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Matrix Analytic Method
In probability theory, the matrix analytic method is a technique to compute the stationary probability distribution of a Markov chain which has a repeating structure (after some point) and a state space which grows unboundedly in no more than one dimension. Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. Method description An M/G/1-type stochastic matrix is one of the form ::P = \begin B_0 & B_1 & B_2 & B_3 & \cdots \\ A_0 & A_1 & A_2 & A_3 & \cdots \\ & A_0 & A_1 & A_2 & \cdots \\ & & A_0 & A_1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end where ''B''''i'' and ''A''''i'' are ''k'' × ''k'' matrices. (Note that unmarked matrix entries represent zeroes.) Such a matrix describes the embedded Markov chain ... [...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] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M/G/1 Type Markov Chain
In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server. The model name is written in Kendall's notation, and is an extension of the M/M/1 queue, where service times must be exponentially distributed. The classic application of the M/G/1 queue is to model performance of a fixed head hard disk. Model definition A queue represented by a M/G/1 queue is a stochastic process whose state space is the set , where the value corresponds to the number of customers in the queue, including any being served. Transitions from state ''i'' to ''i'' + 1 represent the arrival of a new customer: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state ''i'' to ''i'' − 1 represent a customer who has been served, finishing being served and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Geometric Method
In probability theory, the matrix geometric method is a method for the analysis of quasi-birth–death processes, continuous-time Markov chain whose transition rate matrices with a repetitive block structure. The method was developed "largely by Marcel F. Neuts and his students starting around 1975." Method description The method requires a transition rate matrix with tridiagonal block structure as follows ::Q=\begin B_ & B_ \\ B_ & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& A_0 & A_1 & A_2 \\ &&&& \ddots & \ddots & \ddots \end where each of ''B''00, ''B''01, ''B''10, ''A''0, ''A''1 and ''A''2 are matrices. To compute the stationary distribution ''π'' writing ''π'' ''Q'' = 0 the balance equations are considered for sub-vectors ''π''''i'' ::\begin \pi_0 B_ + \pi_1 B_ &= 0\\ \pi_0 B_ + \pi_1 A_1 + \pi_2 A_0 &= 0\\ \pi_1 A_2 + \pi_2 A_1 + \pi_3 A_0 &= 0 \\ & \vdots \\ \pi_ A_2 + \pi_i A_1 + \pi_ A_0 &= 0\\ & \vdots \\ \end Observe that the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedded 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 happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Recurrent
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 happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the Algorithm#Termination, termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A\mathbf=\mathbf by Gaussian elimination). Iterative methods are often the only cho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Reduction
Cyclic reduction is a numerical method for solving large linear systems by repeatedly splitting the problem. Each step eliminates even or odd rows and columns of a matrix and remains in a similar form. The elimination step is relatively expensive but splitting the problem allows parallel computation. Applicability The method only applies to matrices that can be represented as a (block) Toeplitz matrix, such problems often arise in implicit solutions for partial differential equations on a lattice. For example fast solvers for Poisson's equation express the problem as solving a tridiagonal matrix, discretising the solution on a regular grid. Accuracy Systems which have good numerical stability initially tend to get better with each step to a point where a good approximate solution can be given, but because the special matrix form must be preserved pivoting cannot be performed to improve numerical accuracy. Comparison to multigrid The method is not iterative, it seeks an exact sol ... [...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] |
Single Queueing Nodes
Single may refer to: Arts, entertainment, and media * Single (music), a song release Songs * "Single" (Natasha Bedingfield song), 2004 * "Single" (New Kids on the Block and Ne-Yo song), 2008 * "Single" (William Wei song), 2016 * "Single", by Meghan Trainor from the album '' Only 17'' Sports * Single (baseball), the most common type of base hit * Single (cricket), point in cricket * Single (football), Canadian football point * Single-speed bicycle Transportation * Single-cylinder engine, an internal combustion engine design with one cylinder, or a motorcycle using such engine * Single (locomotive), a steam locomotive with a single pair of driving wheels * As a verb: to convert a double-track railway to a single-track railway Other uses * Single (mathematics) (1-tuple), a list or sequence with only one element * Single person, a person who is not in a committed relationship * Single precision, a computer numbering format that occupies one storage location in computer memory a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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