For certain special types of Markov chains, including the birth-death chain, a more powerful relation holds, called the detailed balance equations , eq. 3A.5. This says that not only does the probability of leaving a state equal the probability of entering it (global eqns), but also that the probability of going from any state i to any state j is the same as the probability of going backwards from state j to state i. This latter fact may not be true for general markov chains only satisfying the global equations.
For example, consider a chain with states arranged in a circle, with clockwise transition prob all equal to 1/4 and counterclockwise transition prob all equal to 3/4 (sort of like a birth-death chain with the ends connected). You can verify that the steady-state prob is equal to 1/K for each state (for K states), so that the prob of going clockwise between any two states is 1/(4K), while the prob of going backwards (counterclockwise) between the same two states is 3/(4K). Therefore, this chain does not satisfy the detailed balance equations, although the global balance equations are satisfied (verify this yourself).
This example also illustrates one important property which is associated with chains satisfying the detailed balance equations (but not with more general chains), namely, the property called "reversibility" (this will be a topic in the next few lectures). A reversible chain is one in which the prob of going from any state i to any state j is the same as the prob of going backwards from j to i. You can see that this is satisfied trivially by any chain satisfying the detailed balance equations...so in particular, any birth-death chain is reversible. In contrast, the circular chain given above is not reversible. You will see that reversibility allows us to say many useful things about certain chains, which are not true for more general chains just satisfying the global balance eqns. For example, it tells us that the M/M/1 queue (which can be described by a b-d chain) viewed in "reverse time" is also an M/M/1, and hence the departure process from an M/M/1 is also Poisson.
I hope this helps. If you have more questions, send me e-mail or we can set up an appt sometime next week.
--Won
>|> hi won, >|> >|> i am reading the appendix for chapter 3 and i am confused: >|> >|> i think i am missing the differences between: >|> global balance eq >|> partial balance eq >|> detailed balance eq >|> >|> as far as i understand detailed balance eq holds for birth-death cases only >|> but the rest i couldnoit figure out..could you help me?