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Buzen's Algorithm
In queueing theory, a discipline within the mathematical theory of probability, Buzen's algorithm (or convolution algorithm) is an algorithm for calculating the normalization constant G(''N'') in the Gordon–Newell theorem. This method was first proposed by Jeffrey P. Buzen in his 1971 PhD dissertation and subsequently published in a refereed journal in 1973. Computing G(''N'') is required to compute the stationary probability distribution of a closed queueing network. Performing a naïve computation of the normalizing constant requires enumeration of all states. For a closed network with ''N'' circulating customers and ''M'' service facilities, G(''N'') is the sum of \tbinom individual terms, with each term consisting of ''M'' factors raised to powers whose sum is ''N''. Buzen's algorithm computes G(''N'') using only ''NM'' multiplications and ''NM'' additions. This dramatic improvement opened the door to applying the Gordon-Newell theorem to models of real world computer sy ... [...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 node ... [...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] |
Normalization Constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. Definition In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function. Examples If we start from the simple Gaussian function p(x)=e^, \quad x\in(-\infty,\infty) we have the corresponding Gaussian integral \int_^\infty p(x) \, dx = \int_^\infty e^ \, dx = \sqrt, Now if we use the latter's reciprocal value as a normalizing constant for the former, defining a function \varphi(x) as \varphi(x) = \frac p(x) = \frac e^ so that its integral is unit \int_^\infty \varphi(x) \, dx = \int_^\infty \frac e^ \, dx = 1 then the function \varphi(x) is a probability d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon–Newell Theorem
In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers where customers cannot leave the network. Jackson's theorem cannot be applied to closed networks because the queue length at a node in the closed network is limited by the population of the network. The Gordon–Newell theorem calculates the open network solution and then eliminates the infeasible states by renormalizing the probabilities. Calculation of the normalizing constant makes the treatment more awkward as the whole state space must be enumerated. Buzen's algorithm or mean value analysis can be used to calculate the normalizing constant more efficiently. Definition of a Gordon–Newell network A network of ''m'' interconnected queues is known as a Gordon–Newell network or closed Jackson network if it meets the following conditions: # the network i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeffrey P
Jeffrey may refer to: * Jeffrey (name), including a list of people with the name * ''Jeffrey'' (1995 film), a 1995 film by Paul Rudnick, based on Rudnick's play of the same name * ''Jeffrey'' (2016 film), a 2016 Dominican Republic documentary film *Jeffrey's, Newfoundland and Labrador, Canada *Jeffrey City, Wyoming, United States *Jeffrey Street, Sydney, Australia * Jeffrey's sketch, a sketch on American TV show ''Saturday Night Live'' *'' Nurse Jeffrey'', a spin-off miniseries from the American medical drama series ''House, MD'' *Jeffreys Bay, Western Cape, South Africa People with the surname * Alexander Jeffrey (1806–1874), Scottish solicitor and historian * Charles Jeffrey (footballer) (died 1915), Scottish footballer * E. C. Jeffrey (1866–1952), Canadian-American botanist *Grant Jeffrey (1948–2012), Canadian writer *Hester C. Jeffrey (1842–1934), American activist, suffragist and community organizer *Richard Jeffrey (1926–2002), American philosopher, logician, and pro ... [...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] |
Operations Research (journal)
''Operations Research'' is a bimonthly peer-reviewed academic journal covering operations research that is published by the Institute for Operations Research and the Management Sciences. It was established in 1952 as the ''Journal of the Operations Research Society of America'' and obtained its current name in 1955. The editor-in-chief iJohn Birge(University of Chicago). Abstracting and indexing The journal is abstracted and indexed by ''Mathematical Reviews'', MathSciNet, Science Citation Index Expanded, Scopus, Social Sciences Citation Index, and ''Zentralblatt MATH''. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 2.604. References External links * Mathematics journals Publications est ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentially Distributed
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. Definitions Probability density function The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin \lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoretical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...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 node ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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