Question

Most railroad cars are owned by individual railroad companies. When a car leaves its home​ railroad's trackage, it becomes part of a national pool of cars and can be used by other railroads. The rules governing the use of these pooled cars are designed to eventually return the car to the home trackage. A particular railroad found that each month 17​% of its boxcars on the home trackage left to join the national pool and 31​% of its boxcars in the national pool were returned to the home trackage. If these percentages remain valid for a long period of​ time, what percentage of its boxcars can this railroad expect to have on its home trackage in the long​ run?

Answer 1

The transition matrix is

For limiting probability, we must have

So that

So that

And of course

Solving this we get

So in the long term 64.58% are on the home-track and 35.42% are on national pool

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