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The Poisson point process is often defined on the real number line, where it can be considered a stochastic process. It is used, for example, in queueing theory [15] to model random events distributed in time, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes.
A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution. A Poisson point process can also be defined using these two properties. Namely, we say that a point process is a Poisson point process if the following two ...
In Campbell's work, he presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to G. H. Hardy, which has inspired the result to be sometimes called the Campbell–Hardy theorem. [10] [11]
A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by.
In queueing theory, a discipline within the mathematical theory of probability, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node. D. G. Kendall proposed describing queueing models using three factors written A/S/ c in 1953 [1] where A denotes the time between arrivals to the ...
Queueing theory is one of the major areas of study in the discipline of management science. Through management science, businesses are able to solve a variety of problems using different scientific and mathematical approaches. Queueing analysis is the probabilistic analysis of waiting lines, and thus the results, also referred to as the ...
Point process operation. In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space.
Poisson random measure. Let be some measure space with - finite measure . The Poisson random measure with intensity measure is a family of random variables defined on some probability space such that. i) ∀ ∈ , N {\displaystyle \forall A\in {\mathcal {A}},\quad N_ {A}} is a Poisson random variable with rate.