L25.10 Birth-Death Processes - Part I
We are going to spend the rest of this lecture by looking into an interesting subclass of Markov chains for which steady-state convergence exists-- the class of birth and death processes. So what is a birth and death process? It's a Markov chain whose diagram looks like this-- the states of the Markov chain start from zero and go to some finite integer number m. And if you are at a typical state in the middle, say i. Then next you will either go to the right-- or up by one unit-- or you will go to the left-- or down by one unit-- or you will stay where you are. And this will happen with the following transition probabilities-- here p i-- i function of the state-- here q i. And this one the remaining 1 minus p i minus q i. So it's like keeping track of some animal population. Animals get born. They die. The assumption here is that at any point in time, either one animal gets born, or one dies, or nothing happens. There are no multiple deaths, no births happening at...