Markov Chain

The questions are complete please.

In analysing switching by Business Class customers between airlines the following data has been obtained by British Airways (BA): Next flight by BA Competition Last flight by BA 0. 85 0.15 Competition 0.10 0.90 For example if the last flight by a Business Class customer was by BA the probability that their next flight is by BA is 0.85. Business Class customers make 2 flights a year on average. Currently BA have 30% of the Business Class market. What would you forecast BA's share of the Business Class market to be after two years?An admissions tutor is analysing applications from potential students for a particular undergraduate course at Imperial College (IC). She regards each potential student as being in one of four possible states: State 1: has not applied to IC State 2: has applied to IC but an accept reject decision has not yet been made State 3: has applied to IC and has been rejected State 4: has applied to IC and has been accepted (been made an offer of a place) At the start of the year (month 1 in the admissions year) all potential students are in state 1. Her review of admissions statistics for recent years has identified the following transition matrix for the probability of moving between states each month: 1 From e. e 0.10 0.15 0.7 What percentage of potential students will have been accepted after 3 months have elapsed? Is it possible to work out a meaningful long-run system state or not (and why)? The admissions tutor has control over the elements in one row of the above transition matrix, namely row 2. The elements in this row reflect: transition from 2 to 2: the speed with which applications are processed each month transition from 2 to 3: the proportion of applicants who are rejected each month transition from 2 to 4: the proportion of applicants who are accepted each month. To be more specific, at the start of each month the admissions tutor has to decide the proportion of applicants who should be accepted that month. However she is constrained by a policy decision that, at the end of each month, the total number of rejections should never be more than one-third of the total number of offers, nor should it ever be less than 20% of the total number of offers. Further analysis reveals that applicants who wait longer than 2 months between applying to IC and receiving a decision (reject or accept) almost never choose to come to IC, even if they get an offer of a place. Formulate the problem that the admissions tutor faces each month as a linear program. Comment on any assumptions you have made in so doing.A petrol station owner is considering the effect on his business (Superpet) of a new petrol station (Global) which has opened just down the road. Currently (of the total market shared between Superpet and Global) Superpet has 80% of the market and Global has 20% Analysis over the last week has indicated the following probabilities for customers switching the station they stop at each week: To Superpet Global From Superpet 0.75 @.25 Global 0.55 0.45 What will be the expected market share for Superpet and Global after another two weeks have past? What would be the long-run prediction for the expected market share for Superpet and Global?A company is considering using Markov theory to analyse brand switching between three different brands of floppy disks. Survey data has been gathered and has been used to estimate the following transition matrix for the probability of moving between brands each month: To Brand 1 2 3 From Brand 1 | e.80 0. 10 0. 10 | 0.03 0.95 0.02 31 0.20 0. 05 0.75 The current (month 1) market shares are 45%, 25% and 30% for brands 1, 2 and 3 respectively. What will be the expected market shares after two months have elapsed (i.e. in month 3)? What is the long-run prediction for the expected market share for each of the three brands? Would you expect the actual market share to approach the long-run prediction for the market or not (and why)