Exam 16: Markov Processes

On any particular day,an individual can take one of two routes to work.Route A has a 25% chance of being congested,whereas route B has a 40% chance of being congested.The probability of the individual taking a particular route depends on his previous day's experience.If one day he takes route A and it is not congested,he will take route A again the next day with probability 0.8.If it is congested,he will take route B the next day with probability 0.7.On the other hand,if he takes route B one day and it is not congested,he will take route B again the next day with probability 0.9.Similarly,if route B is congested,he will take route A the next day with probability 0.6. a.Construct the transition matrix for this problem.(Hint: There are four states corresponding to the route taken and the congestion.The transition probabilities are products of the independent probabilities of congestion and next-day choice.) b.What is the long-run proportion of time that route A is taken?

A state is said to be absorbing if the probability of making a transition out of that state is zero.

If the probability of making a transition from a state is 0,then that state is called a(n)

At steady state,

When absorbing states are present,each row of the transition matrix corresponding to an absorbing state will have a single 1 and all other probabilities will be 0.

Markov process trials

A transition probability describes

Transition probabilities indicate that a customer moves,or makes a transition,from a state in a given period to each state in the following period.

A Markov chain cannot consist of all absorbing states.

Precision Craft,Inc.,manufactures ornate pedestal sinks.On any day,the status of a given sink is either: (a)somewhere in the normal manufacturing process,(b)being reworked because of a detected flaw,(c)finished successfully,or (d)scrapped because a flaw could not be corrected.The transition matrix is as follows: ​ ​ a.What is the probability of a sink eventually being finished if it is currently in process? b.What is the probability of a sink eventually being scrapped if it is currently in rework? c.What is the probability that a sink currently in rework will have a "finished" status either tomorrow or the next day? (Hint: There are three ways this can happen.)

All Markov chain transition matrices have the same number of rows as columns.

For Markov processes having the memoryless property,the prior states of the system must be considered in order to predict the future behavior of the system.

The probability of going from state 1 in period 2 to state 4 in period 3 is

When absorbing states are present,each row of the transition matrix corresponding to an absorbing state will have a single 1 and all other probabilities will be 0.

Markov processes use historical probabilities.

The probability of making a transition from state i in a given period to state j in the next period is denoted as