**Description:** A non-homogeneous Markov model is a type of generative model that allows transitions between states to vary over time, unlike homogeneous Markov models where these transitions are constant. In this model, the probability of moving from one state to another depends not only on the current state but also on the time elapsed. This means that the dynamics of the system can change over time, making it more flexible and suitable for representing phenomena where conditions change. Non-homogeneous Markov models are useful in situations where transition probabilities are expected to be affected by external factors or by time itself, allowing for a more realistic representation of complex systems. Their ability to adapt to changes in the environment makes them valuable tools in fields such as economics, biology, and engineering, where processes are not static and can evolve over time.
**History:** Markov models were introduced by Russian mathematician Andrey Markov in the early 20th century, specifically in 1906. However, the notion of non-homogeneity in these models began to develop later as researchers realized that many real systems do not behave consistently over time. Over the decades, numerous research and theoretical developments have expanded the understanding and application of non-homogeneous Markov models, especially in fields such as queue theory, statistics, and machine learning.
**Uses:** Non-homogeneous Markov models are used in various fields, including economics to model market behavior over time, in biology to study population evolution, and in engineering to optimize production processes. They are also applicable in time series data analysis, where conditions may change, such as in weather forecasting or time series analysis.
**Examples:** A practical example of a non-homogeneous Markov model is the analysis of product demand in a market, where consumer preferences may change over time due to trends or seasonal events. Another example is found in biology, where the spread of diseases can be modeled, considering that the infection rate may vary over time due to factors such as population immunity or the emergence of new virus variants.
