What does i and j represent in Markov Chains?

[elvis0206]

in Markov Chains, the conditional probability is
P{Xt+1 = j︱Xt= i} =Pij

I don't know what are i and j represent here. Do they represent the step go to j from i or go to i from j???
can anyone give the explanation to me...thanks a lots!

 

[LouisSheffield]

Yep - It's a homogeneous Markov chain since any notation of time-dependence has been dropped.
see
Probability, Markov chains, queues ... - Google Books

Markov chains are only concerned with present state and possible next states -
like a random walk, where at any time you're at one single place.
Think of a drunk staggering - he's somewhere now, but his next step will be governed by probability.

Your expression essentially reads:
For all time (i.e. homogeneous), The probability of being in state j (conditionally) given we had just been in state i can be termed Pij,
and will be constant (no time dependence) for any occurrence of the transition from state i to j.

You know it's i to j (not j to i) because it was i (conditionally) at t, whereas j is at t+1

 

[prank]

can you go on further, with its physical implementation part