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Global Balance Equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain 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 flux from state i to state j. So the left-hand side represents the total flow from out of state ''i'' into states other than ''i'', while the right-hand side represents the total flow out of all states j \neq i into state i. In general it is computationally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The " =" symbol, which appears in every equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process 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). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilibrium Distribution
In probability theory and statistics, a Markov chain or Markov process is a stochastic process 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). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance ... [...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 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Rate Matrix
In probability theory, a transition-rate matrix (also known as a Q-matrix, intensity matrix, or infinitesimal generator matrix) is an array of numbers describing the instantaneous rate at which a 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 ... transitions between states. In a transition-rate matrix Q (sometimes written A), element q_ (for i \neq j) denotes the rate departing from i and arriving in state j. The rates q_ \geq 0, and the diagonal elements q_ are defined such that :q_ = -\sum_ q_, and therefore the rows of the matrix sum to zero. Up to a global sign, a large class of examples of such matrices is provided by the Laplacian of a directed, weighted graph. The vertices of the graph correspond to the Markov chain's states. Properties The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Time Markov Chain
In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past. For instance, a machine may have two states, ''A'' and ''E''. When it is in state ''A'', there is a 40% chance of it moving to state ''E'' and a 60% chance of it remaining in state ''A''. When it is in state ''E'', there is a 70% chance of it moving to ''A'' and a 30% chance of it staying in ''E''. The sequence of states of the machine is a Markov chain. If we denote the chain by X_0, X_1, X_2, ... then X_0 is the state which the machine starts in and X_ is the random variable describing its state after 10 transitions. The process continues forever, indexed by the natural numbers. An example of a stochastic process which is not a Markov chain is the model of a machine which has states ''A'' and ''E'' and moves to ''A'' from either state with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Whittle (mathematician)
Peter Whittle (27 February 1927 – 10 August 2021) was a mathematician and statistician from New Zealand, working in the fields of stochastic nets, optimal control, time series analysis, stochastic optimisation and stochastic dynamics. From 1967 to 1994, he was the Churchill Professor of Mathematics for Operational Research at the University of Cambridge. Career Whittle was born in Wellington. He graduated from the University of New Zealand in 1947 with a BSc in mathematics and physics and in 1948 with an MSc in mathematics. He then moved to Uppsala, Sweden in 1950 to study for his PhD with Herman Wold (at Uppsala University). His thesis, ''Hypothesis Testing in Time Series'', generalised Wold's autoregressive representation theorem for univariate stationary processes to multivariate processes. Whittle's thesis was published in 1951. A synopsis of Whittle's thesis also appeared as an appendix to the second edition of Wold's book on time-series analysis. Whittle rema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product-form Solution
In probability theory, a product-form solution is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a product (mathematics), product of the metric across the different components. Using capital Pi notation a product-form solution has algebraic form :\text(x_1,x_2,x_3,\ldots,x_n) = B \prod_^n \text(x_i) where ''B'' is some constant. Solutions of this form are of interest as they are computationally inexpensive to evaluate for large values of ''n''. Such solutions in queueing networks are important for finding performance metrics in models of multiprogrammed and time-shared computer systems. Equilibrium distributions The first product-form solutions were found for equilibrium distributions of Markov chains. Trivially, models composed of two or more Independence (probability theory)#Independent random variables, independent sub-components exhibit a product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erol Gelenbe
Sami Erol Gelenbe (born 22 August 1945, Istanbul) is a Turkish and French computer scientist, electronic engineer and applied mathematician, renowned for pioneering work in computer system and network performance. His academic career spans several prestigious institutions and roles, including current positions as Professor at the Institute of Theoretical and Applied Informatics of the Polish Academy of Sciences since 2017, and visiting professorships at King's College London, the I3S Laboratory (French National Centre for Scientific Research, CNRS, Côte d'Azur University, University of Côte d'Azur) and the Abraham de Moivre Laboratory (CNRS, Imperial College London). A Fellow of several national academies, Gelenbe has chaired the Informatics Section of Academia Europaea since 2023. His extensive professorial tenures include roles at the University of Liège, Paris-Saclay University, University Paris-Saclay, Paris Descartes University, University Paris Descartes, New Jersey Inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-network
In queueing theory, a discipline within the mathematical theory of probability, a G-network (generalized queueing network, often called a Gelenbe network) is an open network of G-queues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic re-routing or traffic destruction, as well as a model for neural networks. A G-queue is a network of queues with several types of novel and useful customers: *''positive'' customers, which arrive from other queues or arrive externally as Poisson arrivals, and obey standard service and routing disciplines as in conventional network models, *''negative'' customers, which arrive from another queue, or which arrive externally as Poisson arrivals, and remove (or 'kill') customers in a non-empty queue, representing the need to remove traffic when the network is congested, including the removal of "batches" of customers *"triggers", which arrive from other queues or from outside the network, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
