Assignment Instruction: Review your lecture notes up to where we stopped (Markov Processes) and carefully answer/solve the following problems. Illustrate your solutions and or answers with diagrams wherever applicable.
Problem 1: Management of the New Fangled Softdrink Company believes that the probability of a customer purchasing Red Pop or the company's major completion, Super Cola, is based on the customer's most recent purchase. Suppose the following transition probabilities are appropriate:
|
From
|
To
|
|
Red Pop
|
Super Cola
|
|
Red Pop
|
0.9
|
0.1
|
|
Super Cola
|
0.1
|
0.9
|
a. Show the two-period tree diagram for a customer who last purchased Red Pop. What is the probability that this customer purchases Red Pop on the second purchase?
b. What is the long-run market share for each of these two products?
c. A Red Pop advertising campaign is being planned to increase the probability of attracting Super Cola customers. Management believes that the new campaign will increase the probability of a customer switching from Super Cola to Red Pop is 0.15. What is the projected effect of the advertising campaign on the market shares?
Problem 2: The computer software system during registration at William V.S. Tubman University has been experiencing computer downtime. Let us assume that the trials of an associated Markov Process are defined as 1-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities:
|
From
|
To
|
|
Running
|
Down
|
|
Running
|
0.9
|
0.1
|
|
Down
|
0.3
|
0.7
|
a. If the system is initially running, what is the probability of being down in the next hour operation?
b. What are the steady-state probabilities of the system in the running state and in the down state?
Problem 3: One cause of the downtime in Problem 2 was traced to a specific piece of computer hardware. Management believes that switching to a different hardware component will result in the following transition probabilities:
|
From
|
To
|
|
Running
|
Down
|
|
Running
|
0.95
|
0.05
|
|
Down
|
0.6
|
0.4
|
a. What are the steady-state probabilities of the system in the running and down states?
b. If the cost of the system being down for any period is estimated to be $500 (including lost revenues for time down and maintenance), what is the breakeven cost for the new hardware component on a time-period basis?
Problem 4: Data collected from selected major metropolitan areas in Montserrado County in Liberia show that 2% of individual living the city limits move to the suburbs during a 1-year period while 1% of individuals living in the suburbs move to the city during a 1-year period. Answer the following questions assuming this process is modeled by a Markov process with two states: city and suburbs.
a. Show the matrix of transition probabilities.
b. Compute the steady-state probabilities.
c. In a particular metropolitan area, 40% of the population lives in the city and 60% of the population live in the suburbs. What population changes do your steady-state probabilities project for this metropolitan area?
Problem 5: The purchase patterns of two brands of toothpaste can be expressed as a Markov Process with the following transition probabilities:
|
From
|
To
|
|
Special B
|
MDA
|
|
Special B
|
0.9
|
0.1
|
|
MDA
|
0.05
|
0.95
|
a. Which brand appears to have the most loyal customers? Explain.
b. What are the projected market shares for the two brands?
Problem 6: Suppose that in Problem 8 a new toothpaste brand enters the market such that the following transition probabilities exist:
|
From
|
To
|
|
Special B
|
MDA
|
T-White
|
|
Special B
|
0.8
|
0.1
|
0.1
|
|
MDA
|
0.05
|
0.75
|
0.2
|
|
T-White
|
0.4
|
0.3
|
0.3
|
a. What are the new long-run market shares? Which brand will suffer most from the introduction of the new brand of toothpaste?
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