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Burke's Theorem
In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem (sometimes the Burke's output theorem) is a theorem (stated and demonstrated by Paul J. Burke while working at Bell Telephone Laboratories) asserting that, for the M/M/1 queue, M/M/c queue or M/M/∞ queue in the steady state with arrivals is a Poisson process with rate parameter λ: # The departure process is a Poisson process with rate parameter λ. # At time ''t'' the number of customers in the queue is independent of the departure process prior to time ''t''. Proof Burke first published this theorem along with a proof in 1956. The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955). A second proof of the theorem follows from a more general result published by Reich. The proof offered by Burke shows that the time intervals between successive departures are independently and exponentially distributed with parameter equal to the arrival rate paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queueing Theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the system of Copenhagen Telephone Exchange company, a Danish company. The ideas have since seen applications including telecommunication, traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management. Spelling The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is ''Queueing Systems''. Single queueing nodes A queue, or queueing nod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demonstration Burke Theorem
Demonstration may refer to: * Demonstration (acting), part of the Brechtian approach to acting * Demonstration (military), an attack or show of force on a front where a decision is not sought * Demonstration (political), a political rally or protest * Demonstration (teaching), a method of teaching by example rather than simple explanation * Demonstration Hall, a building on the Michigan State University campus * Mathematical proof * Product demonstration, a sales or marketing presentation such as a: ** Technology demonstration, an incomplete version of product to showcase idea, performance, method or features of the product * Scientific demonstration, a scientific experiment to illustrate principles * Wolfram Demonstrations Project, a repository of computer based educational demonstrations Music * ''Demonstration'' (Landon Pigg album), 2002 * ''Demonstration'' (Tinie Tempah album), 2013 * ''Demonstrations'' EP, the first EP by We Came As Romans * Demonstrate (song), a song by Jo ... [...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] |
Brownian Queue
In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation (sometimes heavy traffic limit theorem or diffusion approximation) is the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman who showed that when the utilisation parameter of an M/M/1 queue is near 1 a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion. Heavy traffic condition Heavy traffic approximations are typically stated for the process ''X''(''t'') describing the number of customers in the system at time ''t''. They are arrived at by considering the model under the limiting values of some model parameters and therefore for the result to be finite the model must be rescaled by a factor ''n'', denoted ::\hat X_n(t) = \frac and the limit of this process is considered as ''n'' → ∞. There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markovian Arrival Processes
In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP) is a mathematical model for the time between job arrivals to a system. The simplest such process is a Poisson process where the time between each arrival is exponentially distributed. The processes were first suggested by Neuts in 1979. Definition A Markov arrival process is defined by two matrices, ''D''0 and ''D''1 where elements of ''D''0 represent hidden transitions and elements of ''D''1 observable transitions. The block matrix ''Q'' below is a transition rate matrix for a continuous-time Markov chain. : Q=\left begin D_&D_&0&0&\dots\\ 0&D_&D_&0&\dots\\ 0&0&D_&D_&\dots\\ \vdots & \vdots & \ddots & \ddots & \ddots \end\right; . The simplest example is a Poisson process where ''D''0 = −''λ'' and ''D''1 = ''λ'' where there is only one possible transition, it is observable, and occurs at rate ''λ''. For ''Q'' to be a valid transitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Detailed Balance
The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reverse process. History The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle.Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press. The arguments in favor of this property are founded upon microscopic reversibility. Tolman, R. C. (1938). ''The Principles of Statistical Mechanics''. Oxford University Press, London, UK. Five years before Boltzmann, James Clerk Maxwell used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason. He compared the idea of detailed balance with other types of balancing (like cyclic balance) and found that "Now it is impossible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov's Criterion
In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem giving a necessary and sufficient condition for a Markov chain or continuous-time Markov chain to be stochastically identical to its time-reversed version. Discrete-time Markov chains The theorem states that an irreducible, positive recurrent, aperiodic Markov chain with transition matrix ''P'' is reversible if and only if its stationary Markov chain satisfies : p_ p_ \cdots p_ p_ = p_ p_ \cdots p_ p_ for all finite sequences of states : j_1, j_2, \ldots, j_n \in S . Here ''pij'' are components of the transition matrix ''P'', and ''S'' is the state space of the chain. Example Consider this figure depicting a section of a Markov chain with states ''i'', ''j'', ''k'' and ''l'' and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop ''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reversed Process
A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed. A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time. A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process. Mathematics In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one, so that for every state there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state and the forward-time evolution of another corresponding state, given by the operator equation: :U_ = \pi \, U_\, \pi Any time-independent structures (e.g. critical points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...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 probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M/M/∞ Queue
In queueing theory, a discipline within the mathematical theory of probability, the M/M/∞ queue is a multi-server queueing model where every arrival experiences immediate service and does not wait. In Kendall's notation it describes a system where arrivals are governed by a Poisson process, there are infinitely many servers, so jobs do not need to wait for a server. Each job has an exponentially distributed service time. It is a limit of the M/M/c queue model where the number of servers ''c'' becomes very large. The model can be used to model bound lazy deletion performance. Model definition An M/M/∞ queue is a stochastic process whose state space is the set where the value corresponds to the number of customers currently being served. Since, the number of servers in parallel is infinite, there is no queue and the number of customers in the systems coincides with the number of customers being served at any moment. * Arrivals occur at rate ''λ'' according to a Poisson proces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M/M/c Queue
In queueing theory, a discipline within , the queue (or Erlang–T model) is a multi-server queueing model. In Kendall's notation it describes a system where arrivals form a single queue and are governed by a , there are servers, and job service times are exponentially distributed. It is a generalization of which considers only a single server. The model with infinitely many servers is the M/M/∞ queue. Model definition An M/M/c queue is a stochastic process whose state space is the set where the value corresponds to the number of customers in the system, including any currently in service. * Arrivals occur at rate according to a Poisson process and move the process from state to +1. * Service times have an exponential distribution with parameter . If there are fewer than jobs, some of the servers will be idle. If there are more than jobs, the jobs queue in a buffer. * The buffer is of infinite size, so there is no limit on the number of customers it can contain. The mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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