10. When running an MCMC algorithm, it is important to know when the transient (or burn-in) period has finished; otherwise, steady-state statistical analyses may not be applicable. In practice this is often done via a visual inspection of the sample path. As an example, run the random walk sampler with normal target distribution N(10,1) and proposal Y ~ N(x, 0.01). Take a sample size of N = 5000. Determine roughly when the process reaches stationarity.6. Consider a continuous flow line consisting of three machines in tandem separated by two storage areas, or buffers, through which a continuous (fluid) stream of items flows from one machine to the next; see Figure 1. Ma Figure 1 A flow line with three machines and two buffers (three-stage flow line). Each machine / = 1, 2, 3 has a specific machine speed v, which is the maximum rate at which it can transfer products from its upstream buffer to its downstream buffer. The lifetime of machine / has an exponential distribution with parameter 1 . The repair of machine / starts immediately after failure and requires an exponential time with parameter #, . All life and repair times are assumed to be independent of each other. Failures are operation-independent. In particular, the failure rate of a "starved" machine (a machine that is idle because it does not receive input from its upstream buffer) is the same as that of a fully operational machine. The first machine has an unlimited supply. Suppose all machine speeds are 1, the buffers are of equal size b, and all machines are identical with parameters / = 1 and / = 2. (a) Implement an event- or process-oriented simulation program for this system. (b) Assess via simulation the average throughput of the system (the long-run amount of fluid that enters/leaves the system per unit of time) as a function of the buffer size b