A certain part of a machine can be in two states: working or undergoing repair. A working part fails during the course of a day with probability a. A part undergoing repair is put into working order during the course of a day with probability b. Let Xn be the state of the part. (a) Show that Xn is a two-state Markov chain by drawing the state transition diagram. (b) Give the one-step transition probability matrix P. (c) Draw the trellis diagram of state transitions. (d) Find the steady state probability for each of the two states.
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Refer back to Exercise 3. $$ \begin{array}{l}{\text { (a) Construct the one-step transition matrix } \mathbf{P} \text { of this chain. }} \\ {\text { (b) Show that } X_{n}=\text { the machine's state (full, part, broken) on the } n \text { th day is a regular Markov }} \\ {\text { chain. }}\end{array} $$ $$ \begin{array}{l}{\text { (c) Determine the steady-state probabilities for this chain. }} \\ {\text { (d) On what proportion of days is the machine fully operational? }} \\ {\text { (e) What is the average number of days between breakdowns? }}\end{array} $$
Markov Chains
Regular Markov Chains and the Steady-State Theorem
A certain factory has two machines and one repair crew. Assume that the probability of any one machine breaking down in a given day is Ě‘. Assume that if the repair crew is working on a machine, the probability that they will complete the repair in one more day is Ě’. For simplicity, ignore the possibility of a repair completion or a breakdown taking place except at the end of a day. Let Xâ‚™ represent the number of machines in operation at the end of the nth day. Assume that the behavior of {Xâ‚™} can be modeled by a Markov chain. (a) Set up the one-step transition matrix P. (b) If the system starts out with both machines operating, what is the probability that both will be operating 2 days later?
Ameer S.
Consider a system with two components. We observe the state of the system every hour. A given component operating at time n has probability p of failing before the next observation at time n + 1. A component that was in a failed condition at time n has a probability r of being repaired by time n + 1, independent of how long the component has been in a failed state. The component failures and repairs are mutually independent events. Let Xn be the number of components in operation at time n. The process {Xn, n = 0,1,...} is a discrete time homogeneous Markov chain with state space I = 0, 1, 2. a) Determine its transition probability matrix, and draw the state diagram. b) Obtain the steady state probability vector, if it exists.
Shu-Ting H.
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