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.
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.
The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are:
In epidemiology, force of infection is the rate at which susceptible individuals acquire an infectious disease. Because it takes account of susceptibility it can be used to compare the rate of transmission between different groups of the population for the same infectious disease, or even between different infectious diseases. That is to say, is directly proportional to ; the effective transmission rate.
The System Average Interruption Duration Index (SAIDI) is commonly used as a reliability index by electric power utilities. SAIDI is the average outage duration for each customer served, and is calculated as:
The System Average Interruption Frequency Index (SAIFI) is commonly used as a reliability index by electric power utilities. SAIFI is the average number of interruptions that a customer would experience, and is calculated as
An outage management system (OMS) is a computer system used by operators of electric distribution systems to assist in restoration of power.
In queueing theory, a discipline within the mathematical theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times have an exponential distribution. The model name is written in Kendall's notation. The model is the most elementary of queueing models and an attractive object of study as closed-form expressions can be obtained for many metrics of interest in this model. An extension of this model with more than one server is the M/M/c queue.
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 or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson. The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area or volume.
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.
In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as
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
In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue.
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. Reliability indices are extensively used in the modern electricity regulation. Another cross-disciplinary index is forced outage rate (FOR), a probability that a particular type of a device is out of order.
