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Lecture

Continuous-Time Markov Chains: Birth and Death Processes

Description

This lecture covers the topic of continuous-time Markov chains, focusing on birth and death processes. The instructor explains the transition probabilities, steady-state analysis, and arrival and departure processes in detail.

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Related concepts (31)

Markov chain

A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC).

Birth–death process

The birth–death process (or birth-and-death process) is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one. It was introduced by William Feller. The model's name comes from a common application, the use of such models to represent the current size of a population where the transitions are literal births and deaths.

1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.

Markov model

In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property.

Poisson point 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.

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Related concepts (31)

Markov chain

Birth–death process

Markov model

Poisson point process