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Palm Probabilities
In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time-average probabilities. A Palm probability or Palm expectation, often denoted P^0(\cdot) or E^0cdot/math>, is a probability or expectation conditioned on a specified event occurring at time 0. Little's formula A simple example of a formula from Palm calculus is Little's law L=\lambda W, which states that the time-average number of users (''L'') in a system is equal to the product of the rate (\lambda) at which users arrive and the Palm-average waiting time (''W'') that a user spends in the system. That is, the average ''W'' gives equal weight to the waiting time of all customers, rather than being the time-average of "the waiting times of the customers currently in the system". Feller's paradox An important example of the use of Palm probabilities is Feller's paradox, often associat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teletraffic Engineering
Teletraffic engineering, telecommunications traffic engineering, or just traffic engineering when in context, is the application of transportation traffic engineering theory to telecommunications. Teletraffic engineers use their knowledge of statistics including queuing theory, the nature of traffic, their practical models, their measurements and simulations to make predictions and to plan telecommunication networks such as a telephone network or the Internet. These tools and knowledge help provide reliable service at lower cost. The field was created by the work of A. K. Erlang for circuit-switched networks but is applicable to packet-switched networks, as they both exhibit Markovian properties, and can hence be modeled by e.g. a Poisson arrival process. The crucial observation in traffic engineering is that in large systems the law of large numbers can be used to make the aggregate properties of a system over a long period of time much more predictable than the behaviour of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conny Palm
Conrad "Conny" Palm (1907–1951) was a Swedish electrical engineer and statistician, known for several contributions to teletraffic engineering and queueing theory. Rolf B. HaugenThe life and work of Conny Palm – some personal comments and experiencesfrom ''Telektronikk'' ( Telenor research journal), 2(3):50–55, 1995 Palm enrolled at the School of Electrical Engineering at the Royal Institute of Technology in Stockholm in 1925, being awarded his M.Sc. (1940) and Ph.D (1943) on a dissertation entitled ''Intensitätsschwankungen im Fernsprechverkehr'' (Intensity Fluctuations in Telephone Traffic). Palm's work was also joint with L. M. Ericsson, cooperating with Christian Jacobæus. He attended Harald Cramér's queueing theory group, met William Feller (1937). Later, Palm was in the Swedish Board for Computing Machinery (''Matematikmaskinnämnden''), where he led the project that developed the first Swedish computer, the BARK (1947–51), informally referred to as CONIAC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ... [...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] |
Pollaczek–Khinchine Formula
In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model. The formula was first published by Felix Pollaczek in 1930 and recast in probabilistic terms by Aleksandr Khinchin two years later. In ruin theory the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt). Mean queue length The formula states that the mean number of customers in system ''L'' is given by : L = \rho + \frac where *\lambda is the arrival rate of the Poisson process *1/\mu is the mean of the service time distribution ''S'' *\rho=\lambda/\mu is the utilization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Process
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. , .Daley, D.J, Vere-Jones, D. (1988). ''An Introduction to the Theory of Point Processes''. Springer, New York. , . Point processes can be used for spatial data analysis,Diggle, P. (2003). ''Statistical Analysis of Spatial Point Patterns'', 2nd edition. Arnold, London. . which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. There are different mathematical interpretations of a point process, such as a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects ... [...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] |
Performance Evaluation
A performance appraisal, also referred to as a performance review, performance evaluation,Muchinsky, P. M. (2012). ''Psychology Applied to Work'' (10th ed.). Summerfield, NC: Hypergraphic Press. (career) development discussion, or employee appraisal, sometimes shortened to "PA", is a periodic and systematic process whereby the job performance of an employee is documented and evaluated. This is done after employees are trained about work and settle into their jobs. Performance appraisals are a part of career development and consist of regular reviews of employee performance within organizations. Performance appraisals are most often conducted by an employee's immediate manager or line manager. While extensively practiced, annual performance reviews have also been criticized as providing feedback too infrequently to be useful, and some critics argue that performance reviews in general do more harm than good. It is an element of the principal-agent framework, that describes the re ... [...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] |
Stochastic Calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyoshi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates. The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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