Exam 16: Markov Processes

Appointments in a medical office are scheduled every 15 minutes. Throughout the day, appointments will be running on time or late, depending on the previous appointment only, according to the following matrix of transition probabilities: Previous Next Appointment Appointment On Time Late On Time .75 .25 Late .30 .70 a.The day begins with the first appointment on time.What are the state probabilities for periods 1, 2, 3 and 4? b.What are the steady state probabilities?

Where is a fundamental matrix, N, used? How is N computed?

A Markov chain cannot consist of all absorbing states.

Steady state probabilities are independent of initial state.

A state i is an absorbing state if p_ii = 0.

A city is served by three cable TV companies: Xcellent Cable, Your Cable, and Zephyr Cable. A survey of 1000 cable subscribers shows this breakdown of customers from the beginning to the end of August. Company on Company on August 31 August 1 Xcellent Your Zephyr Xcellent 300 50 50 Your 10 200 40 Zephyr 40 80 230 a.Construct the transition matrix. b.What was each company's share of the market at the beginning and the end of the month? c.If the current trend continues what will the market shares be?

The probability of reaching an absorbing state is given by the

The matrix of transition probabilities below deals with brand loyalty to Bark Bits and Canine Chow dog food. Current Next Purchase Purchase Bark Bits Canine Chow Bark Bits .75 .25 Canine Chow .20 .80 a.What are the steady state probabilities? b.What is the probability that a customer will switch brands on the next purchase after a large number of periods?

The probability that the system is in state 2 in the 5th period is $\pi$ ₅(2).

Absorbing state probabilities are the same as

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

The probability that a system is in a particular state after a large number of periods is

Two airlines offer conveniently scheduled flights to the airport nearest your corporate headquarters. Historically, flights have been scheduled as reflected in this transition matrix. Current Next Flight Flight Airline A Airline B Airline A .6 .4 Airline B .2 .8 a.If your last flight was on B, what is the probability your next flight will be on A? b.If your last flight was on B, what is the probability your second next flight will be on A? c.What are the steady state probabilities?

In Markov analysis, we are concerned with the probability that the

For a situation with weekly dining at either an Italian or Mexican restaurant,

A television ratings company surveys 100 viewers on March 1 and April 1 to find what was being watched at 6:00 p.m. -- the local NBC affiliate's local news, the CBS affiliate's local news, or "Other" which includes all other channels and not watching TV. The results show MIarch 1 Record of Switches During March to Choice Number NBC CBS Other NBC 30 - 5 10 CBS 40 15 - 5 Other 30 5 5 - a.What are the numbers in each choice for April 1? b.What is the transition matrix? c.What ratings percentages do you predict for May 1?

Why is a computer necessary for some Markov analyses?

Rent-To-Keep rents household furnishings by the month. At the end of a rental month a customer can: a) rent the item for another month, b) buy the item, or c) return the item. The matrix below describes the month-to-month transition probabilities for 32-inch stereo televisions the shop stocks. This Next Month Month Rent Buy Return Rent .72 .10 .18 Buy 0 1 0 Return 0 0 1 What is the probability that a customer who rented a TV this month will eventually buy it?

The fundamental matrix is used to calculate the probability of the process moving into each absorbing state.

Bark Bits Company is planning an advertising campaign to raise the brand loyalty of its customers to .80. a.The former transition matrix is $\left[ \begin{array} { c c } .75 & .25 \\.20 & .80\end{array} \right]$ What is the new one? b.What are the new steady state probabilities? c.If each point of market share increases profit by $15000, what is the most you would pay for the advertising?