers and the number of broken tool changers? 6. Compute the one-step state transition probability matrix, P. Hint: Assume we look at the process at the beginning of each day (start of first shift). 7. Draw the state transition diagram for this process. 8. In steady state, what is the expected number of operating routers? 9. Limited production occurs if there are two or fewer routers are operating. In the long run, what is the percentage of time that production will be limited? 10. If we start with four operating routers on day 1 , what is the expected number of operating routers at the beginning of day five? 11. If three routers are operating today, what is the probability that production will be limited eight days from now