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Kelly's Lemma
In probability theory, Kelly's lemma states that for a stationary continuous-time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the forward-time process. The theorem is named after Frank Kelly. Statement For a continuous time Markov chain with state space ''S'' and transition-rate matrix ''Q'' (with elements ''q''''ij'') if we can find a set of numbers ''q''ij'' and ''π''''i'' summing to 1 where ::\begin \sum_ \pi_i q'_ &= \sum_ q_ \quad \forall i\in S\\ \pi_i q_ &= \pi_jq_' \quad \forall i,j \in S \end then ''q''ij'' are the rates for the reversed process and ''π''''i'' are the stationary distribution for both processes. Proof Given the assumptions made on the ''q''''ij'' and ''π''''i'' we can see :: \sum_ \pi_i q_ = \sum_ \pi_j q'_ = \pi_j \sum_ q_ = -\pi_j q_ so the global balance equation In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous-time Markov Chain
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to the least value of a set of exponential random variables, one for each possible state it can move to, with the parameters determined by the current state. An example of a CTMC with three states \ is as follows: the process makes a transition after the amount of time specified by the holding time—an exponential random variable E_i, where ''i'' is its current state. Each random variable is independent and such that E_0\sim \text(6), E_1\sim \text(12) and E_2\sim \text(18). When a transition is to be made, the process moves according to the jump chain, a discrete-time Markov chain with stochastic matrix: :\begin 0 & \frac & \frac \\ \frac & 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Kelly (mathematician)
__NOTOC__ Francis Patrick Kelly, CBE, FRS (born 28 December 1950) is Professor of the Mathematics of Systems at the Statistical Laboratory, University of Cambridge. He served as Master of Christ's College, Cambridge from 2006 to 2016. Kelly's research interests are in random processes, networks and optimisation, especially in very large-scale systems such as telecommunication or transportation networks. In the 1980s, he worked with colleagues in Cambridge and at British Telecom's Research Labs on Dynamic Alternative Routing in telephone networks, which was implemented in BT's main digital telephone network. He has also worked on the economic theory of pricing to congestion control and fair resource allocation in the internet. From 2003 to 2006 he served as Chief Scientific Advisor to the United Kingdom Department for Transport. Kelly was elected a Fellow of the Royal Society in 1989. In December 2006 he was elected 37th Master of Christ's College, Cambridge. He was appoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Balance Equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ... in and out of states or set of states. Global balance The global balance equations (also known as full balance equations) are a set of equations that characterize the equilibrium distribution (or any stationary distribution) of a Markov chain, when such a distribution exists. For a continuous time Markov chain with state space \mathcal, transition rate from state i to j given by q_ and equilibrium distribution given by \pi, the global balance equations are given by ::\pi_i = \sum_ \pi_j q_, or equivalently :: \pi_i \sum_ q_ = \sum_ \pi_j q_. for all i \in S. Here \pi_i q_ represents the probability f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Processes
Markov (Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics *Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at the University of Stirling *John Markoff (sociologist) (born 1942), American professor of sociology and history at the University of Pittsburgh *Konstantin Markov (1905–1980), Soviet geomorphologist and quaternary geologist Mathematics, science, and technology *Alexander V. Markov (1965-), Russian biologist *Andrey Markov (1856–1922), Russian mathematician *Vladimir Andreevich Markov (1871–1897), Russian mathematician, brother of Andrey Markov (Sr.) *Andrey Markov Jr. (1903–1979), Russian mathematician and son of Andrey Markov *John Markoff (born 1949), American journalist of computer industry and technology *Moisey Markov (1908–1994), Russian physicist Performing arts *Albert Markov, Russian American violinist, composer * Alexan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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