Modems networked to a mainframe computer system have a limited capacity. is the probability that a user dials into the network when a modem connection is available, and 1/4 is the probability that a call is received when all lines are busy. The system can be considered as a binary Markov chain. Draw the state transition diagram of the Markov chain. i) ii) iii) Find the state transition matrix and the probability state vector p(k). Describe the steady-state behaviour of the system, i.e., find the vector (For a binary Markov chain, 1 [β a+BB p² = α] (1-a-B)k a + a+ß 86²)

We will use Markov chain to model weather XYZ city. According to the city’s meteorologist, every day in XYZ is either sunny, cloudy or rainy.  The meteorologist have informed us that the city never has two consecutive sunny days.  If it is sunny one day, then it is equally likely to be either cloudy or rainy the next day.  If it is rainy or cloudy one day, then there is one chance in two that it will be the same the next possibilities.  In the long run, what proportion of days are cloudy, sunny and rainy? Show the transition matrix.

A factory worker will quit with probability 1/2 during her first month, with probability 1/4 during her second month and with probability 1/8 after that. Whenever someone quits, their replacement will start at the beginning of the next month. Model the status of each position as a Markov chain with 3 states. Identify the states and transition matrix. Write down the system of equations determining the long-run proportions. Suppose there are 900 workers in the factory. Find the average number of the workers who have been there for more than 2 months.

Next Generation Red Pink White A given plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each type of these genotypes is crossed with a pink-flowering plant (genotype RW), then the transition matrix is as shown to the right. Red 0.5 0.5 This Pink 0.25 0.5 0.25 Generation White 0 0.5 0.5 Assuming that the plants of each generation are crossed only with pink plants to produce the next generation, show that regardless of the makeup of the first generation, the genotype composition will eventually stabilize at 25% red, 50% pink, and 25% white. (Find the stationary matrix)

Alan and Betty play a series of games with Alan winning each game independently with probability p = 0.6. The overall winner is the first player to win two games in a row.  Define a Markov chain to model the above problem.

At Suburban Community College, 30% of all business majors switched to another major the next semester, while the remaining 70% continued as business majors. Of all non-business majors, 10% switched to a business major the following semester, while the rest did not. Set up these data as a Markov transition matrix. HINT [See Example 1.] (Let 1 business majors, and 2 = non-business majors.) = Calculate the probability that a business major will no longer be a business major in two semesters' time.