What are the Limiting Distribution and Limiting Probabilities

Hi

I have been studying Markov Chains and Queuing Theory. I have completed quite a lot of Markov Chains and am moving onto Queuing Theory, and there is this procedure to convert an M/G/1 queue into an imbedded Markov Chain, and suddenly the all the pages refer to something called the Limiting Distribution of the Markov Chain.

What ????

I flipped back through the book, but could not see a single explaination of what the term "Limiting Distribution" of a Markov Chain is supposed to mean.

I checked through several books, they all use the term but never explain it. They do not even list it in the index.

I can't find it explained anywhere in the Internet either.

Please help me out, **at the earliest**, and explain what in the world "Limiting Distribution" of a Markov Chain is supposed to mean.

I know that if initial 0th distribuiton is **I, **and transition probability matrix is **P**, then the **n**th Distribution is **(I)P**exp**n.**

where **P**exp**n** means P to the power of n.

Please help me out, **at the earliest**, and explain what in the world "Limiting Distribution" of a Markov Chain is supposed to mean.

Note: Definition, Explaination and Example is usually the best way of getting an understanding.

Thanks in Advance

Pankaj