In optics, the refractive index of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of refraction, n1 sin θ1 = n2 sin θ2, where θ1 and θ2 are the angle of incidence and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the critical angle for total internal reflection, their intensity and Brewster's angle.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. 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.
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.
Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system. The term is used for repairable systems while mean time to failure (MTTF) denotes the expected time to failure for a non-repairable system.
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.
Haldane's dilemma, also known as the waiting time problem, is a limit on the speed of beneficial evolution, calculated by J. B. S. Haldane in 1957. Before the invention of DNA sequencing technologies, it was not known how much polymorphism DNA harbored, although alloenzymes were beginning to make it clear that substantial polymorphism existed. This was puzzling because the amount of polymorphism known to exist seemed to exceed the theoretical limits that Haldane calculated, that is, the limits imposed if polymorphisms present in the population generally influence an organism's fitness. Motoo Kimura's landmark paper on neutral theory in 1968 built on Haldane's work to suggest that most molecular evolution is neutral, resolving the dilemma. Although neutral evolution remains the consensus theory among modern biologists, and thus Kimura's resolution of Haldane's dilemma is widely regarded as correct, some biologists argue that adaptive evolution explains a large fraction of substitutions in protein coding sequence, and they propose alternative solutions to Haldane's dilemma.
A diversity index is a method of measuring how many different types there are in a dataset. Diversity indices are statistical representations of different aspects of biodiversity, which are useful simplifications for comparing different communities or sites.
In electromagnetics, directivity is a parameter of an antenna or optical system which measures the degree to which the radiation emitted is concentrated in a single direction. It is the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. Therefore, the directivity of a hypothetical isotropic radiator, a source of electromagnetic waves which radiates the same power in all directions, is 1, or 0 dBi.
The System Average Interruption Duration Index (SAIDI) is commonly used as a reliability index by electric power utilities. SAIDI is the average cumulative outage duration for each customer served, and is calculated as:
The Customer Average Interruption Duration Index (CAIDI) is a reliability index commonly used by electric power utilities. It is related to SAIDI and SAIFI, and is calculated as
A vertex model is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model. This contrasts with a nearest-neighbour model, such as the Ising model, in which the energy, and thus the Boltzmann weight of a statistical microstate is attributed to the bonds connecting two neighbouring particles. The energy associated with a vertex in the lattice of particles is thus dependent on the state of the bonds which connect it to adjacent vertices. It turns out that every solution of the Yang–Baxter equation with spectral parameters in a tensor product of vector spaces yields an exactly-solvable vertex model.
The Momentary Average Interruption Frequency Index (MAIFI) is a reliability index used by electric power utilities. MAIFI is the average number of momentary interruptions that a customer would experience during a given period. Electric power utilities may define momentary interruptions differently, with some considering a momentary interruption to be an outage of less than 1 minute in duration while others may consider a momentary interruption to be an outage of less than 5 minutes in duration.
The Customer Average Interruption Frequency Index (CAIFI) is a popular reliability index used in the reliability analysis of an electric power system. It is designed to show trends in customers interrupted and helps to show the number of customers affected out of the whole customer base.
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1.
In queueing theory, a discipline within the mathematical theory of probability, the M/M/c queue 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 Poisson process, there are c servers, and job service times are exponentially distributed. It is a generalisation of the M/M/1 queue which considers only a single server. The model with infinitely many servers is the M/M/∞ queue.
The Average Service Availability Index (ASAI) is a reliability index commonly used by electric power utilities. ASAI is calculated as
The Average Service Unavailability Index (ASUI) is a reliability index commonly used by electric power utilities. It is calculated as
The Customer Total Average Interruption Duration Index (CTAIDI) is a reliability index associated with electric power distribution. CTAIDI is the average total duration of interruption for customers who had at least one interruption during the period of analysis, and is calculated as:
The fitness-density covariance is a coexistence mechanism that can allow similar species to coexist because they are in different locations. This effect will be the strongest if species are completely segregated, but can also work if their populations overlap somewhat. If a fitness-density covariance is operating, then when a species becomes very rare, its population will shift to predominantly locations with favorable conditions. Similarly, when a species becomes very common, then conditions will worsen where they are most common, and they will spread into areas where conditions are less favorable. This negative feedback can help species avoid being driven extinct by competition, and it can prevent stronger species from becoming too common and crowding out other species.
Reliability index is an attempt to quantitatively assess the reliability of a system using a single numerical value. The set of reliability indices varies depending on the field of engineering, multiple different indices may be used to characterize a single system. In the simple case of an object that cannot be used or repaired once it fails, a useful index is the mean time to failure representing an expectation of the object's service lifetime. Another cross-disciplinary index is forced outage rate (FOR), a probability that a particular type of a device is out of order. Reliability indices are extensively used in the modern electricity regulation.
