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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] |
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] |
Jackson's Theorem (queueing Theory)
In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes Jacksonian network) is a class of queueing network where the equilibrium distribution is particularly simple to compute as the network has a product-form solution. It was the first significant development in the theory of networks of queues, and generalising and applying the ideas of the theorem to search for similar product-form solutions in other networks has been the subject of much research, including ideas used in the development of the Internet. The networks were first identified by James R. Jackson A version from January 1963 is available at http://www.dtic.mil/dtic/tr/fulltext/u2/296776.pdf and his paper was re-printed in the journal ''Management Science’s'' ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’ Jackson was inspired by the work of Burke and Reich, though Jean Walrand notes "product-form results … rea much less immediate res ... [...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] |
Normalizing 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] |
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] |
Mean Value Analysis
In queueing theory, a discipline within the mathematical theory of probability, mean value analysis (MVA) is a recursive technique for computing expected queue lengths, waiting time at queueing nodes and throughput in equilibrium for a closed separable system of queues. The first approximate techniques were published independently by Schweitzer and Bard, followed later by an exact version by Lavenberg and Reiser published in 1980. It is based on the arrival theorem, which states that when one customer in an ''M''-customer closed system arrives at a service facility he/she observes the rest of the system to be in the equilibrium state for a system with ''M'' − 1 customers. Problem setup Consider a closed queueing network of ''K'' M/M/1 queues, with ''M'' customers circulating in the system. Suppose that the customers are indistinguishable from each other, so that the network has a single class of customers. To compute the mean queue length and waiting time at each o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FCFS
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] |
BCMP Network
In queueing theory, a discipline within the mathematical theory of probability, a BCMP network is a class of queueing network for which a product-form equilibrium distribution exists. It is named after the authors of the paper where the network was first described: Baskett, Chandy, Muntz and Palacios. The theorem is a significant extension to a Jackson network allowing virtually arbitrary customer routing and service time distributions, subject to particular service disciplines. The paper is well known, and the theorem was described in 1990 as "one of the seminal achievements in queueing theory in the last 20 years" by J. Michael Harrison and Ruth J. Williams. Definition of a BCMP network A network of ''m'' interconnected queues is known as a BCMP network if each of the queues is of one of the following four types: # FCFS discipline where all customers have the same negative exponential service time distribution. The service rate can be state dependent, so write \scriptstyle f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theorems
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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