If one starts in SS, the expected number of steps before he step into ¯SS¯ is 1/Φ(S)1/Φ(S).Definition 4.3 The normalized conductance of Markov chain, denoted ΦΦ, is defined byΦ=minS⊆V,S≠{},VΦ(S)Φ=minS⊆V,S≠{},VΦ(S)Theorem 4.5 The εε-mixing time of ...
For a Markov chain with a single class which is aperiodic, the steady-state probabilities \pi_{j} satisfy \pi_{j}=\lim_{n\to \infty}\frac{v_{ij}\left(n\right)}{n} where v_{ij}\left(n\right) is the expected value of the number of visits to state j within the first n tr...
Lemma 7.1 (Expected number of steps of Algorithm 7.1). 假设算法 7.1 接收的包含 n 个变量的 2-SAT formula 是可满足的,且允许算法一直运行直到其找到一个 satisfying assignment,那么算法的期望运行步数不超过 n^{2}. 回忆算法 7.1 中,我们实际上设定了算法至多执行 2mn^{2} 步. 应用 Markov's inequali...
The set in (9.9) is the collection of numbers of steps n such that the Markov chain can go from the state i back to itself in n steps. For instance, consider the leftmost Markov chain of Figure 9.1 and assume that 0 < a < 1 and 0 < b < 1. One finds that {n≥ 1|Pn(0, ...
Thus the entropy of the random trajectory T/sub ii/ is the product of the expected number of steps 1/ mu /sub i/ to return to state i and the entropy rate H(X) per step for the stationary Markov chain. A general closed form solution for the entropies H(T/sub ij/) is given by...
First passage times (sometimes also calledhitting times) are stopping times for first visits in a given state (or more generally a set of states). Themean first passage timeτ(s,s′)for two different statess,s′is defined as the expected number of steps it takes to reachs′for the firs...
What is the expected number of steps until it is in state 4? 5. Suppose the chain starts in state 1. What is the probability that the chain will enter state 5 before it enters state 3? Solution: 1. The chain is irreducible, aperiodic, and positive recurrent. 37 2. Solve the system...
If we let Xn denote the number of customers in the system as seen by the nth arrival, it is easy to see that the process{Xn,n≥1} is a Markov chain. To compute the transition probabilities Pij for this Markov chain, let us first note that, as long as there are customers to be ...
A recurrent state is known as positive recurrent if it is expected to return within a finite number of steps and null recurrent otherwise. Ergodicity: a state 'i' is said to be ergodic if it is aperiodic and positive recurrent. If all states in an irreducible Markov chain are ergodic, ...
Since we have an absorbing Markov chain, we calculate the expected time until absorption. The first entry of the vector will output the expected number of steps until closing from the SDR funnel while the second entry will output the expected number if we started from the AE funnel. # Extrac...