Markov process and Stochastic process

Markov chain

In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. Having the Markov Property means for a given process knowledge of the previous states is irrelevant for predicting the probability of subsequent states. In this way a Markov chain is "memoryless", no given state has any causal connection with a previous state.
At each point in time the system may have changed states from the state the system was in the moment before, or the system may have stayed in the same state. The changes of state are called transitions. If a sequence of states has the Markov property, then every future state is conditionally independent of every prior state.

Markov property
In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state and all past states, depends only upon the present state and not on any past states, i.e. it is conditionally independent of the past states (the path of the process) given the present state. A process with the Markov property is usually called a Markov process, and may be described as Markovian. See in particular


Markov chain
Continuous-time Markov process
(In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that satisfies the Markov property and takes values from a set called the state space )
Mathematically, if X(t), t > 0, is a stochastic process, the Markov property states that
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Markov processes are typically termed (time-) homogeneous if
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and otherwise are termed (time-) inhomogeneous (or (time-) nonhomogeneous). Homogeneous Markov processes, usually being simpler than inhomogeneous ones, form the most important class of Markov processes.
In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states. For example, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X, i.e. mathematically,
If Y has the Markov property, then it is a Markovian representation of X. In this case, X is also called a second-order Markov process. Higher-order Markov processes are defined analogously.
An example of a non-Markovian process with a Markovian representation is a moving average time series.
The most famous Markov processes are Markov chains, but many other processes, including Brownian motion (to a close approximation), are Markovian.

Examples of Markov chains
DMP
Memorylessness
Semi-Markov process
Andrey Markov
Continuous-time Markov process
Markov chain
Markov decision process

Poisson process
The Poisson arrival process or Poisson process counts the number of arrivals, each of which has a exponentially distributed time between arrival. In the most general case this can be represented by the rate matrix,
In the homogeneous case this is more simply,
Here every transition is marked.
Markov arrival process
The Markov arrival process (MAP) is a generalisation of the Poisson process by having non-exponential distribution sojourn between arrivals. The homogeneous case has rate matrix,
An arrival is seen every time a transtion occurs that increases the level (a marked transition), e.g. a transition in the D1 sub-matrix. Sub-matricies D0 and D1 have elements of λi,j, the rate of a Poisson process, such that,
and
There are several special cases of the Markov arrival process.


Stochastic process
A stochastic process, or sometimes random process, is the counterpart of a deterministic process (or deterministic system) considered in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as it is the case, for example, for solutions of an ordinary differential equation), in a random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are more possibilities the process might go to, but some paths are more probable and others less.
In the simplest possible case ('discrete time'), a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as functions of one or several deterministic arguments ('inputs', in most cases regarded as 'time') whose values ('outputs') are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same 'type'.[1] Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.
Familiar examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks. Examples of random fields include static images, random terrain (landscapes), or composition variations of an inhomogeneous material.
//(an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state.
Logistics is the art and science of managing and controlling the flow of goods, energy, information and other resources like products, services, and people, from the source of production to the marketplace. It is difficult to accomplish any marketing or manufacturing without logistical support. It involves the integration of information, transportation, inventory, warehousing, material handling, and packaging. The operating responsibility of logistics is the geographical repositioning of raw materials, work in process, and finished inventories where required at the lowest cost possible.
Logistics and Supply Chain services are provided by a wide range of 3rd party suppliers.
A supply chain, logistics network, or supply network is a coordinated system of organizations, people, activities, information and resources involved in moving a product or service in physical or virtual manner from supplier to customer. Supply chain activities (aka value chains or life cycle processes) transform raw materials and components into a finished product that is delivered to the end customer. Supply chains link value chains.[Today,the ever increasing technical complexity of standard consumer goods, combined with the ever increasing size and depth of the global market has meant that the link between consumer and vendor is usually only the final link in a long and complex chain or network of exchanges.
This supply chain begins with the extraction of raw material and includes several production links, for instance; component construction, assembly and merging before moving onto several layers of storage facilities of ever decreasing size and ever more remote geographical locations, and finally reaching the consumer.
Supply chain management (SCM) is the process of planning, implementing, and controlling the operations of the supply chain with the purpose to satisfy customer requirements as efficiently as possible. Supply chain management spans all movement and storage of raw materials, work-in-process inventory, and finished goods from point-of-origin to point-of-consumption. The term supply chain management was coined by consultant Keith Oliver, of strategy consulting firm Booz Allen Hamilton in 1982.
The definition one America professional association put forward is that Supply Chain Management encompasses the planning and management of all activities involved in sourcing, procurement, conversion, and logistics management activities. Importantly, it also includes coordination and collaboration with channel partners, which can be suppliers, intermediaries, third-party service providers, and customers. In essence, Supply Chain Management integrates supply and demand management within and across companies.
Lean manufacturing is a generic process management philosophy derived mostly from the Toyota Production System (TPS)[1] but also from other sources. It is renowned for its focus on reduction of the original Toyota 'seven wastes' in order to improve overall customer value. Lean is often linked with Six Sigma because of that methodology's emphasis on reduction of process variation (or its converse smoothness). Toyota's steady growth from a small player to the most valuable and the biggest car company in the world has focused attention upon how it has achieved this, making "Lean" a hot topic in management science in the first decade of the 21st century.
For many, Lean is the set of TPS 'tools' that assist in the identification and steady elimination of waste (muda), the improvement of quality, and production time and cost reduction. To solve the problem of waste, Lean Manufacturing has several 'tools' at its disposal. These include continuous process improvement (kaizen), the "5 Whys" and mistake-proofing (poka-yoke). In this way it can be seen as taking a very similar approach to other improvement methodologies.
There is a second approach to Lean Manufacturing which is promoted by Toyota in which the focus is upon implementing the 'flow' or smoothness of work (opposite of mura, unevenness) through the system and not upon 'waste reduction' per se. Techniques to improve flow include production levelling, "pull" production (by means of kanban) and the Heijunka box.)