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Markov Additive Process
In applied probability, a Markov additive process (MAP) is a bivariate Markov process where the future states depends only on one of the variables. Definition Finite or countable state space for ''J''(''t'') The process \ is a Markov additive process with continuous time parameter ''t'' if # \ is a Markov process # the conditional distribution of (X(t+s)-X(t), J(t+s)) given (X(t), J(t)) depends only on J(t). The state space of the process is R × ''S'' where ''X''(''t'') takes real values and ''J''(''t'') takes values in some countable set ''S''. General state space for ''J''(''t'') For the case where ''J''(''t'') takes a more general state space the evolution of ''X''(''t'') is governed by ''J''(''t'') in the sense that for any ''f'' and ''g'' we require ::\mathbb E \mathcal F_t= \mathbb E_ (X_s)g(J_s)/math>. Example A fluid queue is a Markov additive process where ''J''(''t'') is a continuous-time Markov chain. Applications Çinlar uses the unique structure of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such research is motivated (to some degree) by applied problems, it is usually the mathematical aspects of the problems that are of most interest to researchers (as is typical of applied mathematics in general). Applied probabilists are particularly concerned with the application of stochastic processes, and probability more generally, to the natural, applied and social sciences, including biology, physics (including astronomy), chemistry, medicine, computer science and information technology, and economics. Another area of interest is in engineering: particularly in areas of uncertainty, risk management, probabilistic design, and Quality assurance. See also *Areas of application: **Ruin theory **Statistical physics **Stoichiometry and mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Process
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Process
An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process (a Lévy process is an additive process with identically distributed increments). An example of an additive process is a Brownian motion with a time-dependent drift. The additive process was introduced by Paul Lévy in 1937. There are applications of the additive process in quantitative finance (this family of processes can capture important features of the implied volatility) and in digital image processing. Definition An additive process is a generalization of a Lévy process obtained relaxing the hypothesis of identically distributed increments. Thanks to this feature an additive process can describe more complex phenomenons than a Lévy process. A stochastic process \_ on \mathbb R^d such that X_0=0 almost surely is an additive process if it satisfy the following hypothesis: #It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Queue
In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term dam theory was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread of wildfires, in ruin theory and to model high speed data networks. The model applies the leaky bucket algorithm to a stochastic source. The model was first introduced by Pat Moran in 1954 where a discrete-time model was considered. Fluid queues allow arrivals to be continuous rather than discrete, as in models like the M/M/1 and M/G/1 queues. Fluid queues have been used to model the performance of a network switch, a router, the IEEE 802.11 protocol, Asynchronous Transfer Mode (the intended technology for B-ISDN), peer-to-peer file sharing, optical burs ... [...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] |
Gamma Process
In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator, is a random process with independent gamma distributed increments. Often written as \Gamma(t;\gamma,\lambda), it is a pure-jump increasing Lévy process with intensity measure \nu(x)=\gamma x^ \exp(-\lambda x), for positive x. Thus jumps whose size lies in the interval ,x+dx) occur as a Poisson process with intensity \nu(x)\,dx. The parameter \gamma controls the rate of jump arrivals and the scaling parameter \lambda inversely controls the jump size. It is assumed that the process starts from a value 0 at ''t'' = 0. The gamma process is sometimes also parameterised in terms of the mean (\mu) and variance (v) of the increase per unit time, which is equivalent to \gamma = \mu^2/v and \lambda = \mu/v. Properties Since we use the Gamma function in these properties, we may write the process at time t as X_t\equiv\Gamma(t;\gamma, \lambda) to eliminate ambiguity. Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weibull Distribution
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice René Fréchet and first applied by to describe a particle size distribution. Definition Standard parameterization The probability density function of a Weibull random variable is : f(x;\lambda,k) = \begin \frac\left(\frac\right)^e^, & x\geq0 ,\\ 0, & x 0 is the '' shape parameter'' and λ > 0 is the '' scale parameter'' of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (''k'' = 1) and the Rayleigh distribution (''k'' = 2 and \lambda = \sqrt\sigma ). If the quantity ''X'' is a "time-to-failure", the Weibull distribution gives a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. Definition In the theory of dynamical systems, the state space of a discrete system defined by a function ''ƒ'' can be modeled as a directed graph where each possible state of the dynamical system is represented by a vertex with a directed edge from ''a'' to ''b'' if and only if ''ƒ''(''a'') = ''b''. This is known as a state diagram. For a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
