$"17-33.$ Customers tend to exhibit loyalty to product brands but may be persuaded through clever marketing and advertising to switch brands. Consider the case of three brands: $A,B,$ and C$.$ Customer "unyielding" loyalty to a given brand is estimated at $75\%,$ giving the competitors only a $25\%$ margin to realize a switch. Competitors launch their advertising campaigns once a year. For brand A customers, the probabilities of switching to brands $B$ and $C$ are $1$ and $\mathrm{.}15,$ respectively. Customers of brand $B$ are likely to switch to A and $C$ with probabilities $\mathrm{.}2$ and $\mathrm{.}05,$ respectively. Brand $C$ customers can switch to brands A and $B$ with equal probabilities.

$($a$)$ Express the situation as a Markov chain.

$($b$)$ In the long run, how much market share will each brand command?

$($c$)$ How long on the average will it take for a brand A customer to switch to brand $B?$ To brand $C?$