Q$\mathrm{.}4.$ The personal computer industry is very fast moving technology that provides motivation for customers to upgrade with new computers every few years. Brand loyalty is very important and companies try to do things to keep their customers happy. However, some current customers may switch to a different company. Three particular brands, Doorway, Bell, and Compaq, hold the major shares of the market. People who own Doorway computers will buy another Doorway in their next purchase $80\%$ of the time, while the rest will switch to the other companies in equal proportions. Owners of Bell computers will buy Bell again $90\%$ of the time, while $5\%$ will buy Doorway and $5\%$ will buy Compaq. About $70\%$ of the Compaq owners will make Compaq their next purchase while $20\%$ will buy Doorway and the rest will buy Bell. $(5$ marks$)$

At the present time, each of the three companies has an equal share of the market.

a$.$ What is the market share for the next period given the initial market shares?

b$.$ If each brand currently has $200,000$ customers who plan to buy a new computer in the next year, how many computers of each type will be purchased?

c$.$ What is the long$-$run market share of each of the three companies?