Abstract
AbstractThis chapter provides the derivations of the results in the previous chapter. It also develops the theory of continuous-time Markov chains.Section 6.1 proves the results on the spreading of rumors. Section 6.2 presents the theory of continuous-time Markov chains that are used to model queueing networks, among many other applications. That section explains the relationships between continuous-time and related discrete-time Markov chains. Sections 6.3 and 6.4 prove the results about product-form networks by using a time-reversal argument.
Highlights
Since αn+1 is the probability that there is one survivor after n + 1 generations, it is the probability that at least one of the X1 children of the root has a survivor after n generations
That is not possible since it listened to some previous nodes that are all red
We saw earlier that a continuous-time Markov chain (CTMC) can be approximated by a discrete-time Markov chain that has a time step 1
Summary
Theorem 6.1 (Spreading of a Message) Let Z be the number of nodes that eventually receive the message. Proof For part (a), let Xn be the number of nodes that are n steps from the root. If Xn = k, we can write Xn+1 = Y1 + · · · + Yk where Yj is the number of children of node j at level n. 1. For instance, assume that each node has three children with probability 0.5 and has no child otherwise. If X1 = k, the probability that none of the k children of the root has a survivor after n generations is (1 − αn)k. P. Proof The probability that node n does not listen to anyone is an = (1 − p)n.
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