Markov processes were introduced by scientists in 1907. Leading mathematicians of that time developed this theory, some still improve it. This system is spreading in other scientific fields. Practical Markov chains are used in various fields where a person needs to arrive in a state of expectation. But in order to clearly understand the system, you need to have knowledge of terms and conditions. The main factor that determines the Markov process, are considered randomness. True, it is not similar to the concept of uncertainty. Certain conditions and variables are inherent in it.
Features of the randomness factor
This condition is subject to static stability, or rather, to its laws, which are not taken into account under uncertainty. In turn, this criterion allows the use of mathematical methods in the theory of Markov processes, as noted by a scientist who studied the dynamics of probabilities. The work he created directly concerned these variables. In turn, the studied and developed random process, which has the concepts of state and transition, as well as used in stochastic and mathematical problems, makes it possible for these models to function. Among other things, it provides an opportunity to improve other important applied theoretical and practical sciences:
- diffusion theory;
- queuing theory;
- theory of reliability and other things;
- chemistry;
- physics;
- Mechanics.
Essential features of an unplanned factor
This Markov process is due to a random function, that is, any value of the argument is considered a given quantity or one that takes a pre-prepared look. Examples are:
- chain vibrations;
- movement speed;
- surface roughness in a given area.
It is also assumed that the fact of a random function is time, that is, indexation occurs. The classification has the form of state and argument. This process can be with discrete as well as continuous states or time. Moreover, the cases are different: everything happens either in one, or in another form, or simultaneously.
A detailed analysis of the concept of randomness
It was quite difficult to build a mathematical model with the necessary performance indicators in an explicitly analytical form. In the future, it became possible to realize this task, because the Markov random process arose. Analyzing this concept in detail, it is necessary to derive a certain theorem. The Markov process is a physical system that has changed its position and state, which were not previously programmed. Thus, it turns out that a random process takes place in it. For example: a space orbit and a ship that is being launched onto it. The result is achieved only due to some inaccuracies and adjustments, without which the specified mode is not implemented. Most processes that occur are inherent in randomness, uncertainty.
In essence, virtually any option that can be considered will be subject to this factor. Aircraft, technical device, dining room, clock - all this is subject to random changes. Moreover, this function is inherent in any ongoing process in the real world. However, while this does not apply to individually tuned parameters, the perturbations that occur are perceived as deterministic.
The concept of Markov random process
The design of a technical or mechanical device, device forces the creator to take into account various factors, in particular uncertainties. The calculation of random fluctuations and disturbances arises at the moment of personal interest, for example, when implementing autopilot. Some processes studied in sciences like physics and mechanics are such.
But pay attention to them and carry out rigorous research should begin at the moment when it is directly needed. The Markov random process has the following definition: the characteristic of the probability of a future species depends on the state in which it is at a given moment in time, and is not related to how the system looked. So, this concept indicates that the result can be predicted, taking into account only probability and forgetting about the background.
Detailed conception
At the moment, the system is in a certain state, it goes over and changes, to predict what will happen next, in fact, is impossible. But, given the likelihood, we can say that the process will be completed in a certain form or save the previous one. That is, the future arises from the present, forgetting about the past. When a system or process transitions to a new state, the background is usually omitted. Probability in Markov processes plays an important role.
For example, a Geiger counter shows the number of particles, which depends on a specific indicator, and not on exactly when it came. The main criterion is the above criterion. In practical application, not only Markov processes can be considered, but also similar ones, for example: airplanes participate in the battle of a system, each of which is indicated by a certain color. In this case, probability is again the main criterion. At what point the preponderance in number will occur, and for what color, is unknown. That is, this factor depends on the state of the system, and not on the sequence of death of aircraft.
Structural Process Analysis
Markov process refers to any state of the system without a probabilistic consequence and without taking into account the background. That is, if you include the future in the present and omit the past. A saturation of a given time with a history will lead to multidimensionality and will lead to complex circuit construction. Therefore, it is better to study these systems with simple circuits with minimal numerical parameters. As a result, these variables are considered decisive and determined by some factors.
An example of Markov processes: a working technical device, which at this moment is operational. In this state of affairs, it is of interest that the device will function for a long period of time. But if you perceive the equipment as debugged, then this option will no longer belong to the process under consideration due to the fact that there is no information about how long the device worked before and whether repairs were made. However, if we supplement these two time variables and include them in the system, then its state can be attributed to Markovsky.
Description of discrete state and time continuity
Models of Markov processes are applied at the moment when it is necessary to neglect the background. For research in practice, discrete, continuous states are most often encountered. Examples of this situation are: the structure of the equipment includes components that can fail during working hours, and this happens as an unplanned, random action. As a result, the state of the system undergoes repair of one or the other element, at which moment some of them will be in good working order or both of them will be debugged, or vice versa, they will be fully operational.
The discrete Markov process is based on probability theory and is also a transition of a system from one state to another. Moreover, this factor occurs instantly, even if accidental breakdowns and repairs occur. To analyze such a process, it is better to use state graphs, that is, geometric diagrams. System states in this case are indicated by various figures: triangles, rectangles, dots, arrows.
Modeling this process
Markov processes with discrete states are possible modifications of systems as a result of a transition that occurs instantly, and which can be numbered. For example, you can build a state graph from the arrows for the nodes, where each will indicate the path of differently directed factors of failure, operational status, etc. In the future, any questions may arise: it seems that not all geometric elements indicate the right direction, because in the process, every node is able to deteriorate. When working, it is important to consider short circuits.
The Markov process with continuous time occurs when the data are not fixed in advance, they happen by chance. Transitions were not previously planned and occur irregularly, at any time. In this case, probability again plays a major role. However, if the current situation relates to the above, then to describe it will be necessary to develop a mathematical model, but it is important to understand the theory of possibility.
Probabilistic Theories
These theories consider probabilistic, having characteristic features like random order, movement and factors, mathematical problems, and not deterministic ones that are defined now and then. The managed Markov process has an opportunity factor and is based on it. Moreover, this system is capable of transitioning to any state instantly under various conditions and time intervals.
To put this theory into practice, you need to have important knowledge of probability and its application. In most cases, everyone is in a state of expectation, which in the general sense is the theory under consideration.
Probability Theory Examples
Examples of Markov processes in this situation may include:
- Cafe;
- ticket offices;
- repair shops;
- stations for various purposes, etc.
As a rule, people daily come across this system, today it is called mass service. At facilities where such a service is present, there is the possibility of requiring various requests that are satisfied in the process.
Hidden process models
Such models are static and copy the work of the original process. In this case, the main feature is the function of observing unknown parameters that must be solved. As a result, these elements can be used in analysis, practice, or for the recognition of various objects. Ordinary Markov processes are based on visible transitions and on probability; in a hidden model, only unknown variables are observed that are affected by the state.
Essential disclosure of hidden Markov models
It also has a probability distribution among other values, as a result, the researcher will see a sequence of symbols and states. Each action has a probability distribution among other values, because of this, a hidden model gives information about the generated successive states. The first notes and references to them appeared in the late sixties of the last century.
Then they began to be used for speech recognition and as analyzers of biological data. In addition, hidden models spread in writing, movements, computer science. Also, these elements imitate the operation of the main process and are static, however, despite this, there are much more distinctive features. In particular, this fact relates to direct observation and sequence generation.
Stationary Markov Process
This condition exists with a homogeneous transition function, as well as with a stationary distribution, which is considered the main and, by definition, random action. The phase space for this process is a finite set, but with this state of affairs, the initial differentiation always exists. Transient probabilities in this process are considered under time conditions or additional elements.
A detailed study of Markov models and processes reveals the question of satisfying equilibrium in various areas of life and society. Given the fact that this industry affects science and mass service, the situation can be corrected by analyzing and predicting the outcome of any events or actions of the same faulty watches or equipment. To fully use the capabilities of the Markov process, it is worthwhile to understand them in detail. After all, this device has found wide application not only in science, but also in games. This system in its pure form is usually not considered, and if used, it is only based on the above models and schemes.